\(\int (A+B x+C x^2) (2+6 x+3 x^2+9 x^3)^{3/2} \, dx\) [92]

Optimal result

Integrand size = 30, antiderivative size = 636 \[ \int \left (A+B x+C x^2\right ) \left (2+6 x+3 x^2+9 x^3\right )^{3/2} \, dx=-\frac {2 (5967 A+1152 B-1582 C) \left (2+6 x+3 x^2+9 x^3\right )^{3/2}}{243243 (1+3 x)}+\frac {2 (108 A-45 B-14 C) x \left (2+6 x+3 x^2+9 x^3\right )^{3/2}}{891 (1+3 x)}-\frac {8 (71955 A+31869 B-23072 C) \left (2+6 x+3 x^2+9 x^3\right )^{3/2}}{3648645 (1+3 x) \left (2+3 x^2\right )}+\frac {4 (64935 A-23418 B-9226 C) x \left (2+6 x+3 x^2+9 x^3\right )^{3/2}}{405405 (1+3 x) \left (2+3 x^2\right )}+\frac {16 (4590 A-1251 B-742 C) \left (2+6 x+3 x^2+9 x^3\right )^{3/2}}{25515 (1+3 x) \left (1+\sqrt {7}+3 x\right ) \left (2+3 x^2\right )}+\frac {2}{351} (9 B-2 C) \left (2+3 x^2\right ) \left (2+6 x+3 x^2+9 x^3\right )^{3/2}+\frac {2 (117 A+9 B-28 C) \left (2+3 x^2\right ) \left (2+6 x+3 x^2+9 x^3\right )^{3/2}}{1287 (1+3 x)}+\frac {2}{135} C (1+3 x) \left (2+3 x^2\right ) \left (2+6 x+3 x^2+9 x^3\right )^{3/2}-\frac {16 (4590 A-1251 B-742 C) \left (1+\sqrt {7}+3 x\right ) \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} \left (2+6 x+3 x^2+9 x^3\right )^{3/2} E\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{3645 \sqrt {3} 7^{3/4} (1+3 x)^{3/2} \left (2+3 x^2\right )^2}+\frac {8 \left (656370 A-71955 \sqrt {7} A-178893 B-31869 \sqrt {7} B-106106 C+23072 \sqrt {7} C\right ) \left (1+\sqrt {7}+3 x\right ) \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} \left (2+6 x+3 x^2+9 x^3\right )^{3/2} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{521235 \sqrt {3} 7^{3/4} (1+3 x)^{3/2} \left (2+3 x^2\right )^2} \] Output:

-2*(5967*A+1152*B-1582*C)*(9*x^3+3*x^2+6*x+2)^(3/2)/(243243+729729*x)+2*(1 
08*A-45*B-14*C)*x*(9*x^3+3*x^2+6*x+2)^(3/2)/(891+2673*x)-8/3648645*(71955* 
A+31869*B-23072*C)*(9*x^3+3*x^2+6*x+2)^(3/2)/(1+3*x)/(3*x^2+2)+4/405405*(6 
4935*A-23418*B-9226*C)*x*(9*x^3+3*x^2+6*x+2)^(3/2)/(1+3*x)/(3*x^2+2)+16/25 
515*(4590*A-1251*B-742*C)*(9*x^3+3*x^2+6*x+2)^(3/2)/(1+3*x)/(1+7^(1/2)+3*x 
)/(3*x^2+2)+2/351*(9*B-2*C)*(3*x^2+2)*(9*x^3+3*x^2+6*x+2)^(3/2)+2*(117*A+9 
*B-28*C)*(3*x^2+2)*(9*x^3+3*x^2+6*x+2)^(3/2)/(1287+3861*x)+2/135*C*(1+3*x) 
*(3*x^2+2)*(9*x^3+3*x^2+6*x+2)^(3/2)-16/76545*(4590*A-1251*B-742*C)*(1+7^( 
1/2)+3*x)*((3*x^2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*(9*x^3+3*x^2+6*x+2)^(3/2)*El 
lipticE(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)/(1+3*x)^(3/2)/(3*x^2+2)^2+8/10945935*(656370*A-71955*7^ 
(1/2)*A-178893*B-31869*7^(1/2)*B-106106*C+23072*7^(1/2)*C)*(1+7^(1/2)+3*x) 
*((3*x^2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*(9*x^3+3*x^2+6*x+2)^(3/2)*InverseJaco 
biAM(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)/(1+3*x)^(3/2)/(3*x^2+2)^2
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.33 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.50 \[ \int \left (A+B x+C x^2\right ) \left (2+6 x+3 x^2+9 x^3\right )^{3/2} \, dx=\frac {2 \left (2+3 x^2\right ) \left ((1+3 x) \sqrt {6+9 x^2} \left (1755 A \left (490+1170 x+2115 x^2+756 x^3+1701 x^4\right )+9 B \left (34916+36954 x+153000 x^2+312795 x^3+119070 x^4+280665 x^5\right )+7 C \left (-21832+16920 x+23778 x^2+146502 x^3+322947 x^4+128304 x^5+312741 x^6\right )\right )+\frac {1144 i (4590 A-1251 B-742 C) (1+3 x) E\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )}{\sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}}}-56 i (71955 A+31869 B-23072 C) \sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right ),\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )\right )}{3648645 \sqrt {6+9 x^2} \sqrt {2+6 x+3 x^2+9 x^3}} \] Input:

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

Output:

(2*(2 + 3*x^2)*((1 + 3*x)*Sqrt[6 + 9*x^2]*(1755*A*(490 + 1170*x + 2115*x^2 
 + 756*x^3 + 1701*x^4) + 9*B*(34916 + 36954*x + 153000*x^2 + 312795*x^3 + 
119070*x^4 + 280665*x^5) + 7*C*(-21832 + 16920*x + 23778*x^2 + 146502*x^3 
+ 322947*x^4 + 128304*x^5 + 312741*x^6)) + ((1144*I)*(4590*A - 1251*B - 74 
2*C)*(1 + 3*x)*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])] - (56*I)*( 
71955*A + 31869*B - 23072*C)*Sqrt[(I*(1 + 3*x))/(I + Sqrt[6])]*EllipticF[A 
rcSin[Sqrt[Sqrt[6] - (3*I)*x]/(2^(3/4)*3^(1/4))], (2*Sqrt[6])/(I + Sqrt[6] 
)]))/(3648645*Sqrt[6 + 9*x^2]*Sqrt[2 + 6*x + 3*x^2 + 9*x^3])
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 16.74 (sec) , antiderivative size = 2883, normalized size of antiderivative = 4.53, 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, 1236, 27, 1236, 27, 1231, 27, 1231, 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.

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

\(\Big \downarrow \) 2526

\(\displaystyle \frac {1}{27} \int 3 (9 A-2 C+(9 B-2 C) x) \left (9 x^3+3 x^2+6 x+2\right )^{3/2}dx+\frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int (9 A-2 C+(9 B-2 C) x) \left (9 x^3+3 x^2+6 x+2\right )^{3/2}dx+\frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}\)

\(\Big \downarrow \) 2490

\(\displaystyle \frac {1}{9} \int \left (\frac {1}{27} (27 (9 A-2 C)-3 (9 B-2 C))+(9 B-2 C) \left (x+\frac {1}{9}\right )\right ) \left (9 \left (x+\frac {1}{9}\right )^3+\frac {17}{3} \left (x+\frac {1}{9}\right )+\frac {110}{81}\right )^{3/2}d\left (x+\frac {1}{9}\right )+\frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}\)

