\(\int \frac {A+B x+C x^2}{(2+6 x+3 x^2+9 x^3)^{5/2}} \, dx\) [96]

Optimal result

Integrand size = 30, antiderivative size = 555 \[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\frac {(1+3 x) (2 (3 A-B-2 C)+(3 A+6 B-2 C) x) \left (2+3 x^2\right )}{126 \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}+\frac {(1+3 x) (9 (17 A-8 B-2 C)+(150 A+111 B-58 C) x) \left (2+3 x^2\right )^2}{1764 \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}-\frac {(615 A-471 B-32 C) (1+3 x) \left (2+3 x^2\right )^3}{12348 \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}-\frac {(5610 A+447 B-1346 C) (1+3 x)^2 \left (2+3 x^2\right )^3}{86436 \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}+\frac {(5610 A+447 B-1346 C) (1+3 x)^3 \left (2+3 x^2\right )^3}{86436 \left (1+\sqrt {7}+3 x\right ) \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}-\frac {(5610 A+447 B-1346 C) (1+3 x)^{5/2} \left (1+\sqrt {7}+3 x\right ) \left (2+3 x^2\right )^2 \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{12348 \sqrt {3} 7^{3/4} \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}+\frac {\left (\left (5610-615 \sqrt {7}\right ) A+\left (447+471 \sqrt {7}\right ) B-2 \left (673-16 \sqrt {7}\right ) C\right ) (1+3 x)^{5/2} \left (1+\sqrt {7}+3 x\right ) \left (2+3 x^2\right )^2 \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{24696 \sqrt {3} 7^{3/4} \left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \] Output:

1/126*(1+3*x)*(6*A-2*B-4*C+(3*A+6*B-2*C)*x)*(3*x^2+2)/(9*x^3+3*x^2+6*x+2)^ 
(5/2)+1/1764*(1+3*x)*(153*A-72*B-18*C+(150*A+111*B-58*C)*x)*(3*x^2+2)^2/(9 
*x^3+3*x^2+6*x+2)^(5/2)-1/12348*(615*A-471*B-32*C)*(1+3*x)*(3*x^2+2)^3/(9* 
x^3+3*x^2+6*x+2)^(5/2)-1/86436*(5610*A+447*B-1346*C)*(1+3*x)^2*(3*x^2+2)^3 
/(9*x^3+3*x^2+6*x+2)^(5/2)+1/86436*(5610*A+447*B-1346*C)*(1+3*x)^3*(3*x^2+ 
2)^3/(1+7^(1/2)+3*x)/(9*x^3+3*x^2+6*x+2)^(5/2)-1/259308*(5610*A+447*B-1346 
*C)*(1+3*x)^(5/2)*(1+7^(1/2)+3*x)*(3*x^2+2)^2*((3*x^2+2)/(1+7^(1/2)+3*x)^2 
)^(1/2)*EllipticE(sin(2*arctan(1/7*(1+3*x)^(1/2)*7^(3/4))),1/14*(98+14*7^( 
1/2))^(1/2))*3^(1/2)*7^(1/4)/(9*x^3+3*x^2+6*x+2)^(5/2)+1/518616*((5610-615 
*7^(1/2))*A+(447+471*7^(1/2))*B-2*(673-16*7^(1/2))*C)*(1+3*x)^(5/2)*(1+7^( 
1/2)+3*x)*(3*x^2+2)^2*((3*x^2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*InverseJacobiAM( 
2*arctan(1/7*(1+3*x)^(1/2)*7^(3/4)),1/14*(98+14*7^(1/2))^(1/2))*3^(1/2)*7^ 
(1/4)/(9*x^3+3*x^2+6*x+2)^(5/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 13.89 (sec) , antiderivative size = 520, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\frac {\frac {B \left (8916+28890 x+70848 x^2+675 x^3+76950 x^4-36207 x^5\right )+6 C \left (886+4548 x+8097 x^2+19965 x^3+7065 x^4+18171 x^5\right )-9 A \left (6850+16854 x+32163 x^2+59970 x^3+29745 x^4+50490 x^5\right )}{2+6 x+3 x^2+9 x^3}+\left (3 \sqrt {2}+i \sqrt {3}\right ) (5610 A+447 B-1346 C) \sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}} \sqrt {2+3 x^2} E\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )+\frac {\sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}} \left (224 i \sqrt {3} C \sqrt {\sqrt {6}-3 i x} \sqrt {2+3 x^2}+3 B \left (1248 i \sqrt {6} \sqrt {\sqrt {6}+3 i x}+3744 \sqrt {\sqrt {6}+3 i x} x-149 i \sqrt {3} \sqrt {\sqrt {6}-3 i x} \sqrt {2+3 x^2}\right )+15 A \left (87 i \sqrt {6} \sqrt {\sqrt {6}+3 i x}+261 \sqrt {\sqrt {6}+3 i x} x-374 i \sqrt {3} \sqrt {\sqrt {6}-3 i x} \sqrt {2+3 x^2}\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right ),\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )}{\sqrt {\sqrt {6}-3 i x}}}{259308 \sqrt {2+6 x+3 x^2+9 x^3}} \] Input:

Integrate[(A + B*x + C*x^2)/(2 + 6*x + 3*x^2 + 9*x^3)^(5/2),x]
 

Output:

((B*(8916 + 28890*x + 70848*x^2 + 675*x^3 + 76950*x^4 - 36207*x^5) + 6*C*( 
886 + 4548*x + 8097*x^2 + 19965*x^3 + 7065*x^4 + 18171*x^5) - 9*A*(6850 + 
16854*x + 32163*x^2 + 59970*x^3 + 29745*x^4 + 50490*x^5))/(2 + 6*x + 3*x^2 
 + 9*x^3) + (3*Sqrt[2] + I*Sqrt[3])*(5610*A + 447*B - 1346*C)*Sqrt[(I*(1 + 
 3*x))/(I + Sqrt[6])]*Sqrt[2 + 3*x^2]*EllipticE[ArcSin[Sqrt[Sqrt[6] - (3*I 
)*x]/(2^(3/4)*3^(1/4))], (2*Sqrt[6])/(I + Sqrt[6])] + (Sqrt[(I*(1 + 3*x))/ 
(I + Sqrt[6])]*((224*I)*Sqrt[3]*C*Sqrt[Sqrt[6] - (3*I)*x]*Sqrt[2 + 3*x^2] 
+ 3*B*((1248*I)*Sqrt[6]*Sqrt[Sqrt[6] + (3*I)*x] + 3744*Sqrt[Sqrt[6] + (3*I 
)*x]*x - (149*I)*Sqrt[3]*Sqrt[Sqrt[6] - (3*I)*x]*Sqrt[2 + 3*x^2]) + 15*A*( 
(87*I)*Sqrt[6]*Sqrt[Sqrt[6] + (3*I)*x] + 261*Sqrt[Sqrt[6] + (3*I)*x]*x - ( 
374*I)*Sqrt[3]*Sqrt[Sqrt[6] - (3*I)*x]*Sqrt[2 + 3*x^2]))*EllipticF[ArcSin[ 
Sqrt[Sqrt[6] - (3*I)*x]/(2^(3/4)*3^(1/4))], (2*Sqrt[6])/(I + Sqrt[6])])/Sq 
rt[Sqrt[6] - (3*I)*x])/(259308*Sqrt[2 + 6*x + 3*x^2 + 9*x^3])
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 30.02 (sec) , antiderivative size = 3718, normalized size of antiderivative = 6.70, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.567, Rules used = {2526, 27, 2490, 2486, 27, 1235, 27, 1235, 27, 1237, 27, 1237, 27, 1269, 1172, 321, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x+C x^2}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2526

\(\displaystyle \frac {1}{27} \int \frac {3 (9 A-2 C+(9 B-2 C) x)}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}dx-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int \frac {9 A-2 C+(9 B-2 C) x}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}dx-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 2490

\(\displaystyle \frac {1}{9} \int \frac {\frac {1}{27} (27 (9 A-2 C)-3 (9 B-2 C))+(9 B-2 C) \left (x+\frac {1}{9}\right )}{\left (9 \left (x+\frac {1}{9}\right )^3+\frac {17}{3} \left (x+\frac {1}{9}\right )+\frac {110}{81}\right )^{5/2}}d\left (x+\frac {1}{9}\right )-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 2486

