Integrand size = 30, antiderivative size = 336 \[ \int \frac {A+B x+C x^2}{\sqrt {2+6 x+3 x^2+9 x^3}} \, dx=\frac {2 C (1+3 x) \left (2+3 x^2\right )}{27 \sqrt {2+6 x+3 x^2+9 x^3}}+\frac {2 (9 B-2 C) (1+3 x) \left (2+3 x^2\right )}{27 \left (1+\sqrt {7}+3 x\right ) \sqrt {2+6 x+3 x^2+9 x^3}}-\frac {2 \sqrt [4]{7} (9 B-2 C) \sqrt {1+3 x} \left (1+\sqrt {7}+3 x\right ) \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{27 \sqrt {3} \sqrt {2+6 x+3 x^2+9 x^3}}+\frac {\left (27 A-9 \left (1-\sqrt {7}\right ) B-2 \left (2+\sqrt {7}\right ) C\right ) \sqrt {1+3 x} \left (1+\sqrt {7}+3 x\right ) \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{27 \sqrt {3} \sqrt [4]{7} \sqrt {2+6 x+3 x^2+9 x^3}} \] Output:
2/27*C*(1+3*x)*(3*x^2+2)/(9*x^3+3*x^2+6*x+2)^(1/2)+2/27*(9*B-2*C)*(1+3*x)* (3*x^2+2)/(1+7^(1/2)+3*x)/(9*x^3+3*x^2+6*x+2)^(1/2)-2/81*7^(1/4)*(9*B-2*C) *(1+3*x)^(1/2)*(1+7^(1/2)+3*x)*((3*x^2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*Ellipti cE(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)/(9*x^3+3*x^2+6*x+2)^(1/2)+1/567*(27*A-9*(1-7^(1/2))*B-2*(2+7^(1/2))* C)*(1+3*x)^(1/2)*(1+7^(1/2)+3*x)*((3*x^2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*Inver seJacobiAM(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^(3/4)/(9*x^3+3*x^2+6*x+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.33 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x+C x^2}{\sqrt {2+6 x+3 x^2+9 x^3}} \, dx=\frac {2 \left (2+3 x^2\right ) \left (C (1+3 x) \sqrt {6+9 x^2}+\left (i+\sqrt {6}\right ) (9 B-2 C) \sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}} E\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )+i (27 A-9 B-4 C) \sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right ),\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )\right )}{27 \sqrt {6+9 x^2} \sqrt {2+6 x+3 x^2+9 x^3}} \] Input:
Integrate[(A + B*x + C*x^2)/Sqrt[2 + 6*x + 3*x^2 + 9*x^3],x]
Output:
(2*(2 + 3*x^2)*(C*(1 + 3*x)*Sqrt[6 + 9*x^2] + (I + Sqrt[6])*(9*B - 2*C)*Sq rt[(I*(1 + 3*x))/(I + Sqrt[6])]*EllipticE[ArcSin[Sqrt[Sqrt[6] - (3*I)*x]/( 2^(3/4)*3^(1/4))], (2*Sqrt[6])/(I + Sqrt[6])] + I*(27*A - 9*B - 4*C)*Sqrt[ (I*(1 + 3*x))/(I + Sqrt[6])]*EllipticF[ArcSin[Sqrt[Sqrt[6] - (3*I)*x]/(2^( 3/4)*3^(1/4))], (2*Sqrt[6])/(I + Sqrt[6])]))/(27*Sqrt[6 + 9*x^2]*Sqrt[2 + 6*x + 3*x^2 + 9*x^3])
Result contains complex when optimal does not.