\(\Big \downarrow \) 2486

\(\displaystyle \frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \int \frac {1}{9} \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 A-9 B-16 C+9 (9 B-2 C) \left (x+\frac {1}{9}\right )\right ) \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}d\left (x+\frac {1}{9}\right )}{6561 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17\right )^{3/2}}+\frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \int \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 A-9 B-16 C+9 (9 B-2 C) \left (x+\frac {1}{9}\right )\right ) \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}d\left (x+\frac {1}{9}\right )}{59049 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17\right )^{3/2}}+\frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2 \int \frac {81}{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}}}} \left (\frac {1053 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A-9 \left (137-819 \sqrt {2}+\left (331-126 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+221 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (6298-6552 \sqrt {2}-221 \left (-55+63 \sqrt {2}\right )^{2/3}+2 \sqrt [3]{-55+63 \sqrt {2}} \left (829+63 \sqrt {2}\right )\right ) C}{\left (-55+63 \sqrt {2}\right )^{2/3}}+9 \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )\right ) \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 )}{1053}+\frac {2}{117} \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 )^{3/2} \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} (9 B-2 C)\right )}{59049 \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 )^{3/2} \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 )^{3/2}}+\frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {1}{13} \int \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {1053 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A-9 \left (137-819 \sqrt {2}+\left (331-126 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+221 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (6298-6552 \sqrt {2}-221 \left (-55+63 \sqrt {2}\right )^{2/3}+2 \sqrt [3]{-55+63 \sqrt {2}} \left (829+63 \sqrt {2}\right )\right ) C}{\left (-55+63 \sqrt {2}\right )^{2/3}}+9 \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )\right ) \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 )+\frac {2}{117} \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 )^{3/2} \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} (9 B-2 C)\right )}{59049 \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 )^{3/2} \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 )^{3/2}}+\frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}+\frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2}{117} (9 B-2 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {2}{99} \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\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}}}} \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}+\frac {2}{891} \int -\frac {2187 \left (\frac {117 \left (187 \left (10963-6930 \sqrt {2}\right )-5 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}-1445 \left (-55+63 \sqrt {2}\right )^{4/3}\right ) A-9 \left (539 \left (10963-6930 \sqrt {2}\right )+85 \left (28149-18487 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+5 \left (23019-34496 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-4 \left (946 \left (10963-6930 \sqrt {2}\right )-85 \left (7997-2282 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-5 \left (82769-62293 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{\left (55-63 \sqrt {2}\right )^2}-\frac {9 \left (702 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (583+546 \sqrt {2}-\left (349-231 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-187 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (4873-4368 \sqrt {2}-187 \left (-55+63 \sqrt {2}\right )^{2/3}+\sqrt [3]{-55+63 \sqrt {2}} \left (977+231 \sqrt {2}\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right ) \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}}{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}}}}}d\left (x+\frac {1}{9}\right )\right )\right )}{59049 \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}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}+\frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2}{117} (9 B-2 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {2}{99} \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\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}}}} \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}-\frac {27}{11} \int \frac {\left (\frac {117 \left (187 \left (10963-6930 \sqrt {2}\right )-5 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}-1445 \left (-55+63 \sqrt {2}\right )^{4/3}\right ) A-9 \left (539 \left (10963-6930 \sqrt {2}\right )+85 \left (28149-18487 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+5 \left (23019-34496 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-4 \left (946 \left (10963-6930 \sqrt {2}\right )-85 \left (7997-2282 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-5 \left (82769-62293 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{\left (55-63 \sqrt {2}\right )^2}-\frac {9 \left (702 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (583+546 \sqrt {2}-\left (349-231 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-187 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (4873-4368 \sqrt {2}-187 \left (-55+63 \sqrt {2}\right )^{2/3}+\sqrt [3]{-55+63 \sqrt {2}} \left (977+231 \sqrt {2}\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right ) \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}}{\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}d\left (x+\frac {1}{9}\right )\right )\right )}{59049 \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}}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}+\frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2}{117} (9 B-2 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {2}{99} \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\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}}}} \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}-\frac {27}{11} \left (-\frac {2}{567} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {351 \left (187 \left (10963-6930 \sqrt {2}\right )-7 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}-2023 \left (-55+63 \sqrt {2}\right )^{4/3}\right ) A-9 \left (2827 \left (10963-6930 \sqrt {2}\right )+357 \left (28149-18487 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+21 \left (23019-34496 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-28 \left (319 \left (10963-6930 \sqrt {2}\right )-51 \left (7997-2282 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3 \left (82769-62293 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{\left (55-63 \sqrt {2}\right )^2}+\frac {63 \left (702 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (583+546 \sqrt {2}-\left (349-231 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-187 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (4873-4368 \sqrt {2}-187 \left (-55+63 \sqrt {2}\right )^{2/3}+\sqrt [3]{-55+63 \sqrt {2}} \left (977+231 \sqrt {2}\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right ) \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 {2 \int -\frac {19683 \left (\frac {17 \left (351 \left (97755910835-69031357794 \sqrt {2}+\left (2309305279-1848851235 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-2023 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-2 \left (9 \left (927759902927-650647310292 \sqrt {2}+\left (141682512127-102226892229 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+7616 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B+7 \left (279565816673-198704047626 \sqrt {2}-\left (27614588339-18906940899 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-13447 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )\right )}{\left (-55+63 \sqrt {2}\right )^{11/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\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}}{17 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}d\left (x+\frac {1}{9}\right )}{137781}\right )\right )\right )}{59049 \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}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}+\frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2}{117} (9 B-2 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {2}{99} \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\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}}}} \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}-\frac {27}{11} \left (\frac {2}{119} \int \frac {\left (\frac {17 \left (351 \left (97755910835-69031357794 \sqrt {2}+\left (2309305279-1848851235 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-2023 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-2 \left (9 \left (927759902927-650647310292 \sqrt {2}+\left (141682512127-102226892229 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+7616 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B+7 \left (279565816673-198704047626 \sqrt {2}-\left (27614588339-18906940899 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-13447 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )\right )}{\left (-55+63 \sqrt {2}\right )^{11/3}}-9 \sqrt [3]{55+63 \sqrt {2}} \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\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}}{\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}d\left (x+\frac {1}{9}\right )-\frac {2}{567} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {351 \left (187 \left (10963-6930 \sqrt {2}\right )-7 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}-2023 \left (-55+63 \sqrt {2}\right )^{4/3}\right ) A-9 \left (2827 \left (10963-6930 \sqrt {2}\right )+357 \left (28149-18487 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+21 \left (23019-34496 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-28 \left (319 \left (10963-6930 \sqrt {2}\right )-51 \left (7997-2282 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3 \left (82769-62293 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{\left (55-63 \sqrt {2}\right )^2}+\frac {63 \left (702 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (583+546 \sqrt {2}-\left (349-231 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-187 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (4873-4368 \sqrt {2}-187 \left (-55+63 \sqrt {2}\right )^{2/3}+\sqrt [3]{-55+63 \sqrt {2}} \left (977+231 \sqrt {2}\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right ) \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}\right )\right )\right )}{59049 \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}}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}+\frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2}{117} (9 B-2 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {2}{99} \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\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}}}} \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}-\frac {27}{11} \left (\frac {2}{119} \left (\frac {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}}}} \left (5 \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )+\frac {17 \left (5967 \left (97755910835-69031357794 \sqrt {2}+\left (2309305279-1848851235 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-2023 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-34 \left (9 \left (927759902927-650647310292 \sqrt {2}+\left (141682512127-102226892229 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+7616 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B+7 \left (279565816673-198704047626 \sqrt {2}-\left (27614588339-18906940899 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-13447 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )\right )}{\left (-55+63 \sqrt {2}\right )^{11/3}}\right )-459 \sqrt [3]{55+63 \sqrt {2}} \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\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}}{2295}-\frac {2 \int -\frac {19683 \left (\frac {5 \left (351 \left (1054034 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+85 \left (300219272986226-212410496567241 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (603038973719408-425366875701999 \sqrt {2}-13214 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (7237682 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+\left (359965143612389117-254381136334002090 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (42401177437380145-30058275256783389 \sqrt {2}-14932 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (3804566 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+\left (39328086103508708-27911217636057105 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (4684320714360610-3261427178331861 \sqrt {2}-363838 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\right )}{\left (-55+63 \sqrt {2}\right )^{16/3}}-\frac {9 \left (1755 \left (\left (365689357169816-258076362987135 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4114 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (158950 \left (216237169-151947180 \sqrt {2}\right )-\left (21511138657048-15180962528655 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (85 \left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+484924 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2865214 \left (216237169-151947180 \sqrt {2}\right )-\left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (85 \left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-253154 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (66946 \left (216237169-151947180 \sqrt {2}\right )-\left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{14/3}}\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 )}{98415}\right )-\frac {2}{567} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {351 \left (187 \left (10963-6930 \sqrt {2}\right )-7 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}-2023 \left (-55+63 \sqrt {2}\right )^{4/3}\right ) A-9 \left (2827 \left (10963-6930 \sqrt {2}\right )+357 \left (28149-18487 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+21 \left (23019-34496 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-28 \left (319 \left (10963-6930 \sqrt {2}\right )-51 \left (7997-2282 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3 \left (82769-62293 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{\left (55-63 \sqrt {2}\right )^2}+\frac {63 \left (702 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (583+546 \sqrt {2}-\left (349-231 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-187 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (4873-4368 \sqrt {2}-187 \left (-55+63 \sqrt {2}\right )^{2/3}+\sqrt [3]{-55+63 \sqrt {2}} \left (977+231 \sqrt {2}\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right ) \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}\right )\right )\right )}{59049 \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}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}+\frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2}{117} (9 B-2 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {2}{99} \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\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}}}} \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}-\frac {27}{11} \left (\frac {2}{119} \left (\frac {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} \left (5 \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )+\frac {17 \left (5967 \left (97755910835-69031357794 \sqrt {2}+\left (2309305279-1848851235 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-2023 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-34 \left (9 \left (927759902927-650647310292 \sqrt {2}+\left (141682512127-102226892229 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+7616 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B+7 \left (279565816673-198704047626 \sqrt {2}-\left (27614588339-18906940899 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-13447 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )\right )}{\left (-55+63 \sqrt {2}\right )^{11/3}}\right )-459 \sqrt [3]{55+63 \sqrt {2}} \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{2295}+\frac {1}{5} \int \frac {\frac {5 \left (351 \left (1054034 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+85 \left (300219272986226-212410496567241 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (603038973719408-425366875701999 \sqrt {2}-13214 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (7237682 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+\left (359965143612389117-254381136334002090 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (42401177437380145-30058275256783389 \sqrt {2}-14932 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (3804566 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+\left (39328086103508708-27911217636057105 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (4684320714360610-3261427178331861 \sqrt {2}-363838 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\right )}{\left (-55+63 \sqrt {2}\right )^{16/3}}-\frac {9 \left (1755 \left (\left (365689357169816-258076362987135 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4114 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (158950 \left (216237169-151947180 \sqrt {2}\right )-\left (21511138657048-15180962528655 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (85 \left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+484924 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2865214 \left (216237169-151947180 \sqrt {2}\right )-\left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (85 \left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-253154 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (66946 \left (216237169-151947180 \sqrt {2}\right )-\left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{14/3}}}{\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 )\right )-\frac {2}{567} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {351 \left (187 \left (10963-6930 \sqrt {2}\right )-7 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}-2023 \left (-55+63 \sqrt {2}\right )^{4/3}\right ) A-9 \left (2827 \left (10963-6930 \sqrt {2}\right )+357 \left (28149-18487 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+21 \left (23019-34496 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-28 \left (319 \left (10963-6930 \sqrt {2}\right )-51 \left (7997-2282 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3 \left (82769-62293 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{\left (55-63 \sqrt {2}\right )^2}+\frac {63 \left (702 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (583+546 \sqrt {2}-\left (349-231 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-187 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (4873-4368 \sqrt {2}-187 \left (-55+63 \sqrt {2}\right )^{2/3}+\sqrt [3]{-55+63 \sqrt {2}} \left (977+231 \sqrt {2}\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right ) \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}\right )\right )\right )}{59049 \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}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}+\frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2}{117} (9 B-2 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {2}{99} \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\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}}}} \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}-\frac {27}{11} \left (\frac {2}{119} \left (\frac {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} \left (5 \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )+\frac {17 \left (5967 \left (97755910835-69031357794 \sqrt {2}+\left (2309305279-1848851235 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-2023 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-34 \left (9 \left (927759902927-650647310292 \sqrt {2}+\left (141682512127-102226892229 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+7616 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B+7 \left (279565816673-198704047626 \sqrt {2}-\left (27614588339-18906940899 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-13447 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )\right )}{\left (-55+63 \sqrt {2}\right )^{11/3}}\right )-459 \sqrt [3]{55+63 \sqrt {2}} \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{2295}+\frac {1}{5} \left (\frac {\left (1755 \left (3891096 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-\left (5304207166185673480966686514-3750638480215016331435753105 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-714 \left (5199117516103504118197363-3676329168653289198258516 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1734 \left (507277663515742309600465-358699705339967204431071 \sqrt {2}\right )-\left (319861903200759064062218366-226176374935736653293284265 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (103400154 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-85 \left (1763498740210641524844248021-1246981761108019432196391537 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-42 \left (7778029268428879625119041445-5499899644691154592604704143 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (204 \left (49291411381413617816814271-34854232919728553411543778 \sqrt {2}\right )-\left (970781527584383282142145019-686446661226686242306630437 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (78851916 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-85 \left (1234496826168734906831902028-872920343283302321271571923 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-42 \left (239859432717908128798025065-169604557814763501002510322 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2027751918471261908540117646-1433842748867338910706535146 \sqrt {2}-\left (1504721595665833847734603748-1063997900005765284832117923 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\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 )^{31/3}}-\frac {\left (1755 \left (\left (365689357169816-258076362987135 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4114 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (158950 \left (216237169-151947180 \sqrt {2}\right )-\left (21511138657048-15180962528655 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (85 \left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+484924 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2865214 \left (216237169-151947180 \sqrt {2}\right )-\left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (85 \left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-253154 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (66946 \left (216237169-151947180 \sqrt {2}\right )-\left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\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 )^{14/3}}\right )\right )-\frac {2}{567} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {351 \left (187 \left (10963-6930 \sqrt {2}\right )-7 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}-2023 \left (-55+63 \sqrt {2}\right )^{4/3}\right ) A-9 \left (2827 \left (10963-6930 \sqrt {2}\right )+357 \left (28149-18487 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+21 \left (23019-34496 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-28 \left (319 \left (10963-6930 \sqrt {2}\right )-51 \left (7997-2282 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3 \left (82769-62293 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{\left (55-63 \sqrt {2}\right )^2}+\frac {63 \left (702 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (583+546 \sqrt {2}-\left (349-231 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-187 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (4873-4368 \sqrt {2}-187 \left (-55+63 \sqrt {2}\right )^{2/3}+\sqrt [3]{-55+63 \sqrt {2}} \left (977+231 \sqrt {2}\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right ) \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}\right )\right )\right )}{59049 \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}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}+\frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2}{117} (9 B-2 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {2}{99} \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\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}}}} \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}-\frac {27}{11} \left (\frac {2}{119} \left (\frac {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} \left (5 \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )+\frac {17 \left (5967 \left (97755910835-69031357794 \sqrt {2}+\left (2309305279-1848851235 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-2023 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-34 \left (9 \left (927759902927-650647310292 \sqrt {2}+\left (141682512127-102226892229 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+7616 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B+7 \left (279565816673-198704047626 \sqrt {2}-\left (27614588339-18906940899 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-13447 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )\right )}{\left (-55+63 \sqrt {2}\right )^{11/3}}\right )-459 \sqrt [3]{55+63 \sqrt {2}} \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{2295}+\frac {1}{5} \left (\frac {2 i \sqrt {2} \left (1755 \left (3891096 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-\left (5304207166185673480966686514-3750638480215016331435753105 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-714 \left (5199117516103504118197363-3676329168653289198258516 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1734 \left (507277663515742309600465-358699705339967204431071 \sqrt {2}\right )-\left (319861903200759064062218366-226176374935736653293284265 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (103400154 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-85 \left (1763498740210641524844248021-1246981761108019432196391537 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-42 \left (7778029268428879625119041445-5499899644691154592604704143 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (204 \left (49291411381413617816814271-34854232919728553411543778 \sqrt {2}\right )-\left (970781527584383282142145019-686446661226686242306630437 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (78851916 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-85 \left (1234496826168734906831902028-872920343283302321271571923 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-42 \left (239859432717908128798025065-169604557814763501002510322 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2027751918471261908540117646-1433842748867338910706535146 \sqrt {2}-\left (1504721595665833847734603748-1063997900005765284832117923 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\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 )^{61/6} \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 {i \sqrt {2} \left (1755 \left (\left (365689357169816-258076362987135 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4114 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (158950 \left (216237169-151947180 \sqrt {2}\right )-\left (21511138657048-15180962528655 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (85 \left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+484924 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2865214 \left (216237169-151947180 \sqrt {2}\right )-\left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (85 \left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-253154 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (66946 \left (216237169-151947180 \sqrt {2}\right )-\left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\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 )^{29/6} \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 )\right )-\frac {2}{567} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {351 \left (187 \left (10963-6930 \sqrt {2}\right )-7 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}-2023 \left (-55+63 \sqrt {2}\right )^{4/3}\right ) A-9 \left (2827 \left (10963-6930 \sqrt {2}\right )+357 \left (28149-18487 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+21 \left (23019-34496 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-28 \left (319 \left (10963-6930 \sqrt {2}\right )-51 \left (7997-2282 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3 \left (82769-62293 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{\left (55-63 \sqrt {2}\right )^2}+\frac {63 \left (702 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (583+546 \sqrt {2}-\left (349-231 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-187 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (4873-4368 \sqrt {2}-187 \left (-55+63 \sqrt {2}\right )^{2/3}+\sqrt [3]{-55+63 \sqrt {2}} \left (977+231 \sqrt {2}\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right ) \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}\right )\right )\right )}{59049 \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}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}+\frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2}{117} (9 B-2 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {2}{99} \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\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}}}} \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}-\frac {27}{11} \left (\frac {2}{119} \left (\frac {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} \left (5 \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )+\frac {17 \left (5967 \left (97755910835-69031357794 \sqrt {2}+\left (2309305279-1848851235 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-2023 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-34 \left (9 \left (927759902927-650647310292 \sqrt {2}+\left (141682512127-102226892229 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+7616 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B+7 \left (279565816673-198704047626 \sqrt {2}-\left (27614588339-18906940899 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-13447 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )\right )}{\left (-55+63 \sqrt {2}\right )^{11/3}}\right )-459 \sqrt [3]{55+63 \sqrt {2}} \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{2295}+\frac {1}{5} \left (\frac {2 i \sqrt {2} \left (1755 \left (3891096 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-\left (5304207166185673480966686514-3750638480215016331435753105 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-714 \left (5199117516103504118197363-3676329168653289198258516 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1734 \left (507277663515742309600465-358699705339967204431071 \sqrt {2}\right )-\left (319861903200759064062218366-226176374935736653293284265 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (103400154 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-85 \left (1763498740210641524844248021-1246981761108019432196391537 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-42 \left (7778029268428879625119041445-5499899644691154592604704143 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (204 \left (49291411381413617816814271-34854232919728553411543778 \sqrt {2}\right )-\left (970781527584383282142145019-686446661226686242306630437 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (78851916 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-85 \left (1234496826168734906831902028-872920343283302321271571923 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-42 \left (239859432717908128798025065-169604557814763501002510322 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2027751918471261908540117646-1433842748867338910706535146 \sqrt {2}-\left (1504721595665833847734603748-1063997900005765284832117923 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\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 )^{61/6} \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 {i \sqrt {2} \left (1755 \left (\left (365689357169816-258076362987135 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4114 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (158950 \left (216237169-151947180 \sqrt {2}\right )-\left (21511138657048-15180962528655 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (85 \left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+484924 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2865214 \left (216237169-151947180 \sqrt {2}\right )-\left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (85 \left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-253154 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (66946 \left (216237169-151947180 \sqrt {2}\right )-\left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\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 )^{29/6} \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 )\right )-\frac {2}{567} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {351 \left (187 \left (10963-6930 \sqrt {2}\right )-7 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}-2023 \left (-55+63 \sqrt {2}\right )^{4/3}\right ) A-9 \left (2827 \left (10963-6930 \sqrt {2}\right )+357 \left (28149-18487 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+21 \left (23019-34496 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-28 \left (319 \left (10963-6930 \sqrt {2}\right )-51 \left (7997-2282 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3 \left (82769-62293 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{\left (55-63 \sqrt {2}\right )^2}+\frac {63 \left (702 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (583+546 \sqrt {2}-\left (349-231 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-187 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (4873-4368 \sqrt {2}-187 \left (-55+63 \sqrt {2}\right )^{2/3}+\sqrt [3]{-55+63 \sqrt {2}} \left (977+231 \sqrt {2}\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right ) \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}\right )\right )\right )}{59049 \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}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2}{135} C \left (9 x^3+3 x^2+6 x+2\right )^{5/2}+\frac {\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2} \left (\frac {2}{117} (9 B-2 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {2}{99} \left (1053 A-\left (117-\frac {1683}{\sqrt [3]{-55+63 \sqrt {2}}}+99 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (104+\frac {187}{\sqrt [3]{-55+63 \sqrt {2}}}-11 \sqrt [3]{-55+63 \sqrt {2}}\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}}}} \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}-\frac {27}{11} \left (\frac {2}{119} \left (\frac {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} \left (5 \left (\sqrt [3]{10963+6930 \sqrt {2}} \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )+\frac {17 \left (5967 \left (97755910835-69031357794 \sqrt {2}+\left (2309305279-1848851235 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-2023 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A-34 \left (9 \left (927759902927-650647310292 \sqrt {2}+\left (141682512127-102226892229 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+7616 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B+7 \left (279565816673-198704047626 \sqrt {2}-\left (27614588339-18906940899 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-13447 \left (1476145-1071819 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right )\right )}{\left (-55+63 \sqrt {2}\right )^{11/3}}\right )-459 \sqrt [3]{55+63 \sqrt {2}} \left (1755 \left (289+77 \sqrt [3]{-55+63 \sqrt {2}}-17 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) A+18 \left (5440-4543 \sqrt [3]{-55+63 \sqrt {2}}-320 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-14 \left (9605+847 \sqrt [3]{-55+63 \sqrt {2}}-565 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C\right ) \left (x+\frac {1}{9}\right )\right )}{2295}+\frac {1}{5} \left (\frac {2 i \sqrt {2} \left (1755 \left (3891096 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-\left (5304207166185673480966686514-3750638480215016331435753105 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-714 \left (5199117516103504118197363-3676329168653289198258516 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{2/3} \left (1734 \left (507277663515742309600465-358699705339967204431071 \sqrt {2}\right )-\left (319861903200759064062218366-226176374935736653293284265 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (103400154 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-85 \left (1763498740210641524844248021-1246981761108019432196391537 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-42 \left (7778029268428879625119041445-5499899644691154592604704143 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (204 \left (49291411381413617816814271-34854232919728553411543778 \sqrt {2}\right )-\left (970781527584383282142145019-686446661226686242306630437 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (78851916 \left (1929625603378312917043-1364449848061003738650 \sqrt {2}\right )-85 \left (1234496826168734906831902028-872920343283302321271571923 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-42 \left (239859432717908128798025065-169604557814763501002510322 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}-5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2027751918471261908540117646-1433842748867338910706535146 \sqrt {2}-\left (1504721595665833847734603748-1063997900005765284832117923 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\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 )^{61/6} \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 {i \sqrt {2} \left (1755 \left (\left (365689357169816-258076362987135 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-4114 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (158950 \left (216237169-151947180 \sqrt {2}\right )-\left (21511138657048-15180962528655 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) A-2 \left (9 \left (85 \left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+484924 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (2865214 \left (216237169-151947180 \sqrt {2}\right )-\left (105011697013009-74561570661501 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) B+7 \left (85 \left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-253154 \left (31038388975-21980036547 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}+5 \left (-55+63 \sqrt {2}\right )^{2/3} \left (66946 \left (216237169-151947180 \sqrt {2}\right )-\left (89844430514122-63276342470649 \sqrt {2}\right ) \sqrt [3]{55+63 \sqrt {2}}\right )\right ) C\right )\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 )^{29/6} \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 )\right )-\frac {2}{567} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {351 \left (187 \left (10963-6930 \sqrt {2}\right )-7 \left (10963-6930 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}-2023 \left (-55+63 \sqrt {2}\right )^{4/3}\right ) A-9 \left (2827 \left (10963-6930 \sqrt {2}\right )+357 \left (28149-18487 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+21 \left (23019-34496 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-28 \left (319 \left (10963-6930 \sqrt {2}\right )-51 \left (7997-2282 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-3 \left (82769-62293 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{\left (55-63 \sqrt {2}\right )^2}+\frac {63 \left (702 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (583+546 \sqrt {2}-\left (349-231 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-187 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (4873-4368 \sqrt {2}-187 \left (-55+63 \sqrt {2}\right )^{2/3}+\sqrt [3]{-55+63 \sqrt {2}} \left (977+231 \sqrt {2}\right )\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right ) \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}\right )\right )\right )}{59049 \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}}\)