\(\displaystyle \frac {6561 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17\right )^{5/2} \int \frac {81 A-9 B-16 C+9 (9 B-2 C) \left (x+\frac {1}{9}\right )}{9 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2}}d\left (x+\frac {1}{9}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17\right )^{5/2} \int \frac {81 A-9 B-16 C+9 (9 B-2 C) \left (x+\frac {1}{9}\right )}{\left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2}}d\left (x+\frac {1}{9}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \int \frac {177147 \left (\frac {9 \left (68 \left (55-63 \sqrt {2}\right )-2023 \sqrt [3]{-55+63 \sqrt {2}}-7 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A-9 \left (32 \left (55-63 \sqrt {2}\right )-119 \left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-7 \left (26+7 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (36 \left (55-63 \sqrt {2}\right )-119 \left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+7 \left (81-56 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{55-63 \sqrt {2}}+\frac {63 \left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{2 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}d\left (x+\frac {1}{9}\right )}{11160261\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \int \frac {\frac {9 \left (68 \left (55-63 \sqrt {2}\right )-2023 \sqrt [3]{-55+63 \sqrt {2}}-7 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A-9 \left (32 \left (55-63 \sqrt {2}\right )-119 \left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-7 \left (26+7 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (36 \left (55-63 \sqrt {2}\right )-119 \left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+7 \left (81-56 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{55-63 \sqrt {2}}+\frac {63 \left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}}{\left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}d\left (x+\frac {1}{9}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {\sqrt [3]{2 \left (126-55 \sqrt {2}\right )} \left (\frac {17 \left (\left (2329250-501417 \sqrt {2}+119 \left (13988-10395 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1156 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A+\left (2885386-2280852 \sqrt {2}-14 \left (31697-13797 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-544 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) B-2 \left (7 \left (82772-44163 \sqrt {2}\right )+7 \left (19378-16569 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+68 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{5/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{3213 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \int -\frac {531441 \left (\frac {17 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}+9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-4 \left (170+91 \sqrt [3]{-55+63 \sqrt {2}}-10 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{17 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{3720087\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {\sqrt [3]{2 \left (126-55 \sqrt {2}\right )} \left (\frac {17 \left (\left (2329250-501417 \sqrt {2}+119 \left (13988-10395 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1156 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A+\left (2885386-2280852 \sqrt {2}-14 \left (31697-13797 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-544 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) B-2 \left (7 \left (82772-44163 \sqrt {2}\right )+7 \left (19378-16569 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+68 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{5/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{3213 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}-\frac {\sqrt [3]{126-55 \sqrt {2}} \int \frac {\frac {17 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}+9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-4 \left (170+91 \sqrt [3]{-55+63 \sqrt {2}}-10 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{119\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {\sqrt [3]{2 \left (126-55 \sqrt {2}\right )} \left (\frac {17 \left (\left (2329250-501417 \sqrt {2}+119 \left (13988-10395 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1156 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A+\left (2885386-2280852 \sqrt {2}-14 \left (31697-13797 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-544 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) B-2 \left (7 \left (82772-44163 \sqrt {2}\right )+7 \left (19378-16569 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+68 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{5/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{3213 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}-\frac {\sqrt [3]{126-55 \sqrt {2}} \left (-\frac {2 \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17} \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )-\frac {289 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}\right )}{1377 \left (17-\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2}}-\frac {2 \left (-55+63 \sqrt {2}\right )^{2/3} \int \frac {81 \left (\frac {17 \left (5 \left (1891505 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-\left (331288425140012287-234295996498988886 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-1309 \left (-55+63 \sqrt {2}\right )^{2/3} \left (5 \left (4476595998547-3164320515510 \sqrt {2}\right )+102 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )+17 \left (26928 \left (4476595998547-3164320515510 \sqrt {2}\right )+\left (2624147278399273-1855494888173073 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right ) A+2 \left (408211835915503836-288548059168325880 \sqrt {2}+2235415 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4 \left (136780967660832035-96720827514330852 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-24844441872956282 \sqrt [3]{10963+6930 \sqrt {2}}+17573382900702720 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-1547 \left (-55+63 \sqrt {2}\right )^{2/3} \left (5 \left (4476595998547-3164320515510 \sqrt {2}\right )+102 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B-6 \left (409733878555009816-289623928143599280 \sqrt {2}+515865 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-\left (101877402480989545-72046170467021898 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+6420875367334283 \sqrt [3]{10963+6930 \sqrt {2}}-4539640729381695 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-357 \left (-55+63 \sqrt {2}\right )^{2/3} \left (5 \left (4476595998547-3164320515510 \sqrt {2}\right )+102 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )}{\left (55-63 \sqrt {2}\right )^6}-9 \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )-\frac {289 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}\right ) \left (x+\frac {1}{9}\right )\right )}{34 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{729 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{119\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {\sqrt [3]{2 \left (126-55 \sqrt {2}\right )} \left (\frac {17 \left (\left (2329250-501417 \sqrt {2}+119 \left (13988-10395 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1156 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A+\left (2885386-2280852 \sqrt {2}-14 \left (31697-13797 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-544 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) B-2 \left (7 \left (82772-44163 \sqrt {2}\right )+7 \left (19378-16569 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+68 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{5/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{3213 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}-\frac {\sqrt [3]{126-55 \sqrt {2}} \left (-\frac {2 \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17} \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )-\frac {289 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}\right )}{1377 \left (17-\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \int \frac {\frac {17 \left (5 \left (1891505 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-\left (331288425140012287-234295996498988886 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-1309 \left (-55+63 \sqrt {2}\right )^{2/3} \left (5 \left (4476595998547-3164320515510 \sqrt {2}\right )+102 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )+17 \left (26928 \left (4476595998547-3164320515510 \sqrt {2}\right )+\left (2624147278399273-1855494888173073 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right ) A+2 \left (408211835915503836-288548059168325880 \sqrt {2}+2235415 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4 \left (136780967660832035-96720827514330852 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-24844441872956282 \sqrt [3]{10963+6930 \sqrt {2}}+17573382900702720 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-1547 \left (-55+63 \sqrt {2}\right )^{2/3} \left (5 \left (4476595998547-3164320515510 \sqrt {2}\right )+102 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B-6 \left (409733878555009816-289623928143599280 \sqrt {2}+515865 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-\left (101877402480989545-72046170467021898 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+6420875367334283 \sqrt [3]{10963+6930 \sqrt {2}}-4539640729381695 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-357 \left (-55+63 \sqrt {2}\right )^{2/3} \left (5 \left (4476595998547-3164320515510 \sqrt {2}\right )+102 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )}{\left (55-63 \sqrt {2}\right )^6}-9 \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )-\frac {289 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}\right ) \left (x+\frac {1}{9}\right )}{\left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{153 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{119\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {\sqrt [3]{2 \left (126-55 \sqrt {2}\right )} \left (\frac {17 \left (\left (2329250-501417 \sqrt {2}+119 \left (13988-10395 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1156 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A+\left (2885386-2280852 \sqrt {2}-14 \left (31697-13797 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-544 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) B-2 \left (7 \left (82772-44163 \sqrt {2}\right )+7 \left (19378-16569 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+68 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{5/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{3213 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}-\frac {\sqrt [3]{126-55 \sqrt {2}} \left (-\frac {2 \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17} \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )-\frac {289 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}\right )}{1377 \left (17-\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {4 \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (\left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-\left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}{27 \left (-55+63 \sqrt {2}\right )^{25/3} \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}-\frac {2 \left (-55+63 \sqrt {2}\right )^{2/3} \int \frac {81 \left (\frac {5 \left (1902011127302884490975677838-1344927288062447117333513208 \sqrt {2}-15862632 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (1054953198582887301263302-745962937283155901939973 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+30078395853283026760883881 \sqrt [3]{10963+6930 \sqrt {2}}-21268655470306285901210736 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}+17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (780332485236953371166917-551779749363892856331483 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-578 \left (7321169178835506223276-5176895982118527452247 \sqrt {2}+\left (669789678574280540567-473613641395734611772 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A-4 \left (306439439956380590283385960-216684100789942391585186508 \sqrt {2}-671925 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+350169666759651702584974499 \sqrt [3]{55+63 \sqrt {2}}-247607346358480899744702630 \sqrt {2} \sqrt [3]{55+63 \sqrt {2}}+82833660710544730853928605 \sqrt [3]{10963+6930 \sqrt {2}}-58572154411625457848595231 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}+17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2430122932910187202454495-1718356392457074654364899 \sqrt {2}-\left (693837788715553360547413-490618972810803175211250 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (62373181346708852082920-44104233826570810418316 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right ) B-2 \left (920477541854322232638316150-650877781905829557814091112 \sqrt {2}-8513940 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-494388156771461234554537864 \sqrt [3]{55+63 \sqrt {2}}+349584697131360461731703610 \sqrt {2} \sqrt [3]{55+63 \sqrt {2}}-20104740397307087734588335 \sqrt [3]{10963+6930 \sqrt {2}}+14216148921663377987591916 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (3430963406497262421676180-2426072773127871439383714 \sqrt {2}-\left (741890397894108922002693-524597182006964109166935 