Integrand size = 25, antiderivative size = 977 \[ \int \left (A+B x+C x^2\right ) \left (2-4 x^2+3 x^3\right )^p \, dx =\text {Too large to display} \] Output:
C*(3*x^3-4*x^2+2)^(p+1)/(9*p+9)-1/729*(81*A+36*B+32*C-(16+(179-9*345^(1/2) )^(2/3))*(9*B+8*C)/(179-9*345^(1/2))^(1/3))*(4-16/(179-9*345^(1/2))^(1/3)- (179-9*345^(1/2))^(1/3)-9*x)*(3*x^3-4*x^2+2)^p*AppellF1(p+1,-p,-p,2+p,-2*I *(179-9*345^(1/2))^(1/3)*(4-16/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/2))^( 1/3)-9*x)/(48*I-16*3^(1/2)+(3*I+3^(1/2))*(179-9*345^(1/2))^(2/3)),-2*I*(17 9-9*345^(1/2))^(1/3)*(4-16/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/2))^(1/3) -9*x)/(48*I+16*3^(1/2)+(3*I-3^(1/2))*(179-9*345^(1/2))^(2/3)))/(p+1)/((1+2 *I*(179-9*345^(1/2))^(1/3)*(4-16/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/2)) ^(1/3)-9*x)/(48*I+16*3^(1/2)+(3*I-3^(1/2))*(179-9*345^(1/2))^(2/3)))^p)/(( 1+2*I*(179-9*345^(1/2))^(1/3)*(4-16/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/ 2))^(1/3)-9*x)/(48*I-16*3^(1/2)+(3*I+3^(1/2))*(179-9*345^(1/2))^(2/3)))^p) +1/729*(9*B+8*C)*(4-16/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/2))^(1/3)-9*x )^2*(3*x^3-4*x^2+2)^p*AppellF1(2+p,-p,-p,3+p,-2*I*(179-9*345^(1/2))^(1/3)* (4-16/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/2))^(1/3)-9*x)/(48*I-16*3^(1/2 )+(3*I+3^(1/2))*(179-9*345^(1/2))^(2/3)),-2*I*(179-9*345^(1/2))^(1/3)*(4-1 6/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/2))^(1/3)-9*x)/(48*I+16*3^(1/2)+(3 *I-3^(1/2))*(179-9*345^(1/2))^(2/3)))/(2+p)/((1+2*I*(179-9*345^(1/2))^(1/3 )*(4-16/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/2))^(1/3)-9*x)/(48*I+16*3^(1 /2)+(3*I-3^(1/2))*(179-9*345^(1/2))^(2/3)))^p)/((1+2*I*(179-9*345^(1/2))^( 1/3)*(4-16/(179-9*345^(1/2))^(1/3)-(179-9*345^(1/2))^(1/3)-9*x)/(48*I-1...
\[ \int \left (A+B x+C x^2\right ) \left (2-4 x^2+3 x^3\right )^p \, dx=\int \left (A+B x+C x^2\right ) \left (2-4 x^2+3 x^3\right )^p \, dx \] Input:
Integrate[(A + B*x + C*x^2)*(2 - 4*x^2 + 3*x^3)^p,x]
Output:
Integrate[(A + B*x + C*x^2)*(2 - 4*x^2 + 3*x^3)^p, x]
Time = 2.96 (sec) , antiderivative size = 1308, normalized size of antiderivative = 1.34, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2526, 2490, 2486, 27, 1269, 1179, 150}
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.
| \(\displaystyle \int \left (3 x^3-4 x^2+2\right )^p \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{9} \int (9 A+(9 B+8 C) x) \left (3 x^3-4 x^2+2\right )^pdx+\frac {C \left (3 x^3-4 x^2+2\right )^{p+1}}{9 (p+1)}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {1}{9} \int \left (\frac {1}{9} (81 A+4 (9 B+8 C))+(9 B+8 C) \left (x-\frac {4}{9}\right )\right ) \left (3 \left (x-\frac {4}{9}\right )^3-\frac {16}{9} \left (x-\frac {4}{9}\right )+\frac {358}{243}\right )^pd\left (x-\frac {4}{9}\right )+\frac {C \left (3 x^3-4 x^2+2\right )^{p+1}}{9 (p+1)}\) |
| \(\Big \downarrow \) 2486 |
| \(\displaystyle \frac {1}{9} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^{-p} \left (3 \left (x-\frac {4}{9}\right )^3-\frac {16}{9} \left (x-\frac {4}{9}\right )+\frac {358}{243}\right )^p \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^{-p} \int \frac {1}{9} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^p \left (81 A+36 B+32 C+9 (9 B+8 C) \left (x-\frac {4}{9}\right )\right ) \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^pd\left (x-\frac {4}{9}\right )+\frac {C \left (3 x^3-4 x^2+2\right )^{p+1}}{9 (p+1)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{81} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^{-p} \left (3 \left (x-\frac {4}{9}\right )^3-\frac {16}{9} \left (x-\frac {4}{9}\right )+\frac {358}{243}\right )^p \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^{-p} \int \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^p \left (81 A+36 B+32 C+9 (9 B+8 C) \left (x-\frac {4}{9}\right )\right ) \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^pd\left (x-\frac {4}{9}\right )+\frac {C \left (3 x^3-4 x^2+2\right )^{p+1}}{9 (p+1)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{81} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^{-p} \left (3 \left (x-\frac {4}{9}\right )^3-\frac {16}{9} \left (x-\frac {4}{9}\right )+\frac {358}{243}\right )^p \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^{-p} \left (\left (81 A-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)}{\sqrt [3]{179-9 \sqrt {345}}}+36 B+32 C\right ) \int \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^p \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^pd\left (x-\frac {4}{9}\right )+3 (9 B+8 C) \int \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^{p+1} \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^pd\left (x-\frac {4}{9}\right )\right )+\frac {C \left (3 x^3-4 x^2+2\right )^{p+1}}{9 (p+1)}\) |
| \(\Big \downarrow \) 1179 |
| \(\displaystyle \frac {1}{81} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^{-p} \left (3 \left (x-\frac {4}{9}\right )^3-\frac {16}{9} \left (x-\frac {4}{9}\right )+\frac {358}{243}\right )^p \left (\frac {1}{3} \left (81 A+36 B+32 C-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)}{\sqrt [3]{179-9 \sqrt {345}}}\right ) \left (1-\frac {2 i \sqrt [3]{179-9 \sqrt {345}} \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^{-p} \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^p \int \left (1-\frac {6 i \sqrt [3]{179-9 \sqrt {345}} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^p \left (1-\frac {6 i \sqrt [3]{179-9 \sqrt {345}} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^p \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^pd\left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right ) \left (1-\frac {2 i \sqrt [3]{179-9 \sqrt {345}} \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^{-p}+(9 B+8 C) \left (1-\frac {2 i \sqrt [3]{179-9 \sqrt {345}} \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^{-p} \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^p \int \left (1-\frac {6 i \sqrt [3]{179-9 \sqrt {345}} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^p \left (1-\frac {6 i \sqrt [3]{179-9 \sqrt {345}} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^p \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^{p+1}d\left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right ) \left (1-\frac {2 i \sqrt [3]{179-9 \sqrt {345}} \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^{-p}\right ) \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^{-p}+\frac {C \left (3 x^3-4 x^2+2\right )^{p+1}}{9 (p+1)}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {1}{81} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^{-p} \left (3 \left (x-\frac {4}{9}\right )^3-\frac {16}{9} \left (x-\frac {4}{9}\right )+\frac {358}{243}\right )^p \left (\frac {\left (81 A+36 B+32 C-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)}{\sqrt [3]{179-9 \sqrt {345}}}\right ) \left (1-\frac {2 i \sqrt [3]{179-9 \sqrt {345}} \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^{-p} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^{p+1} \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^p \operatorname {AppellF1}\left (p+1,-p,-p,p+2,\frac {6 i \sqrt [3]{179-9 \sqrt {345}} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}},\frac {6 i \sqrt [3]{179-9 \sqrt {345}} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right ) \left (1-\frac {2 i \sqrt [3]{179-9 \sqrt {345}} \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^{-p}}{3 (p+1)}+\frac {(9 B+8 C) \left (1-\frac {2 i \sqrt [3]{179-9 \sqrt {345}} \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^{-p} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )^{p+2} \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^p \operatorname {AppellF1}\left (p+2,-p,-p,p+3,\frac {6 i \sqrt [3]{179-9 \sqrt {345}} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}},\frac {6 i \sqrt [3]{179-9 \sqrt {345}} \left (3 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{3 \sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right ) \left (1-\frac {2 i \sqrt [3]{179-9 \sqrt {345}} \left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right )}{16 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (179-9 \sqrt {345}\right )^{2/3}}\right )^{-p}}{p+2}\right ) \left (9 \left (x-\frac {4}{9}\right )^2-\frac {\left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}+\frac {1}{9} \left (-16+\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+\left (179-9 \sqrt {345}\right )^{2/3}\right )\right )^{-p}+\frac {C \left (3 x^3-4 x^2+2\right )^{p+1}}{9 (p+1)}\) |