Input:

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

Output:

(2*C*(2 + 6*x + 3*x^2 + 9*x^3)^(5/2))/135 + ((110 + 459*(1/9 + x) + 729*(1 
/9 + x)^3)^(3/2)*((2*(9*B - 2*C)*((17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 6 
3*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)^(5/2))/117 + ((2*(1053*A - (117 
 - 1683/(-55 + 63*Sqrt[2])^(1/3) + 99*(-55 + 63*Sqrt[2])^(1/3))*B - 2*(104 
 + 187/(-55 + 63*Sqrt[2])^(1/3) - 11*(-55 + 63*Sqrt[2])^(1/3))*C)*Sqrt[(17 
 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/3) + 9*(1/9 + x)]*(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)^ 
(5/2))/99 - (27*((-2*Sqrt[(17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2 
])^(1/3) + 9*(1/9 + x)]*((351*(187*(10963 - 6930*Sqrt[2]) - 7*(10963 - 693 
0*Sqrt[2])*(-55 + 63*Sqrt[2])^(2/3) - 2023*(-55 + 63*Sqrt[2])^(4/3))*A - 9 
*(2827*(10963 - 6930*Sqrt[2]) + 357*(28149 - 18487*Sqrt[2])*(-55 + 63*Sqrt 
[2])^(1/3) + 21*(23019 - 34496*Sqrt[2])*(-55 + 63*Sqrt[2])^(2/3))*B - 28*( 
319*(10963 - 6930*Sqrt[2]) - 51*(7997 - 2282*Sqrt[2])*(-55 + 63*Sqrt[2])^( 
1/3) - 3*(82769 - 62293*Sqrt[2])*(-55 + 63*Sqrt[2])^(2/3))*C)/(55 - 63*Sqr 
t[2])^2 + (63*(702*(55 - 63*Sqrt[2] + 17*(-55 + 63*Sqrt[2])^(1/3))*A + 9*( 
583 + 546*Sqrt[2] - (349 - 231*Sqrt[2])*(-55 + 63*Sqrt[2])^(1/3) - 187*(-5 
5 + 63*Sqrt[2])^(2/3))*B - 2*(4873 - 4368*Sqrt[2] - 187*(-55 + 63*Sqrt[...
 

