Integrand size = 30, antiderivative size = 438 \[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{3/2}} \, dx=\frac {(1+3 x) (2 (3 A-B-2 C)+(3 A+6 B-2 C) x) \left (2+3 x^2\right )}{42 \left (2+6 x+3 x^2+9 x^3\right )^{3/2}}-\frac {(17 A-8 B-2 C) (1+3 x) \left (2+3 x^2\right )^2}{98 \left (2+6 x+3 x^2+9 x^3\right )^{3/2}}+\frac {(17 A-8 B-2 C) (1+3 x)^2 \left (2+3 x^2\right )^2}{98 \left (1+\sqrt {7}+3 x\right ) \left (2+6 x+3 x^2+9 x^3\right )^{3/2}}-\frac {(17 A-8 B-2 C) (1+3 x)^{3/2} \left (1+\sqrt {7}+3 x\right ) \left (2+3 x^2\right ) \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{14 \sqrt {3} 7^{3/4} \left (2+6 x+3 x^2+9 x^3\right )^{3/2}}+\frac {\left (3 \left (17+\sqrt {7}\right ) A-6 \left (4-\sqrt {7}\right ) B-2 \left (3+\sqrt {7}\right ) C\right ) (1+3 x)^{3/2} \left (1+\sqrt {7}+3 x\right ) \left (2+3 x^2\right ) \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{84 \sqrt {3} 7^{3/4} \left (2+6 x+3 x^2+9 x^3\right )^{3/2}} \] Output:
1/42*(1+3*x)*(6*A-2*B-4*C+(3*A+6*B-2*C)*x)*(3*x^2+2)/(9*x^3+3*x^2+6*x+2)^( 3/2)-1/98*(17*A-8*B-2*C)*(1+3*x)*(3*x^2+2)^2/(9*x^3+3*x^2+6*x+2)^(3/2)+1/9 8*(17*A-8*B-2*C)*(1+3*x)^2*(3*x^2+2)^2/(1+7^(1/2)+3*x)/(9*x^3+3*x^2+6*x+2) ^(3/2)-1/294*(17*A-8*B-2*C)*(1+3*x)^(3/2)*(1+7^(1/2)+3*x)*(3*x^2+2)*((3*x^ 2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*EllipticE(sin(2*arctan(1/7*(1+3*x)^(1/2)*7^( 3/4))),1/14*(98+14*7^(1/2))^(1/2))*3^(1/2)*7^(1/4)/(9*x^3+3*x^2+6*x+2)^(3/ 2)+1/1764*(3*(17+7^(1/2))*A-6*(4-7^(1/2))*B-2*(3+7^(1/2))*C)*(1+3*x)^(3/2) *(1+7^(1/2)+3*x)*(3*x^2+2)*((3*x^2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*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)/(9*x^3+3*x^2+6*x+2)^(3/2)
Result contains complex when optimal does not.
Time = 10.53 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{3/2}} \, dx=\frac {\sqrt {6+9 x^2} \left (2 C \left (-8-7 x+9 x^2\right )+2 B \left (17+21 x+36 x^2\right )-3 A \left (20-7 x+51 x^2\right )\right )+\frac {3 i (17 A-2 (4 B+C)) (1+3 x) \left (2+3 x^2\right ) E\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )}{\sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}}}+7 i (3 A+6 B-2 C) \sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}} \left (2+3 x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right ),\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )}{294 \sqrt {6+9 x^2} \sqrt {2+6 x+3 x^2+9 x^3}} \] Input:
Integrate[(A + B*x + C*x^2)/(2 + 6*x + 3*x^2 + 9*x^3)^(3/2),x]
Output:
(Sqrt[6 + 9*x^2]*(2*C*(-8 - 7*x + 9*x^2) + 2*B*(17 + 21*x + 36*x^2) - 3*A* (20 - 7*x + 51*x^2)) + ((3*I)*(17*A - 2*(4*B + C))*(1 + 3*x)*(2 + 3*x^2)*E llipticE[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])] + (7*I)*(3*A + 6*B - 2*C)* Sqrt[(I*(1 + 3*x))/(I + Sqrt[6])]*(2 + 3*x^2)*EllipticF[ArcSin[Sqrt[Sqrt[6 ] - (3*I)*x]/(2^(3/4)*3^(1/4))], (2*Sqrt[6])/(I + Sqrt[6])])/(294*Sqrt[6 + 9*x^2]*Sqrt[2 + 6*x + 3*x^2 + 9*x^3])
Result contains complex when optimal does not.