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+2 \left (93677746982935297439275-66240360462632765908212 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right ) C}{\left (-55+63 \sqrt {2}\right )^{28/3}}-\frac {18 \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-2 \left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (55-63 \sqrt {2}\right )^9}\right )}{2 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{243 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{153 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{119\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {\sqrt [3]{2 \left (126-55 \sqrt {2}\right )} \left (\frac {17 \left (\left (2329250-501417 \sqrt {2}+119 \left (13988-10395 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1156 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A+\left (2885386-2280852 \sqrt {2}-14 \left (31697-13797 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-544 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) B-2 \left (7 \left (82772-44163 \sqrt {2}\right )+7 \left (19378-16569 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+68 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{5/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{3213 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}-\frac {\sqrt [3]{126-55 \sqrt {2}} \left (-\frac {2 \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17} \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )-\frac {289 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}\right )}{1377 \left (17-\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {4 \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (\left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-\left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}{27 \left (-55+63 \sqrt {2}\right )^{25/3} \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \int \frac {\frac {5 \left (1902011127302884490975677838-1344927288062447117333513208 \sqrt {2}-15862632 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (1054953198582887301263302-745962937283155901939973 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+30078395853283026760883881 \sqrt [3]{10963+6930 \sqrt {2}}-21268655470306285901210736 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}+17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (780332485236953371166917-551779749363892856331483 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-578 \left (7321169178835506223276-5176895982118527452247 \sqrt {2}+\left (669789678574280540567-473613641395734611772 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A-2 \left (2 \left (306439439956380590283385960-216684100789942391585186508 \sqrt {2}-671925 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+350169666759651702584974499 \sqrt [3]{55+63 \sqrt {2}}-247607346358480899744702630 \sqrt {2} \sqrt [3]{55+63 \sqrt {2}}+82833660710544730853928605 \sqrt [3]{10963+6930 \sqrt {2}}-58572154411625457848595231 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}+17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2430122932910187202454495-1718356392457074654364899 \sqrt {2}-\left (693837788715553360547413-490618972810803175211250 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (62373181346708852082920-44104233826570810418316 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right ) B+\left (920477541854322232638316150-650877781905829557814091112 \sqrt {2}-8513940 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-494388156771461234554537864 \sqrt [3]{55+63 \sqrt {2}}+349584697131360461731703610 \sqrt {2} \sqrt [3]{55+63 \sqrt {2}}-20104740397307087734588335 \sqrt [3]{10963+6930 \sqrt {2}}+14216148921663377987591916 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (3430963406497262421676180-2426072773127871439383714 \sqrt {2}-\left (741890397894108922002693-524597182006964109166935 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+2 \left (93677746982935297439275-66240360462632765908212 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{28/3}}-\frac {18 \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-2 \left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (55-63 \sqrt {2}\right )^9}}{\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{3 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{153 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{119\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {\sqrt [3]{2 \left (126-55 \sqrt {2}\right )} \left (\frac {17 \left (\left (2329250-501417 \sqrt {2}+119 \left (13988-10395 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1156 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A+\left (2885386-2280852 \sqrt {2}-14 \left (31697-13797 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-544 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) B-2 \left (7 \left (82772-44163 \sqrt {2}\right )+7 \left (19378-16569 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+68 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{5/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{3213 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}-\frac {\sqrt [3]{126-55 \sqrt {2}} \left (-\frac {2 \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17} \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )-\frac {289 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}\right )}{1377 \left (17-\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {4 \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (\left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-\left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}{27 \left (-55+63 \sqrt {2}\right )^{25/3} \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {\left (5 \left (7711259249342034331540551713816401444467986016-5452683706393466091688175106537932137299420066 \sqrt {2}+1175652 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+5202 \left (3013041993039644902966886774356750694308522-2130542425289165678586957566437978265377827 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-805577598936782021705281372332071291561571391 \sqrt [3]{10963+6930 \sqrt {2}}+569629382979067382117244250912754520232212303 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (2672578208549350787748003395517624928477393-1889798174571446947172794711888173965105016 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+306 \left (449347719939814928406612930511657815952984-317736819879490171907747805891204004627671 \sqrt {2}-\left (42212972401091014524514717989893507598364-29849079038692187752899631198902118087743 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+4 \left (11782445396249377169230236103969695291331446759-8331447038702096367472565767988905703973810630 \sqrt {2}-9350595 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+306 \left (23048647555740695760571392136623258656466545-16297854983829155985278433350946191353771401 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+2218498065526499849500416512746169197670126141 \sqrt [3]{10963+6930 \sqrt {2}}-1568715026188554352471420048789142297660901060 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (7812436492524583911367102687376238089051845-5524226821438115814218592936888010727741109 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-9 \left (2713028889836854119338366421580628232637841-1918401125593279495691302565543493025575810 \sqrt {2}-2 \left (53075282307413412963153521742904012076473-37529892034185175031134338325457235761100 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-6 \left (3173558438211395567949026036480029199543174708-2244044692102804293641894022394269759347864550 \sqrt {2}-1167560 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+3947435809155102797670670702823924227943692900 \sqrt [3]{55+63 \sqrt {2}}-2791258628962151341704724767578146403684943522 \sqrt {2} \sqrt [3]{55+63 \sqrt {2}}+179485343237855158869454044049048901217505467 \sqrt [3]{10963+6930 \sqrt {2}}-126915303328106685159239220021955429240094175 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (34 \left (642519576768834523056575808829993291975058-454329949775884423936783497444418712592165 \sqrt {2}\right )+\left (1652319889364632947563275101003447296244235-1168366598467025851582901678036774545795908 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (2250534349908721721820757961446221398370032-1591368100101396839247954199069900729609170 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right ) C\right ) \int \frac {1}{\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{55/3}}-\frac {2 \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-2 \left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right ) \int \frac {\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}{\sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{\left (55-63 \sqrt {2}\right )^9}\right )}{3 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{153 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{119\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {\sqrt [3]{2 \left (126-55 \sqrt {2}\right )} \left (\frac {17 \left (\left (2329250-501417 \sqrt {2}+119 \left (13988-10395 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1156 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A+\left (2885386-2280852 \sqrt {2}-14 \left (31697-13797 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-544 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) B-2 \left (7 \left (82772-44163 \sqrt {2}\right )+7 \left (19378-16569 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+68 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{5/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{3213 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}-\frac {\sqrt [3]{126-55 \sqrt {2}} \left (-\frac {2 \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17} \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )-\frac {289 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}\right )}{1377 \left (17-\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {4 \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (\left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-\left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}{27 \left (-55+63 \sqrt {2}\right )^{25/3} \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {2 i \sqrt {2} \left (5 \left (7711259249342034331540551713816401444467986016-5452683706393466091688175106537932137299420066 \sqrt {2}+1175652 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+5202 \left (3013041993039644902966886774356750694308522-2130542425289165678586957566437978265377827 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-805577598936782021705281372332071291561571391 \sqrt [3]{10963+6930 \sqrt {2}}+569629382979067382117244250912754520232212303 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (2672578208549350787748003395517624928477393-1889798174571446947172794711888173965105016 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+306 \left (449347719939814928406612930511657815952984-317736819879490171907747805891204004627671 \sqrt {2}-\left (42212972401091014524514717989893507598364-29849079038692187752899631198902118087743 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+4 \left (11782445396249377169230236103969695291331446759-8331447038702096367472565767988905703973810630 \sqrt {2}-9350595 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+306 \left (23048647555740695760571392136623258656466545-16297854983829155985278433350946191353771401 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+2218498065526499849500416512746169197670126141 \sqrt [3]{10963+6930 \sqrt {2}}-1568715026188554352471420048789142297660901060 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (7812436492524583911367102687376238089051845-5524226821438115814218592936888010727741109 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-9 \left (2713028889836854119338366421580628232637841-1918401125593279495691302565543493025575810 \sqrt {2}-2 \left (53075282307413412963153521742904012076473-37529892034185175031134338325457235761100 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-6 \left (3173558438211395567949026036480029199543174708-2244044692102804293641894022394269759347864550 \sqrt {2}-1167560 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+3947435809155102797670670702823924227943692900 \sqrt [3]{55+63 \sqrt {2}}-2791258628962151341704724767578146403684943522 \sqrt {2} \sqrt [3]{55+63 \sqrt {2}}+179485343237855158869454044049048901217505467 \sqrt [3]{10963+6930 \sqrt {2}}-126915303328106685159239220021955429240094175 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (34 \left (642519576768834523056575808829993291975058-454329949775884423936783497444418712592165 \sqrt {2}\right )+\left (1652319889364632947563275101003447296244235-1168366598467025851582901678036774545795908 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (2250534349908721721820757961446221398370032-1591368100101396839247954199069900729609170 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right ) C\right ) \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}} \int \frac {1}{\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{2 \sqrt {3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}+1} \sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\sqrt {3} \left (17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )\right )}+1}}d\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}}{9 \left (-55+63 \sqrt {2}\right )^{109/6} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}-\frac {2 i \sqrt {2} \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-2 \left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \int \frac {\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\sqrt {3} \left (17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )\right )}+1}}{\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{2 \sqrt {3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}+1}}d\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}}{9 \left (55-63 \sqrt {2}\right )^9 \sqrt [6]{-55+63 \sqrt {2}} \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}}}\right )}{3 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{153 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{119\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {\sqrt [3]{2 \left (126-55 \sqrt {2}\right )} \left (\frac {17 \left (\left (2329250-501417 \sqrt {2}+119 \left (13988-10395 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1156 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A+\left (2885386-2280852 \sqrt {2}-14 \left (31697-13797 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-544 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) B-2 \left (7 \left (82772-44163 \sqrt {2}\right )+7 \left (19378-16569 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+68 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{5/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{3213 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}-\frac {\sqrt [3]{126-55 \sqrt {2}} \left (-\frac {2 \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17} \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )-\frac {289 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}\right )}{1377 \left (17-\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {4 \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (\left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-\left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}{27 \left (-55+63 \sqrt {2}\right )^{25/3} \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {2 i \sqrt {2} \left (5 \left (7711259249342034331540551713816401444467986016-5452683706393466091688175106537932137299420066 \sqrt {2}+1175652 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+5202 \left (3013041993039644902966886774356750694308522-2130542425289165678586957566437978265377827 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-805577598936782021705281372332071291561571391 \sqrt [3]{10963+6930 \sqrt {2}}+569629382979067382117244250912754520232212303 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (2672578208549350787748003395517624928477393-1889798174571446947172794711888173965105016 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+306 \left (449347719939814928406612930511657815952984-317736819879490171907747805891204004627671 \sqrt {2}-\left (42212972401091014524514717989893507598364-29849079038692187752899631198902118087743 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+4 \left (11782445396249377169230236103969695291331446759-8331447038702096367472565767988905703973810630 \sqrt {2}-9350595 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+306 \left (23048647555740695760571392136623258656466545-16297854983829155985278433350946191353771401 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+2218498065526499849500416512746169197670126141 \sqrt [3]{10963+6930 \sqrt {2}}-1568715026188554352471420048789142297660901060 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (7812436492524583911367102687376238089051845-5524226821438115814218592936888010727741109 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-9 \left (2713028889836854119338366421580628232637841-1918401125593279495691302565543493025575810 \sqrt {2}-2 \left (53075282307413412963153521742904012076473-37529892034185175031134338325457235761100 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-6 \left (3173558438211395567949026036480029199543174708-2244044692102804293641894022394269759347864550 \sqrt {2}-1167560 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+3947435809155102797670670702823924227943692900 \sqrt [3]{55+63 \sqrt {2}}-2791258628962151341704724767578146403684943522 \sqrt {2} \sqrt [3]{55+63 \sqrt {2}}+179485343237855158869454044049048901217505467 \sqrt [3]{10963+6930 \sqrt {2}}-126915303328106685159239220021955429240094175 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (34 \left (642519576768834523056575808829993291975058-454329949775884423936783497444418712592165 \sqrt {2}\right )+\left (1652319889364632947563275101003447296244235-1168366598467025851582901678036774545795908 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (2250534349908721721820757961446221398370032-1591368100101396839247954199069900729609170 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right ) C\right ) \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}\right ),\frac {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}{17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )}\right )}{9 \left (-55+63 \sqrt {2}\right )^{109/6} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}-\frac {2 i \sqrt {2} \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-2 \left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \int \frac {\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\sqrt {3} \left (17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )\right )}+1}}{\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{2 \sqrt {3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}+1}}d\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}}{9 \left (55-63 \sqrt {2}\right )^9 \sqrt [6]{-55+63 \sqrt {2}} \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}}}\right )}{3 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{153 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{119\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {729 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{5/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{5/2} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{63\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {\sqrt [3]{2 \left (126-55 \sqrt {2}\right )} \left (\frac {17 \left (\left (2329250-501417 \sqrt {2}+119 \left (13988-10395 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1156 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) A+\left (2885386-2280852 \sqrt {2}-14 \left (31697-13797 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-544 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) B-2 \left (7 \left (82772-44163 \sqrt {2}\right )+7 \left (19378-16569 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+68 \left (-55+63 \sqrt {2}\right )^{5/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{5/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{3213 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}-\frac {\sqrt [3]{126-55 \sqrt {2}} \left (-\frac {2 \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17} \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (5 \left (289-154 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-\left (680+364 \sqrt [3]{-55+63 \sqrt {2}}-40 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (85-126 \sqrt [3]{-55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )-\frac {289 \left (5 \left (5940727-14435883 \sqrt {2}+\left (64539497-45513216 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-578 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+\left (522122950-355292658 \sqrt {2}-\left (84172078-64304100 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+1360 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (61314065-47496897 \sqrt {2}+\left (26502823-18140220 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-170 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )}{\left (-55+63 \sqrt {2}\right )^{8/3}}\right )}{1377 \left (17-\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {4 \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (\left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-\left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}{27 \left (-55+63 \sqrt {2}\right )^{25/3} \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}-\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {2 i \sqrt {2} \left (5 \left (7711259249342034331540551713816401444467986016-5452683706393466091688175106537932137299420066 \sqrt {2}+1175652 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+5202 \left (3013041993039644902966886774356750694308522-2130542425289165678586957566437978265377827 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-805577598936782021705281372332071291561571391 \sqrt [3]{10963+6930 \sqrt {2}}+569629382979067382117244250912754520232212303 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (2672578208549350787748003395517624928477393-1889798174571446947172794711888173965105016 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+306 \left (449347719939814928406612930511657815952984-317736819879490171907747805891204004627671 \sqrt {2}-\left (42212972401091014524514717989893507598364-29849079038692187752899631198902118087743 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+4 \left (11782445396249377169230236103969695291331446759-8331447038702096367472565767988905703973810630 \sqrt {2}-9350595 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+306 \left (23048647555740695760571392136623258656466545-16297854983829155985278433350946191353771401 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+2218498065526499849500416512746169197670126141 \sqrt [3]{10963+6930 \sqrt {2}}-1568715026188554352471420048789142297660901060 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (7812436492524583911367102687376238089051845-5524226821438115814218592936888010727741109 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-9 \left (2713028889836854119338366421580628232637841-1918401125593279495691302565543493025575810 \sqrt {2}-2 \left (53075282307413412963153521742904012076473-37529892034185175031134338325457235761100 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-6 \left (3173558438211395567949026036480029199543174708-2244044692102804293641894022394269759347864550 \sqrt {2}-1167560 \left (358653490479588661109809191599641461523-253606315211600293972654979264882356770 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+3947435809155102797670670702823924227943692900 \sqrt [3]{55+63 \sqrt {2}}-2791258628962151341704724767578146403684943522 \sqrt {2} \sqrt [3]{55+63 \sqrt {2}}+179485343237855158869454044049048901217505467 \sqrt [3]{10963+6930 \sqrt {2}}-126915303328106685159239220021955429240094175 \sqrt {2} \sqrt [3]{10963+6930 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3} \left (34 \left (642519576768834523056575808829993291975058-454329949775884423936783497444418712592165 \sqrt {2}\right )+\left (1652319889364632947563275101003447296244235-1168366598467025851582901678036774545795908 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (2250534349908721721820757961446221398370032-1591368100101396839247954199069900729609170 \sqrt {2}\right ) \sqrt [3]{10963+6930 \sqrt {2}}\right )\right ) C\right ) \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}\right ),\frac {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}{17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )}\right )}{9 \left (-55+63 \sqrt {2}\right )^{109/6} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}-\frac {2 i \sqrt {2} \left (935 \left (25721 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )+103173 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-21 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (6069 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (357-19 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) A+2 \left (586381 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4641 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1341249 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (78897-1994 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) B-2 \left (2802433 \left (13391262501501986695-9469096553934155943 \sqrt {2}\right )[phone] \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3213 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (928557 \left (92934404277039361-65713256081466840 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (13391262501501986695-9469096553934155943 \sqrt {2}\right ) \left (54621-2417 \sqrt [3]{10963+6930 \sqrt {2}}\right )\right )\right ) C\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} E\left (\arcsin \left (\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}\right )|\frac {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}{17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )}\right )}{9 \left (55-63 \sqrt {2}\right )^9 \sqrt [6]{-55+63 \sqrt {2}} \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}}}\right )}{3 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{153 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{119\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{126\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{5/2}}-\frac {2 C}{81 \left (9 x^3+3 x^2+6 x+2\right )^{3/2}}\)