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 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1236

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

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 = 0.78 (sec) , antiderivative size = 523, normalized size of antiderivative = 0.82

method	result	size
elliptic	\(\frac {6 C \,x^{6} \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{5}+\left (\frac {18 B}{13}+\frac {32 C}{65}\right ) x^{5} \sqrt {9 x^{3}+3 x^{2}+6 x +2}+\left (\frac {84 B}{143}+\frac {18 A}{11}+\frac {886 C}{715}\right ) x^{4} \sqrt {9 x^{3}+3 x^{2}+6 x +2}+\left (\frac {662 B}{429}+\frac {8 A}{11}+\frac {10852 C}{19305}\right ) x^{3} \sqrt {9 x^{3}+3 x^{2}+6 x +2}+\left (\frac {6800 B}{9009}+\frac {470 A}{231}+\frac {5284 C}{57915}\right ) x^{2} \sqrt {9 x^{3}+3 x^{2}+6 x +2}+\left (\frac {8212 B}{45045}+\frac {260 A}{231}+\frac {752 C}{11583}\right ) x \sqrt {9 x^{3}+3 x^{2}+6 x +2}+\left (\frac {9976 B}{57915}+\frac {140 A}{297}-\frac {43664 C}{521235}\right ) \sqrt {9 x^{3}+3 x^{2}+6 x +2}+\frac {2 \left (\frac {232 A}{693}-\frac {119104 B}{135135}+\frac {21104 C}{173745}\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 {272 A}{63}-\frac {1112 B}{945}-\frac {848 C}{1215}\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}}\)	\(523\)
risch	\(\frac {2 \left (2189187 C \,x^{6}+2525985 B \,x^{5}+898128 x^{5} C +2985255 x^{4} A +1071630 x^{4} B +2260629 C \,x^{4}+1326780 x^{3} A +2815155 B \,x^{3}+1025514 C \,x^{3}+3711825 A \,x^{2}+1377000 B \,x^{2}+166446 C \,x^{2}+2053350 A x +332586 B x +118440 C x +859950 A +314244 B -152824 C \right ) \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{3648645}+\frac {16 \left (656370 A -178893 B -106106 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 )}{1216215 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {464 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 )}{693 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}-\frac {238208 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 )}{135135 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {42208 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 )}{173745 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\)	\(674\)
default	\(\text {Expression too large to display}\)	\(1292\)

Input:

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

Output:

6/5*C*x^6*(9*x^3+3*x^2+6*x+2)^(1/2)+(18/13*B+32/65*C)*x^5*(9*x^3+3*x^2+6*x 
+2)^(1/2)+(84/143*B+18/11*A+886/715*C)*x^4*(9*x^3+3*x^2+6*x+2)^(1/2)+(662/ 
429*B+8/11*A+10852/19305*C)*x^3*(9*x^3+3*x^2+6*x+2)^(1/2)+(6800/9009*B+470 
/231*A+5284/57915*C)*x^2*(9*x^3+3*x^2+6*x+2)^(1/2)+(8212/45045*B+260/231*A 
+752/11583*C)*x*(9*x^3+3*x^2+6*x+2)^(1/2)+(9976/57915*B+140/297*A-43664/52 
1235*C)*(9*x^3+3*x^2+6*x+2)^(1/2)+2*(232/693*A-119104/135135*B+21104/17374 
5*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)*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))+2*(272/63*A-111 
2/945*B-848/1215*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))*El 
lipticE(((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 = 0.07 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.23 \[ \int \left (A+B x+C x^2\right ) \left (2+6 x+3 x^2+9 x^3\right )^{3/2} \, dx=-\frac {16}{6567561} \, {\left (39663 \, A + 205407 \, B - 54460 \, C\right )} {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right ) - \frac {16}{25515} \, {\left (4590 \, A - 1251 \, B - 742 \, C\right )} {\rm weierstrassZeta}\left (-\frac {68}{27}, -\frac {440}{729}, {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right )\right ) + \frac {2}{3648645} \, {\left (2189187 \, C x^{6} + 56133 \, {\left (45 \, B + 16 \, C\right )} x^{5} + 5103 \, {\left (585 \, A + 210 \, B + 443 \, C\right )} x^{4} + 189 \, {\left (7020 \, A + 14895 \, B + 5426 \, C\right )} x^{3} + 9 \, {\left (412425 \, A + 153000 \, B + 18494 \, C\right )} x^{2} + 18 \, {\left (114075 \, A + 18477 \, B + 6580 \, C\right )} x + 859950 \, A + 314244 \, B - 152824 \, C\right )} \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2} \] Input:

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

Output:

-16/6567561*(39663*A + 205407*B - 54460*C)*weierstrassPInverse(-68/27, -44 
0/729, x + 1/9) - 16/25515*(4590*A - 1251*B - 742*C)*weierstrassZeta(-68/2 
7, -440/729, weierstrassPInverse(-68/27, -440/729, x + 1/9)) + 2/3648645*( 
2189187*C*x^6 + 56133*(45*B + 16*C)*x^5 + 5103*(585*A + 210*B + 443*C)*x^4 
 + 189*(7020*A + 14895*B + 5426*C)*x^3 + 9*(412425*A + 153000*B + 18494*C) 
*x^2 + 18*(114075*A + 18477*B + 6580*C)*x + 859950*A + 314244*B - 152824*C 
)*sqrt(9*x^3 + 3*x^2 + 6*x + 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 14.63 (sec) , antiderivative size = 2455, normalized size of antiderivative = 3.86 \[ \int \left (A+B x+C x^2\right ) \left (2+6 x+3 x^2+9 x^3\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

(280*A)/(297*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (19952*B)/(57915*(6*x + 3* 
x^2 + 9*x^3 + 2)^(1/2)) - (87328*C)/(521235*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2 
)) + (320*A*x)/(63*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (17176*B*x)/(12285*( 
6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) - (5888*C*x)/(15795*(6*x + 3*x^2 + 9*x^3 + 
 2)^(1/2)) + (8480*A*x^2)/(693*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (4916*A* 
x^3)/(231*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (1838*A*x^4)/(77*(6*x + 3*x^2 
 + 9*x^3 + 2)^(1/2)) + (2334*A*x^5)/(77*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + 
 (126*A*x^6)/(11*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (162*A*x^7)/(11*(6*x + 
 3*x^2 + 9*x^3 + 2)^(1/2)) + (421648*B*x^2)/(135135*(6*x + 3*x^2 + 9*x^3 + 
 2)^(1/2)) + (437488*B*x^3)/(45045*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (215 
296*B*x^4)/(15015*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (17734*B*x^5)/(1001*( 
6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (3426*B*x^6)/(143*(6*x + 3*x^2 + 9*x^3 + 
 2)^(1/2)) + (1350*B*x^7)/(143*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (162*B*x 
^8)/(13*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (11144*C*x^2)/(34749*(6*x + 3*x 
^2 + 9*x^3 + 2)^(1/2)) + (64432*C*x^3)/(57915*(6*x + 3*x^2 + 9*x^3 + 2)^(1 
/2)) + (25904*C*x^4)/(3861*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (70316*C*x^5 
)/(6435*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (6062*C*x^6)/(429*(6*x + 3*x^2 
+ 9*x^3 + 2)^(1/2)) + (14178*C*x^7)/(715*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) 
+ (522*C*x^8)/(65*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (54*C*x^9)/(5*(6*x + 
3*x^2 + 9*x^3 + 2)^(1/2)) - (544*A*ellipticE(asin((x/((2^(1/2)*3^(1/2)*...
 

Reduce [F]

\[ \int \left (A+B x+C x^2\right ) \left (2+6 x+3 x^2+9 x^3\right )^{3/2} \, dx=\frac {18 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, a \,x^{4}}{11}+\frac {8 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, a \,x^{3}}{11}+\frac {470 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, a \,x^{2}}{231}+\frac {260 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, a x}{231}+\frac {1324 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, a}{693}+\frac {18 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, b \,x^{5}}{13}+\frac {84 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, b \,x^{4}}{143}+\frac {662 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, b \,x^{3}}{429}+\frac {6800 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, b \,x^{2}}{9009}+\frac {8212 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, b x}{45045}-\frac {29728 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, b}{135135}+\frac {6 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c \,x^{6}}{5}+\frac {32 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c \,x^{5}}{65}+\frac {886 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c \,x^{4}}{715}+\frac {10852 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c \,x^{3}}{19305}+\frac {5284 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c \,x^{2}}{57915}+\frac {752 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c x}{11583}-\frac {54976 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c}{173745}-\frac {920 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) a}{231}+\frac {13304 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) b}{45045}+\frac {47456 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) c}{57915}-\frac {136 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}\, x^{2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) a}{7}+\frac {556 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}\, x^{2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) b}{105}+\frac {424 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}\, x^{2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) c}{135} \] Input:

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

Output:

(2*(995085*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*a*x**4 + 442260*sqrt(9*x**3 + 3 
*x**2 + 6*x + 2)*a*x**3 + 1237275*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*a*x**2 + 
 684450*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*a*x + 1161810*sqrt(9*x**3 + 3*x**2 
 + 6*x + 2)*a + 841995*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*b*x**5 + 357210*sqr 
t(9*x**3 + 3*x**2 + 6*x + 2)*b*x**4 + 938385*sqrt(9*x**3 + 3*x**2 + 6*x + 
2)*b*x**3 + 459000*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*b*x**2 + 110862*sqrt(9* 
x**3 + 3*x**2 + 6*x + 2)*b*x - 133776*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*b + 
729729*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*c*x**6 + 299376*sqrt(9*x**3 + 3*x** 
2 + 6*x + 2)*c*x**5 + 753543*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*c*x**4 + 3418 
38*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*c*x**3 + 55482*sqrt(9*x**3 + 3*x**2 + 6 
*x + 2)*c*x**2 + 39480*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*c*x - 192416*sqrt(9 
*x**3 + 3*x**2 + 6*x + 2)*c - 2421900*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/ 
(9*x**3 + 3*x**2 + 6*x + 2),x)*a + 179604*int(sqrt(9*x**3 + 3*x**2 + 6*x + 
 2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*b + 498288*int(sqrt(9*x**3 + 3*x**2 + 6 
*x + 2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*c - 11814660*int((sqrt(9*x**3 + 3*x 
**2 + 6*x + 2)*x**2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*a + 3220074*int((sqrt( 
9*x**3 + 3*x**2 + 6*x + 2)*x**2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*b + 190990 
8*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**2)/(9*x**3 + 3*x**2 + 6*x + 2),x 
)*c))/1216215