Time = 5.02 (sec) , antiderivative size = 1435, normalized size of antiderivative = 3.28, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {2526, 27, 2490, 2486, 27, 1235, 27, 1237, 27, 1269, 1172, 321, 327}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (9 x^3+3 x^2+6 x+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{27} \int \frac {3 (9 A-2 C+(9 B-2 C) x)}{\left (9 x^3+3 x^2+6 x+2\right )^{3/2}}dx-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {9 A-2 C+(9 B-2 C) x}{\left (9 x^3+3 x^2+6 x+2\right )^{3/2}}dx-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {1}{9} \int \frac {\frac {1}{27} (27 (9 A-2 C)-3 (9 B-2 C))+(9 B-2 C) \left (x+\frac {1}{9}\right )}{\left (9 \left (x+\frac {1}{9}\right )^3+\frac {17}{3} \left (x+\frac {1}{9}\right )+\frac {110}{81}\right )^{3/2}}d\left (x+\frac {1}{9}\right )-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 2486 |
| \(\displaystyle \frac {81 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17\right )^{3/2} \int \frac {81 A-9 B-16 C+9 (9 B-2 C) \left (x+\frac {1}{9}\right )}{9 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}d\left (x+\frac {1}{9}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2}}-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {9 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17\right )^{3/2} \int \frac {81 A-9 B-16 C+9 (9 B-2 C) \left (x+\frac {1}{9}\right )}{\left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}}d\left (x+\frac {1}{9}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2}}-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {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 \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} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{21\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \int \frac {177147 \left (\frac {9 \left (68 \left (55-63 \sqrt {2}\right )-289 \sqrt [3]{-55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{5/3}\right ) A-9 \left (32 \left (55-63 \sqrt {2}\right )-17 \left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-\left (26+7 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (36 \left (55-63 \sqrt {2}\right )-17 \left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+\left (81-56 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{55-63 \sqrt {2}}+\frac {9 \left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{2 \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{3720087\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2}}-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {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 \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} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{21\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \int \frac {\frac {9 \left (68 \left (55-63 \sqrt {2}\right )-289 \sqrt [3]{-55+63 \sqrt {2}}-\left (-55+63 \sqrt {2}\right )^{5/3}\right ) A-9 \left (32 \left (55-63 \sqrt {2}\right )-17 \left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-\left (26+7 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (36 \left (55-63 \sqrt {2}\right )-17 \left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}+\left (81-56 \sqrt {2}\right ) \left (-55+63 \sqrt {2}\right )^{2/3}\right ) C}{55-63 \sqrt {2}}+\frac {9 \left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}}{\left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )^{3/2} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{42\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2}}-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {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 \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} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{21\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (-\frac {4 \left (-55+63 \sqrt {2}\right )^{2/3} \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 