Time = 2.01 (sec) , antiderivative size = 869, normalized size of antiderivative = 2.59, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2526, 27, 2490, 2486, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {9 x^3+3 x^2+6 x+2}} \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{27} \int \frac {3 (9 A-2 C+(9 B-2 C) x)}{\sqrt {9 x^3+3 x^2+6 x+2}}dx+\frac {2}{27} C \sqrt {9 x^3+3 x^2+6 x+2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {9 A-2 C+(9 B-2 C) x}{\sqrt {9 x^3+3 x^2+6 x+2}}dx+\frac {2}{27} C \sqrt {9 x^3+3 x^2+6 x+2}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {1}{9} \int \frac {\frac {1}{27} (27 (9 A-2 C)-3 (9 B-2 C))+(9 B-2 C) \left (x+\frac {1}{9}\right )}{\sqrt {9 \left (x+\frac {1}{9}\right )^3+\frac {17}{3} \left (x+\frac {1}{9}\right )+\frac {110}{81}}}d\left (x+\frac {1}{9}\right )+\frac {2}{27} C \sqrt {9 x^3+3 x^2+6 x+2}\) |
| \(\Big \downarrow \) 2486 |
| \(\displaystyle \frac {\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17} \int \frac {81 A-9 B-16 C+9 (9 B-2 C) \left (x+\frac {1}{9}\right )}{9 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110}}+\frac {2}{27} C \sqrt {9 x^3+3 x^2+6 x+2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17} \int \frac {81 A-9 B-16 C+9 (9 B-2 C) \left (x+\frac {1}{9}\right )}{\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{9 \sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110}}+\frac {2}{27} C \sqrt {9 x^3+3 x^2+6 x+2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17} \left (\left (81 A-\frac {\left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) (9 B-2 C)}{\sqrt [3]{63 \sqrt {2}-55}}-9 B-16 C\right ) \int \frac {1}{\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )+(9 B-2 C) \int \frac {\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}{\sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )\right )}{9 \sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110}}+\frac {2}{27} C \sqrt {9 x^3+3 x^2+6 x+2}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2}{27} \sqrt {9 x^3+3 x^2+6 x+2} C+\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} \left (\frac {2 i \sqrt {2} \sqrt [6]{-55+63 \sqrt {2}} \left (81 A-9 B-\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (9 B-2 C)}{\sqrt [3]{-55+63 \sqrt {2}}}-16 C\right ) \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}} \int \frac {1}{\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{2 \sqrt {3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}+1} \sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\sqrt {3} \left (17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )\right )}+1}}d\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}}{9 \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} (9 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 )}{9 \sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{27} \sqrt {9 x^3+3 x^2+6 x+2} C+\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} \left (\frac {2 i \sqrt {2} \sqrt [6]{-55+63 \sqrt {2}} \left (81 A-9 B-\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (9 B-2 C)}{\sqrt [3]{-55+63 \sqrt {2}}}-16 C\right ) \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}\right ),\frac {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}{17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )}\right )}{9 \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} (9 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 )}{9 \sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{27} \sqrt {9 x^3+3 x^2+6 x+2} C+\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} \left (\frac {i \sqrt {2} (9 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 (81 A-9 B-\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (9 B-2 C)}{\sqrt [3]{-55+63 \sqrt {2}}}-16 C\right ) \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}\right ),\frac {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}{17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )}\right )}{9 \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 )}{9 \sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110}}\) |