Input:

Int[(A + B*x + C*x^2)/(2 + 6*x + 3*x^2 + 9*x^3)^(5/2),x]
 

Output:

(-2*C)/(81*(2 + 6*x + 3*x^2 + 9*x^3)^(3/2)) + (729*((17 - (-55 + 63*Sqrt[2 
])^(2/3))/(-55 + 63*Sqrt[2])^(1/3) + 9*(1/9 + x))^(5/2)*(17 + 289/(-55 + 6 
3*Sqrt[2])^(2/3) + (-55 + 63*Sqrt[2])^(2/3) - (9*(17 - (-55 + 63*Sqrt[2])^ 
(2/3))*(1/9 + x))/(-55 + 63*Sqrt[2])^(1/3) + 81*(1/9 + x)^2)^(5/2)*(((126 
- 55*Sqrt[2])^(1/3)*(2*(17*A - 8*B - 2*C) + ((9*(55 - 63*Sqrt[2] + 17*(-55 
 + 63*Sqrt[2])^(1/3))*A + 9*(26 + 7*Sqrt[2] - (8 - 7*Sqrt[2])*(-55 + 63*Sq 
rt[2])^(1/3))*B - 2*(81 - 56*Sqrt[2] + (9 + 7*Sqrt[2])*(-55 + 63*Sqrt[2])^ 
(1/3))*C)*(1/9 + x))/(-55 + 63*Sqrt[2])^(2/3)))/(63*2^(2/3)*(17 + (-55 + 6 
3*Sqrt[2])^(2/3))*((17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/3 
) + 9*(1/9 + x))^(3/2)*(17 + 289/(-55 + 63*Sqrt[2])^(2/3) + (-55 + 63*Sqrt 
[2])^(2/3) - (9*(17 - (-55 + 63*Sqrt[2])^(2/3))*(1/9 + x))/(-55 + 63*Sqrt[ 
2])^(1/3) + 81*(1/9 + x)^2)^(3/2)) + ((126 - 55*Sqrt[2])^(1/3)*(((2*(126 - 
 55*Sqrt[2]))^(1/3)*((17*((2329250 - 501417*Sqrt[2] + 119*(13988 - 10395*S 
qrt[2])*(-55 + 63*Sqrt[2])^(1/3) + 1156*(-55 + 63*Sqrt[2])^(5/3))*A + (288 
5386 - 2280852*Sqrt[2] - 14*(31697 - 13797*Sqrt[2])*(-55 + 63*Sqrt[2])^(1/ 
3) - 544*(-55 + 63*Sqrt[2])^(5/3))*B - 2*(7*(82772 - 44163*Sqrt[2]) + 7*(1 
9378 - 16569*Sqrt[2])*(-55 + 63*Sqrt[2])^(1/3) + 68*(-55 + 63*Sqrt[2])^(5/ 
3))*C))/(-55 + 63*Sqrt[2])^(5/3) - 9*(55 + 63*Sqrt[2])^(1/3)*(5*(289 - 154 
*(-55 + 63*Sqrt[2])^(1/3) - 17*(-55 + 63*Sqrt[2])^(2/3))*A - (680 + 364*(- 
55 + 63*Sqrt[2])^(1/3) - 40*(-55 + 63*Sqrt[2])^(2/3))*B - 2*(85 - 126*(...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2486