A-8 B-2 C)}{\left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}-\frac {2 \left (-55+63 \sqrt {2}\right )^{2/3} \int -\frac {2187 \left (55 A+26 B-18 C+9 (17 A-8 B-2 C) \left (x+\frac {1}{9}\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}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{243 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right )}\right )}{42\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2}}-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {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 \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} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{21\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {18 \left (-55+63 \sqrt {2}\right )^{2/3} \int \frac {55 A+26 B-18 C+9 (17 A-8 B-2 C) \left (x+\frac {1}{9}\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}}}} \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 )}{289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}}-\frac {4 \left (-55+63 \sqrt {2}\right )^{2/3} (17 A-8 B-2 C) \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 (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}\right )}{42\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2}}-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {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 \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} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{21\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {18 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\left (55 A+26 B-18 C-\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (17 A-2 (4 B+C))}{\sqrt [3]{-55+63 \sqrt {2}}}\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 )+(17 A-8 B-2 C) \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 )\right )}{289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}}-\frac {4 \left (-55+63 \sqrt {2}\right )^{2/3} (17 A-8 B-2 C) \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 (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}\right )}{42\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2}}-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {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 \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} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{21\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {18 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {2 i \sqrt {2} \sqrt [6]{-55+63 \sqrt {2}} \left (55 A+26 B-18 C-\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (17 A-2 (4 B+C))}{\sqrt [3]{-55+63 \sqrt {2}}}\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 \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} (17 A-8 B-2 C) \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 \sqrt [6]{-55+63 \sqrt {2}} \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}}}\right )}{289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}}-\frac {4 \left (-55+63 \sqrt {2}\right )^{2/3} (17 A-8 B-2 C) \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 (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}\right )}{42\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2}}-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {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 \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} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{21\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {18 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {2 i \sqrt {2} \sqrt [6]{-55+63 \sqrt {2}} \left (55 A+26 B-18 C-\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (17 A-2 (4 B+C))}{\sqrt [3]{-55+63 \sqrt {2}}}\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 \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} (17 A-8 B-2 C) \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 \sqrt [6]{-55+63 \sqrt {2}} \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}}}\right )}{289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}}-\frac {4 \left (-55+63 \sqrt {2}\right )^{2/3} (17 A-8 B-2 C) \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 (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}\right )}{42\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2}}-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {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 \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} \left (\frac {\sqrt [3]{126-55 \sqrt {2}} \left (2 (17 A-8 B-2 C)+\frac {\left (9 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (26+7 \sqrt {2}-\left (8-7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (81-56 \sqrt {2}+\left (9+7 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3}}\right )}{21\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}+\frac {\sqrt [3]{126-55 \sqrt {2}} \left (\frac {18 \left (-55+63 \sqrt {2}\right )^{2/3} \left (\frac {i \sqrt {2} (17 A-8 B-2 C) \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 \sqrt [6]{-55+63 \sqrt {2}} \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}}}+\frac {2 i \sqrt {2} \sqrt [6]{-55+63 \sqrt {2}} \left (55 A+26 B-18 C-\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (17 A-2 (4 B+C))}{\sqrt [3]{-55+63 \sqrt {2}}}\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 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}\right )}{289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}}-\frac {4 \left (-55+63 \sqrt {2}\right )^{2/3} (17 A-8 B-2 C) \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 (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}\right )}{42\ 2^{2/3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}\right )}{\left (729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110\right )^{3/2}}-\frac {2 C}{27 \sqrt {9 x^3+3 x^2+6 x+2}}\) |
Input:
Int[(A + B*x + C*x^2)/(2 + 6*x + 3*x^2 + 9*x^3)^(3/2),x]
Output:
(-2*C)/(27*Sqrt[2 + 6*x + 3*x^2 + 9*x^3]) + (9*((17 - (-55 + 63*Sqrt[2])^( 2/3))/(-55 + 63*Sqrt[2])^(1/3) + 9*(1/9 + x))^(3/2)*(17 + 289/(-55 + 63*Sq rt[2])^(2/3) + (-55 + 63*Sqrt[2])^(2/3) - (9*(17 - (-55 + 63*Sqrt[2])^(2/3 ))*(1/9 + x))/(-55 + 63*Sqrt[2])^(1/3) + 81*(1/9 + x)^2)^(3/2)*(((126 - 55 *Sqrt[2])^(1/3)*(2*(17*A - 8*B - 2*C) + ((9*(55 - 63*Sqrt[2] + 17*(-55 + 6 3*Sqrt[2])^(1/3))*A + 9*(26 + 7*Sqrt[2] - (8 - 7*Sqrt[2])*(-55 + 63*Sqrt[2 ])^(1/3))*B - 2*(81 - 56*Sqrt[2] + (9 + 7*Sqrt[2])*(-55 + 63*Sqrt[2])^(1/3 ))*C)*(1/9 + x))/(-55 + 63*Sqrt[2])^(2/3)))/(21*2^(2/3)*(17 + (-55 + 63*Sq rt[2])^(2/3))*Sqrt[(17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/3 ) + 9*(1/9 + x)]*Sqrt[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]) + ((126 - 55*Sqrt[2])^(1/3)*((-4*(-55 + 63*Sqrt [2])^(2/3)*(17*A - 8*B - 2*C)*Sqrt[17 + 289/(-55 + 63*Sqrt[2])^(2/3) + (-5 5 + 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])/((289 - 17*(-55 + 63*Sqrt[2])^(2/3 ) + (-55 + 63*Sqrt[2])^(4/3))*Sqrt[(17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/3) + 9*(1/9 + x)]) + (18*(-55 + 63*Sqrt[2])^(2/3)*(((I/9)*S qrt[2]*(17*A - 8*B - 2*C)*Sqrt[(17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*S qrt[2])^(1/3) + 9*(1/9 + x)]*EllipticE[ArcSin[((-55 + 63*Sqrt[2])^(1/6)*Sq rt[(-I)*(((-55 + 63*Sqrt[2])^(2/3)*(1 + I*Sqrt[3]) + (17*I)*(I + Sqrt[3...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
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]
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]
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]
Result contains complex when optimal does not.
Time = 0.83 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.86
| method | result | size |