Input:
Int[(A + B*x + C*x^2)/Sqrt[2 + 6*x + 3*x^2 + 9*x^3],x]
Output:
(2*C*Sqrt[2 + 6*x + 3*x^2 + 9*x^3])/27 + (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]*(((I/9)*Sqrt[2]*(9*B - 2*C)*Sqrt[(17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/3) + 9* (1/9 + x)]*EllipticE[ArcSin[((-55 + 63*Sqrt[2])^(1/6)*Sqrt[(-I)*(((-55 + 6 3*Sqrt[2])^(2/3)*(1 + I*Sqrt[3]) + (17*I)*(I + Sqrt[3]))/(-55 + 63*Sqrt[2] )^(1/3) + 18*(1/9 + x))])/(3^(1/4)*Sqrt[2*(17 + (-55 + 63*Sqrt[2])^(2/3))] )], (2*(17 + (-55 + 63*Sqrt[2])^(2/3)))/(17 + (17*I)*Sqrt[3] + (-55 + 63*S qrt[2])^(2/3)*(1 - I*Sqrt[3]))])/((-55 + 63*Sqrt[2])^(1/6)*Sqrt[((-I)*((17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/3) + 9*(1/9 + x)))/((-5 5 + 63*Sqrt[2])^(2/3)*(3*I - Sqrt[3]) - 17*(3*I + Sqrt[3]))]) + (((2*I)/9) *Sqrt[2]*(-55 + 63*Sqrt[2])^(1/6)*(81*A - 9*B - ((17 - (-55 + 63*Sqrt[2])^ (2/3))*(9*B - 2*C))/(-55 + 63*Sqrt[2])^(1/3) - 16*C)*Sqrt[((-I)*((17 - (-5 5 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/3) + 9*(1/9 + x)))/((-55 + 63 *Sqrt[2])^(2/3)*(3*I - Sqrt[3]) - 17*(3*I + Sqrt[3]))]*EllipticF[ArcSin[(( -55 + 63*Sqrt[2])^(1/6)*Sqrt[(-I)*(((-55 + 63*Sqrt[2])^(2/3)*(1 + I*Sqrt[3 ]) + (17*I)*(I + Sqrt[3]))/(-55 + 63*Sqrt[2])^(1/3) + 18*(1/9 + x))])/(3^( 1/4)*Sqrt[2*(17 + (-55 + 63*Sqrt[2])^(2/3))])], (2*(17 + (-55 + 63*Sqrt[2] )^(2/3)))/(17 + (17*I)*Sqrt[3] + (-55 + 63*Sqrt[2])^(2/3)*(1 - 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[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 = 0.73 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.99
| method | result | size |
| elliptic | \(\frac {2 C \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{27}+\frac {2 \left (A -\frac {2 C}{9}\right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {2 \left (B -\frac {2 C}{9}\right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(332\) |
| risch | \(\frac {2 C \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{27}+\frac {2 \left (9 B -2 C \right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{9 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {2 A \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}-\frac {4 C \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{9 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(454\) |
| default | \(\frac {2 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 )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {2 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}}}\, \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}}+C \left (\frac {2 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{27}-\frac {4 \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 )}{9 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}-\frac {4 \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 )}{9 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\right )\) | \(626\) |
Input:
int((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/27*C*(9*x^3+3*x^2+6*x+2)^(1/2)+2*(A-2/9*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*(B-2/9*C)*(-1/3*I*6^(1/2)+1/3)*((x+1/3)/(-1/3*I *6^(1/2)+1/3))^(1/2)*((x-1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2)*((x+1/ 3*I*6^(1/2))/(-1/3+1/3*I*6^(1/2)))^(1/2)/(9*x^3+3*x^2+6*x+2)^(1/2)*((-1/3- 1/3*I*6^(1/2))*EllipticE(((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2),((-1/3+1/3*I *6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2))+1/3*I*6^(1/2)*EllipticF(((x+1/3)/(- 1/3*I*6^(1/2)+1/3))^(1/2),((-1/3+1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2 )))
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.17 \[ \int \frac {A+B x+C x^2}{\sqrt {2+6 x+3 x^2+9 x^3}} \, dx=\frac {2}{243} \, {\left (81 \, A - 9 \, B - 16 \, C\right )} {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right ) - \frac {2}{27} \, {\left (9 \, B - 2 \, C\right )} {\rm weierstrassZeta}\left (-\frac {68}{27}, -\frac {440}{729}, {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right )\right ) + \frac {2}{27} \, \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2} C \] Input:
integrate((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(1/2),x, algorithm="fricas")
Output:
2/243*(81*A - 9*B - 16*C)*weierstrassPInverse(-68/27, -440/729, x + 1/9) - 2/27*(9*B - 2*C)*weierstrassZeta(-68/27, -440/729, weierstrassPInverse(-6 8/27, -440/729, x + 1/9)) + 2/27*sqrt(9*x^3 + 3*x^2 + 6*x + 2)*C
\[ \int \frac {A+B x+C x^2}{\sqrt {2+6 x+3 x^2+9 x^3}} \, dx=\int \frac {A + B x + C x^{2}}{\sqrt {\left (3 x + 1\right ) \left (3 x^{2} + 2\right )}}\, dx \] Input:
integrate((C*x**2+B*x+A)/(9*x**3+3*x**2+6*x+2)**(1/2),x)
Output:
Integral((A + B*x + C*x**2)/sqrt((3*x + 1)*(3*x**2 + 2)), x)
\[ \int \frac {A+B x+C x^2}{\sqrt {2+6 x+3 x^2+9 x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(1/2),x, algorithm="maxima")
Output:
integrate((C*x^2 + B*x + A)/sqrt(9*x^3 + 3*x^2 + 6*x + 2), x)
\[ \int \frac {A+B x+C x^2}{\sqrt {2+6 x+3 x^2+9 x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(1/2),x, algorithm="giac")
Output:
integrate((C*x^2 + B*x + A)/sqrt(9*x^3 + 3*x^2 + 6*x + 2), x)
Time = 12.76 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.95 \[ \int \frac {A+B x+C x^2}{\sqrt {2+6 x+3 x^2+9 x^3}} \, dx=\text {Too large to display} \] Input:
int((A + B*x + C*x^2)/(6*x + 3*x^2 + 9*x^3 + 2)^(1/2),x)
Output:
(2*A*((6^(1/2)*1i)/3 + 1/3)*((x + (6^(1/2)*1i)/3)/((6^(1/2)*1i)/3 - 1/3))^ (1/2)*(-(x - (6^(1/2)*1i)/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)*((x + 1/3)/((6^ (1/2)*1i)/3 + 1/3))^(1/2)*ellipticF(asin(((x + 1/3)/((6^(1/2)*1i)/3 + 1/3) )^(1/2)), -((6^(1/2)*1i)/3 + 1/3)/((6^(1/2)*1i)/3 - 1/3)))/(6*x + 3*x^2 + 9*x^3 + 2)^(1/2) - (C*((4*((6^(1/2)*1i)/3 + 1/3)*(((6^(1/2)*1i)/3 - 1/3)*e llipticE(asin(((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)), -((6^(1/2)*1i)/3 + 1/3)/((6^(1/2)*1i)/3 - 1/3)) - (6^(1/2)*ellipticF(asin(((x + 1/3)/((6^(1 /2)*1i)/3 + 1/3))^(1/2)), -((6^(1/2)*1i)/3 + 1/3)/((6^(1/2)*1i)/3 - 1/3))* 1i)/3)*((x + (6^(1/2)*1i)/3)/((6^(1/2)*1i)/3 - 1/3))^(1/2)*(-(x - (6^(1/2) *1i)/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)*((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^( 1/2))/(9*((2*x)/3 + x^2/3 + x^3 + 2/9)^(1/2)) - (2*((2*x)/3 + x^2/3 + x^3 + 2/9)^(1/2))/3 + (4*((6^(1/2)*1i)/3 + 1/3)*((x + (6^(1/2)*1i)/3)/((6^(1/2 )*1i)/3 - 1/3))^(1/2)*(-(x - (6^(1/2)*1i)/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2) *((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)*ellipticF(asin(((x + 1/3)/((6^(1 /2)*1i)/3 + 1/3))^(1/2)), -((6^(1/2)*1i)/3 + 1/3)/((6^(1/2)*1i)/3 - 1/3))) /(9*((2*x)/3 + x^2/3 + x^3 + 2/9)^(1/2)))*((2*x)/3 + x^2/3 + x^3 + 2/9)^(1 /2))/(6*x + 3*x^2 + 9*x^3 + 2)^(1/2) + (2*B*((6^(1/2)*1i)/3 + 1/3)*(((6^(1 /2)*1i)/3 - 1/3)*ellipticE(asin(((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)), -((6^(1/2)*1i)/3 + 1/3)/((6^(1/2)*1i)/3 - 1/3)) - (6^(1/2)*ellipticF(asin (((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)), -((6^(1/2)*1i)/3 + 1/3)/((6...
\[ \int \frac {A+B x+C x^2}{\sqrt {2+6 x+3 x^2+9 x^3}} \, dx=\frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}\, b}{3}+\left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) a -\left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) b -\frac {9 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}\, x^{2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) b}{2}+\left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}\, x^{2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) c \] Input:
int((C*x^2+B*x+A)/(9*x^3+3*x^2+6*x+2)^(1/2),x)
Output:
(2*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*b + 6*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*a - 6*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2 )/(9*x**3 + 3*x**2 + 6*x + 2),x)*b - 27*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*b + 6*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*c)/6