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S 
ymbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]} 
, Simp[(a + b*x + d*x^3)^p/(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x 
]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/ 
3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p)   Int[(e + f*x)^m*Sim 
p[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2 
*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/ 
3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4* 
b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

rule 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

rule 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp 
[1/(n*Coeff[Qn, x, n])   Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x 
, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.36 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.80

method	result	size
elliptic	\(\frac {\left (\left (\frac {4 B}{3969}+\frac {C}{3969}-\frac {17 A}{7938}\right ) x^{2}+\left (\frac {B}{1701}-\frac {C}{5103}+\frac {A}{3402}\right ) x +\frac {17 B}{35721}-\frac {8 C}{35721}-\frac {10 A}{11907}\right ) \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{\left (x^{3}+\frac {1}{3} x^{2}+\frac {2}{3} x +\frac {2}{9}\right )^{2}}-\frac {18 \left (\left (\frac {935 A}{86436}+\frac {149 B}{172872}-\frac {673 C}{259308}\right ) x^{2}+\left (\frac {205 A}{74088}-\frac {157 B}{74088}-\frac {4 C}{27783}\right ) x +\frac {815 A}{172872}+\frac {5 B}{43218}-\frac {835 C}{777924}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {2 \left (\frac {145 A}{19208}+\frac {52 B}{2401}-\frac {187 C}{28812}\right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {2 \left (\frac {935 A}{9604}+\frac {149 B}{19208}-\frac {673 C}{28812}\right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}\)	\(446\)
risch	\(-\frac {151470 x^{5} A +12069 B \,x^{5}-36342 x^{5} C +89235 x^{4} A -25650 x^{4} B -14130 C \,x^{4}+179910 x^{3} A -225 B \,x^{3}-39930 C \,x^{3}+96489 A \,x^{2}-23616 B \,x^{2}-16194 C \,x^{2}+50562 A x -9630 B x -9096 C x +20550 A -2972 B -1772 C}{86436 \left (9 x^{3}+3 x^{2}+6 x +2\right )^{\frac {3}{2}}}+\frac {\left (5610 A +447 B -1346 C \right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{28812 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {145 A \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{9604 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {104 B \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{2401 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}-\frac {187 C \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{14406 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\)	\(674\)
default	\(\text {Expression too large to display}\)	\(1127\)

Input:

int((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

((4/3969*B+1/3969*C-17/7938*A)*x^2+(1/1701*B-1/5103*C+1/3402*A)*x+17/35721 
*B-8/35721*C-10/11907*A)*(9*x^3+3*x^2+6*x+2)^(1/2)/(x^3+1/3*x^2+2/3*x+2/9) 
^2-18*((935/86436*A+149/172872*B-673/259308*C)*x^2+(205/74088*A-157/74088* 
B-4/27783*C)*x+815/172872*A+5/43218*B-835/777924*C)/(9*x^3+3*x^2+6*x+2)^(1 
/2)+2*(145/19208*A+52/2401*B-187/28812*C)*(-1/3*I*6^(1/2)+1/3)*((x+1/3)/(- 
1/3*I*6^(1/2)+1/3))^(1/2)*((x-1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2)*( 
(x+1/3*I*6^(1/2))/(-1/3+1/3*I*6^(1/2)))^(1/2)/(9*x^3+3*x^2+6*x+2)^(1/2)*El 
lipticF(((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2),((-1/3+1/3*I*6^(1/2))/(-1/3-1 
/3*I*6^(1/2)))^(1/2))+2*(935/9604*A+149/19208*B-673/28812*C)*(-1/3*I*6^(1/ 
2)+1/3)*((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2)*((x-1/3*I*6^(1/2))/(-1/3-1/3* 
I*6^(1/2)))^(1/2)*((x+1/3*I*6^(1/2))/(-1/3+1/3*I*6^(1/2)))^(1/2)/(9*x^3+3* 
x^2+6*x+2)^(1/2)*((-1/3-1/3*I*6^(1/2))*EllipticE(((x+1/3)/(-1/3*I*6^(1/2)+ 
1/3))^(1/2),((-1/3+1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2))+1/3*I*6^(1/ 
2)*EllipticF(((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2),((-1/3+1/3*I*6^(1/2))/(- 
1/3-1/3*I*6^(1/2)))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=-\frac {5 \, {\left (81 \, {\left (339 \, A - 2157 \, B + 404 \, C\right )} x^{6} + 54 \, {\left (339 \, A - 2157 \, B + 404 \, C\right )} x^{5} + 117 \, {\left (339 \, A - 2157 \, B + 404 \, C\right )} x^{4} + 72 \, {\left (339 \, A - 2157 \, B + 404 \, C\right )} x^{3} + 48 \, {\left (339 \, A - 2157 \, B + 404 \, C\right )} x^{2} + 24 \, {\left (339 \, A - 2157 \, B + 404 \, C\right )} x + 1356 \, A - 8628 \, B + 1616 \, C\right )} {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right ) + 9 \, {\left (81 \, {\left (5610 \, A + 447 \, B - 1346 \, C\right )} x^{6} + 54 \, {\left (5610 \, A + 447 \, B - 1346 \, C\right )} x^{5} + 117 \, {\left (5610 \, A + 447 \, B - 1346 \, C\right )} x^{4} + 72 \, {\left (5610 \, A + 447 \, B - 1346 \, C\right )} x^{3} + 48 \, {\left (5610 \, A + 447 \, B - 1346 \, C\right )} x^{2} + 24 \, {\left (5610 \, A + 447 \, B - 1346 \, C\right )} x + 22440 \, A + 1788 \, B - 5384 \, C\right )} {\rm weierstrassZeta}\left (-\frac {68}{27}, -\frac {440}{729}, {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right )\right ) + 9 \, {\left (27 \, {\left (5610 \, A + 447 \, B - 1346 \, C\right )} x^{5} + 45 \, {\left (1983 \, A - 570 \, B - 314 \, C\right )} x^{4} + 15 \, {\left (11994 \, A - 15 \, B - 2662 \, C\right )} x^{3} + 3 \, {\left (32163 \, A - 7872 \, B - 5398 \, C\right )} x^{2} + 6 \, {\left (8427 \, A - 1605 \, B - 1516 \, C\right )} x + 20550 \, A - 2972 \, B - 1772 \, C\right )} \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2}}{777924 \, {\left (81 \, x^{6} + 54 \, x^{5} + 117 \, x^{4} + 72 \, x^{3} + 48 \, x^{2} + 24 \, x + 4\right )}} \] Input:

integrate((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(5/2),x, algorithm="fricas")
 

Output:

-1/777924*(5*(81*(339*A - 2157*B + 404*C)*x^6 + 54*(339*A - 2157*B + 404*C 
)*x^5 + 117*(339*A - 2157*B + 404*C)*x^4 + 72*(339*A - 2157*B + 404*C)*x^3 
 + 48*(339*A - 2157*B + 404*C)*x^2 + 24*(339*A - 2157*B + 404*C)*x + 1356* 
A - 8628*B + 1616*C)*weierstrassPInverse(-68/27, -440/729, x + 1/9) + 9*(8 
1*(5610*A + 447*B - 1346*C)*x^6 + 54*(5610*A + 447*B - 1346*C)*x^5 + 117*( 
5610*A + 447*B - 1346*C)*x^4 + 72*(5610*A + 447*B - 1346*C)*x^3 + 48*(5610 
*A + 447*B - 1346*C)*x^2 + 24*(5610*A + 447*B - 1346*C)*x + 22440*A + 1788 
*B - 5384*C)*weierstrassZeta(-68/27, -440/729, weierstrassPInverse(-68/27, 
 -440/729, x + 1/9)) + 9*(27*(5610*A + 447*B - 1346*C)*x^5 + 45*(1983*A - 
570*B - 314*C)*x^4 + 15*(11994*A - 15*B - 2662*C)*x^3 + 3*(32163*A - 7872* 
B - 5398*C)*x^2 + 6*(8427*A - 1605*B - 1516*C)*x + 20550*A - 2972*B - 1772 
*C)*sqrt(9*x^3 + 3*x^2 + 6*x + 2))/(81*x^6 + 54*x^5 + 117*x^4 + 72*x^3 + 4 
8*x^2 + 24*x + 4)
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\int \frac {A + B x + C x^{2}}{\left (\left (3 x + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((C*x**2+B*x+A)/(9*x**3+3*x**2+6*x+2)**(5/2),x)
 

Output:

Integral((A + B*x + C*x**2)/((3*x + 1)*(3*x**2 + 2))**(5/2), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(5/2),x, algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/(9*x^3 + 3*x^2 + 6*x + 2)^(5/2), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(5/2),x, algorithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)/(9*x^3 + 3*x^2 + 6*x + 2)^(5/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\int \frac {C\,x^2+B\,x+A}{{\left (9\,x^3+3\,x^2+6\,x+2\right )}^{5/2}} \,d x \] Input:

int((A + B*x + C*x^2)/(6*x + 3*x^2 + 9*x^3 + 2)^(5/2),x)
 

Output:

int((A + B*x + C*x^2)/(6*x + 3*x^2 + 9*x^3 + 2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\text {too large to display} \] Input:

int((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(5/2),x)
 

Output:

( - 2*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*b + 1458*int(sqrt(9*x**3 + 3*x**2 + 
6*x + 2)/(729*x**9 + 729*x**8 + 1701*x**7 + 1485*x**6 + 1458*x**5 + 1026*x 
**4 + 540*x**3 + 252*x**2 + 72*x + 8),x)*a*x**6 + 972*int(sqrt(9*x**3 + 3* 
x**2 + 6*x + 2)/(729*x**9 + 729*x**8 + 1701*x**7 + 1485*x**6 + 1458*x**5 + 
 1026*x**4 + 540*x**3 + 252*x**2 + 72*x + 8),x)*a*x**5 + 2106*int(sqrt(9*x 
**3 + 3*x**2 + 6*x + 2)/(729*x**9 + 729*x**8 + 1701*x**7 + 1485*x**6 + 145 
8*x**5 + 1026*x**4 + 540*x**3 + 252*x**2 + 72*x + 8),x)*a*x**4 + 1296*int( 
sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(729*x**9 + 729*x**8 + 1701*x**7 + 1485*x* 
*6 + 1458*x**5 + 1026*x**4 + 540*x**3 + 252*x**2 + 72*x + 8),x)*a*x**3 + 8 
64*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(729*x**9 + 729*x**8 + 1701*x**7 + 
1485*x**6 + 1458*x**5 + 1026*x**4 + 540*x**3 + 252*x**2 + 72*x + 8),x)*a*x 
**2 + 432*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(729*x**9 + 729*x**8 + 1701* 
x**7 + 1485*x**6 + 1458*x**5 + 1026*x**4 + 540*x**3 + 252*x**2 + 72*x + 8) 
,x)*a*x + 72*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(729*x**9 + 729*x**8 + 17 
01*x**7 + 1485*x**6 + 1458*x**5 + 1026*x**4 + 540*x**3 + 252*x**2 + 72*x + 
 8),x)*a - 1458*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(729*x**9 + 729*x**8 + 
 1701*x**7 + 1485*x**6 + 1458*x**5 + 1026*x**4 + 540*x**3 + 252*x**2 + 72* 
x + 8),x)*b*x**6 - 972*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(729*x**9 + 729 
*x**8 + 1701*x**7 + 1485*x**6 + 1458*x**5 + 1026*x**4 + 540*x**3 + 252*x** 
2 + 72*x + 8),x)*b*x**5 - 2106*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(729...