| elliptic | \(-\frac {18 \left (\left (-\frac {2 B}{147}-\frac {C}{294}+\frac {17 A}{588}\right ) x^{2}+\left (-\frac {B}{126}+\frac {C}{378}-\frac {A}{252}\right ) x -\frac {17 B}{2646}+\frac {4 C}{1323}+\frac {5 A}{441}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {2 \left (\frac {3 B}{98}-\frac {5 C}{147}+\frac {6 A}{49}\right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {2 \left (-\frac {6 B}{49}-\frac {3 C}{98}+\frac {51 A}{196}\right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(377\) |
| risch | \(-\frac {153 A \,x^{2}-72 B \,x^{2}-18 C \,x^{2}-21 A x -42 B x +14 C x +60 A -34 B +16 C}{294 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {\left (153 A -72 B -18 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 )}{294 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {12 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 )}{49 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {3 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 )}{49 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}-\frac {10 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 )}{147 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(620\) |
| default | \(\text {Expression too large to display}\) | \(998\) |
Input:
int((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-18*((-2/147*B-1/294*C+17/588*A)*x^2+(-1/126*B+1/378*C-1/252*A)*x-17/2646* B+4/1323*C+5/441*A)/(9*x^3+3*x^2+6*x+2)^(1/2)+2*(3/98*B-5/147*C+6/49*A)*(- 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*(-6/49*B-3/98*C+51/1 96*A)*(-1/3*I*6^(1/2)+1/3)*((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2)*((x-1/3*I* 6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2)*((x+1/3*I*6^(1/2))/(-1/3+1/3*I*6^(1/2 )))^(1/2)/(9*x^3+3*x^2+6*x+2)^(1/2)*((-1/3-1/3*I*6^(1/2))*EllipticE(((x+1/ 3)/(-1/3*I*6^(1/2)+1/3))^(1/2),((-1/3+1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2))) ^(1/2))+1/3*I*6^(1/2)*EllipticF(((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2),((-1/ 3+1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2)))
Time = 0.07 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.46 \[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{3/2}} \, dx=\frac {{\left (9 \, {\left (55 \, A + 26 \, B - 18 \, C\right )} x^{3} + 3 \, {\left (55 \, A + 26 \, B - 18 \, C\right )} x^{2} + 6 \, {\left (55 \, A + 26 \, B - 18 \, C\right )} x + 110 \, A + 52 \, B - 36 \, C\right )} {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right ) - 9 \, {\left (9 \, {\left (17 \, A - 8 \, B - 2 \, C\right )} x^{3} + 3 \, {\left (17 \, A - 8 \, B - 2 \, C\right )} x^{2} + 6 \, {\left (17 \, A - 8 \, B - 2 \, C\right )} x + 34 \, A - 16 \, B - 4 \, C\right )} {\rm weierstrassZeta}\left (-\frac {68}{27}, -\frac {440}{729}, {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right )\right ) - 3 \, {\left (9 \, {\left (17 \, A - 8 \, B - 2 \, C\right )} x^{2} - 7 \, {\left (3 \, A + 6 \, B - 2 \, C\right )} x + 60 \, A - 34 \, B + 16 \, C\right )} \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2}}{882 \, {\left (9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2\right )}} \] Input:
integrate((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(3/2),x, algorithm="fricas")
Output:
1/882*((9*(55*A + 26*B - 18*C)*x^3 + 3*(55*A + 26*B - 18*C)*x^2 + 6*(55*A + 26*B - 18*C)*x + 110*A + 52*B - 36*C)*weierstrassPInverse(-68/27, -440/7 29, x + 1/9) - 9*(9*(17*A - 8*B - 2*C)*x^3 + 3*(17*A - 8*B - 2*C)*x^2 + 6* (17*A - 8*B - 2*C)*x + 34*A - 16*B - 4*C)*weierstrassZeta(-68/27, -440/729 , weierstrassPInverse(-68/27, -440/729, x + 1/9)) - 3*(9*(17*A - 8*B - 2*C )*x^2 - 7*(3*A + 6*B - 2*C)*x + 60*A - 34*B + 16*C)*sqrt(9*x^3 + 3*x^2 + 6 *x + 2))/(9*x^3 + 3*x^2 + 6*x + 2)
\[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{3/2}} \, dx=\int \frac {A + B x + C x^{2}}{\left (\left (3 x + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((C*x**2+B*x+A)/(9*x**3+3*x**2+6*x+2)**(3/2),x)
Output:
Integral((A + B*x + C*x**2)/((3*x + 1)*(3*x**2 + 2))**(3/2), x)
\[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{3/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \] 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)
\[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{3/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \] 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)
Time = 12.90 (sec) , antiderivative size = 692, normalized size of antiderivative = 1.58 \[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
int((A + B*x + C*x^2)/(6*x + 3*x^2 + 9*x^3 + 2)^(3/2),x)
Output:
(2*((x + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)*((x + (2^(1/2)*3^(1/2) *1i)/3)/((2^(1/2)*3^(1/2)*1i)/3 - 1/3))^(1/2)*(-(x - (2^(1/2)*3^(1/2)*1i)/ 3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)*((2^(1/2)*3^(1/2)*1i)/3 + 1/3)*el lipticPi(((2^(1/2)*3^(1/2)*1i)/3 + 1/3)/((6^(1/2)*1i)/3 + 1/3), asin(((x + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)), -((2^(1/2)*3^(1/2)*1i)/3 + 1 /3)/((2^(1/2)*3^(1/2)*1i)/3 - 1/3))*(A/14 - B/42 - C/21 + (6^(1/2)*B*1i)/4 2 - (6^(1/2)*C*1i)/126 + (6^(1/2)*A*1i)/84))/(((6^(1/2)*1i)/3 + 1/3)*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (2*((x + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3) )^(1/2)*((x + (2^(1/2)*3^(1/2)*1i)/3)/((2^(1/2)*3^(1/2)*1i)/3 - 1/3))^(1/2 )*(-(x - (2^(1/2)*3^(1/2)*1i)/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)*((2 ^(1/2)*3^(1/2)*1i)/3 + 1/3)*ellipticPi(-((2^(1/2)*3^(1/2)*1i)/3 + 1/3)/((6 ^(1/2)*1i)/3 - 1/3), asin(((x + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2) ), -((2^(1/2)*3^(1/2)*1i)/3 + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 - 1/3))*(B/42 - A/14 + C/21 + (6^(1/2)*B*1i)/42 - (6^(1/2)*C*1i)/126 + (6^(1/2)*A*1i)/84) )/(((6^(1/2)*1i)/3 - 1/3)*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2)) + (2^(1/2)*3^(1 /2)*((x + 1/3)/((2^(1/2)*3^(1/2)*1i)/3 + 1/3))^(1/2)*(ellipticE(asin(((2^( 1/2)*3^(1/2)*(x - (2^(1/2)*3^(1/2)*1i)/3)*1i)/4)^(1/2)), (2^(1/2)*3^(1/2)* 2i)/(3*((2^(1/2)*3^(1/2)*1i)/3 + 1/3))) - (2^(1/2)*3^(1/2)*sin(2*asin(((2^ (1/2)*3^(1/2)*(x - (2^(1/2)*3^(1/2)*1i)/3)*1i)/4)^(1/2)))*1i)/(3*((2^(1/2) *3^(1/2)*1i)/3 + 1/3)*((x - (2^(1/2)*3^(1/2)*1i)/3)/((2^(1/2)*3^(1/2)*1...
\[ \int \frac {A+B x+C x^2}{\left (2+6 x+3 x^2+9 x^3\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(3/2),x)
Output:
(6*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*a*x - 6*sqrt(9*x**3 + 3*x**2 + 6*x + 2) *b*x - 4*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*b + 243*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**3)/(81*x**6 + 54*x**5 + 117*x**4 + 72*x**3 + 48*x**2 + 24*x + 4),x)*a*x**3 + 81*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**3)/(81*x**6 + 54*x**5 + 117*x**4 + 72*x**3 + 48*x**2 + 24*x + 4),x)*a*x**2 + 162*int((s qrt(9*x**3 + 3*x**2 + 6*x + 2)*x**3)/(81*x**6 + 54*x**5 + 117*x**4 + 72*x* *3 + 48*x**2 + 24*x + 4),x)*a*x + 54*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)* x**3)/(81*x**6 + 54*x**5 + 117*x**4 + 72*x**3 + 48*x**2 + 24*x + 4),x)*a - 243*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**3)/(81*x**6 + 54*x**5 + 117*x **4 + 72*x**3 + 48*x**2 + 24*x + 4),x)*b*x**3 - 81*int((sqrt(9*x**3 + 3*x* *2 + 6*x + 2)*x**3)/(81*x**6 + 54*x**5 + 117*x**4 + 72*x**3 + 48*x**2 + 24 *x + 4),x)*b*x**2 - 162*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**3)/(81*x** 6 + 54*x**5 + 117*x**4 + 72*x**3 + 48*x**2 + 24*x + 4),x)*b*x - 54*int((sq rt(9*x**3 + 3*x**2 + 6*x + 2)*x**3)/(81*x**6 + 54*x**5 + 117*x**4 + 72*x** 3 + 48*x**2 + 24*x + 4),x)*b - 486*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x* *2)/(81*x**6 + 54*x**5 + 117*x**4 + 72*x**3 + 48*x**2 + 24*x + 4),x)*b*x** 3 - 162*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**2)/(81*x**6 + 54*x**5 + 11 7*x**4 + 72*x**3 + 48*x**2 + 24*x + 4),x)*b*x**2 - 324*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**2)/(81*x**6 + 54*x**5 + 117*x**4 + 72*x**3 + 48*x**2 + 24*x + 4),x)*b*x - 108*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**2)/(81...