Input:
Int[(A + B*x + C*x^2)*(2 - 4*x^2 + 3*x^3)^p,x]
Output:
(C*(2 - 4*x^2 + 3*x^3)^(1 + p))/(9*(1 + p)) + ((358/243 - (16*(-4/9 + x))/ 9 + 3*(-4/9 + x)^3)^p*(((81*A + 36*B + 32*C - ((16 + (179 - 9*Sqrt[345])^( 2/3))*(9*B + 8*C))/(179 - 9*Sqrt[345])^(1/3))*((16 + (179 - 9*Sqrt[345])^( 2/3))/(3*(179 - 9*Sqrt[345])^(1/3)) + 3*(-4/9 + x))^(1 + p)*((-16 + 256/(1 79 - 9*Sqrt[345])^(2/3) + (179 - 9*Sqrt[345])^(2/3))/9 - ((16 + (179 - 9*S qrt[345])^(2/3))*(-4/9 + x))/(179 - 9*Sqrt[345])^(1/3) + 9*(-4/9 + x)^2)^p *AppellF1[1 + p, -p, -p, 2 + p, ((6*I)*(179 - 9*Sqrt[345])^(1/3)*((16 + (1 79 - 9*Sqrt[345])^(2/3))/(3*(179 - 9*Sqrt[345])^(1/3)) + 3*(-4/9 + x)))/(1 6*(3*I + Sqrt[3]) + (3*I - Sqrt[3])*(179 - 9*Sqrt[345])^(2/3)), ((6*I)*(17 9 - 9*Sqrt[345])^(1/3)*((16 + (179 - 9*Sqrt[345])^(2/3))/(3*(179 - 9*Sqrt[ 345])^(1/3)) + 3*(-4/9 + x)))/(16*(3*I - Sqrt[3]) + (3*I + Sqrt[3])*(179 - 9*Sqrt[345])^(2/3))])/(3*(1 + p)*(1 - ((2*I)*(179 - 9*Sqrt[345])^(1/3)*(( 16 + (179 - 9*Sqrt[345])^(2/3))/(179 - 9*Sqrt[345])^(1/3) + 9*(-4/9 + x))) /(16*(3*I + Sqrt[3]) + (3*I - Sqrt[3])*(179 - 9*Sqrt[345])^(2/3)))^p*(1 - ((2*I)*(179 - 9*Sqrt[345])^(1/3)*((16 + (179 - 9*Sqrt[345])^(2/3))/(179 - 9*Sqrt[345])^(1/3) + 9*(-4/9 + x)))/(16*(3*I - Sqrt[3]) + (3*I + Sqrt[3])* (179 - 9*Sqrt[345])^(2/3)))^p) + ((9*B + 8*C)*((16 + (179 - 9*Sqrt[345])^( 2/3))/(3*(179 - 9*Sqrt[345])^(1/3)) + 3*(-4/9 + x))^(2 + p)*((-16 + 256/(1 79 - 9*Sqrt[345])^(2/3) + (179 - 9*Sqrt[345])^(2/3))/9 - ((16 + (179 - 9*S qrt[345])^(2/3))*(-4/9 + x))/(179 - 9*Sqrt[345])^(1/3) + 9*(-4/9 + x)^2...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) ^p) Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d - e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m , p}, x]
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]
\[\int \left (C \,x^{2}+B x +A \right ) \left (3 x^{3}-4 x^{2}+2\right )^{p}d x\]
Input:
int((C*x^2+B*x+A)*(3*x^3-4*x^2+2)^p,x)
Output:
int((C*x^2+B*x+A)*(3*x^3-4*x^2+2)^p,x)
\[ \int \left (A+B x+C x^2\right ) \left (2-4 x^2+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 4 \, x^{2} + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-4*x^2+2)^p,x, algorithm="fricas")
Output:
integral((C*x^2 + B*x + A)*(3*x^3 - 4*x^2 + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2-4 x^2+3 x^3\right )^p \, dx=\int \left (A + B x + C x^{2}\right ) \left (3 x^{3} - 4 x^{2} + 2\right )^{p}\, dx \] Input:
integrate((C*x**2+B*x+A)*(3*x**3-4*x**2+2)**p,x)
Output:
Integral((A + B*x + C*x**2)*(3*x**3 - 4*x**2 + 2)**p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2-4 x^2+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 4 \, x^{2} + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-4*x^2+2)^p,x, algorithm="maxima")
Output:
integrate((C*x^2 + B*x + A)*(3*x^3 - 4*x^2 + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2-4 x^2+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 4 \, x^{2} + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-4*x^2+2)^p,x, algorithm="giac")
Output:
integrate((C*x^2 + B*x + A)*(3*x^3 - 4*x^2 + 2)^p, x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (2-4 x^2+3 x^3\right )^p \, dx=\int \left (C\,x^2+B\,x+A\right )\,{\left (3\,x^3-4\,x^2+2\right )}^p \,d x \] Input:
int((A + B*x + C*x^2)*(3*x^3 - 4*x^2 + 2)^p,x)
Output:
int((A + B*x + C*x^2)*(3*x^3 - 4*x^2 + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2-4 x^2+3 x^3\right )^p \, dx=\text {too large to display} \] Input:
int((C*x^2+B*x+A)*(3*x^3-4*x^2+2)^p,x)
Output:
(729*(3*x**3 - 4*x**2 + 2)**p*a*p**2*x - 324*(3*x**3 - 4*x**2 + 2)**p*a*p* *2 + 1215*(3*x**3 - 4*x**2 + 2)**p*a*p*x - 540*(3*x**3 - 4*x**2 + 2)**p*a* p + 486*(3*x**3 - 4*x**2 + 2)**p*a*x - 216*(3*x**3 - 4*x**2 + 2)**p*a + 72 9*(3*x**3 - 4*x**2 + 2)**p*b*p**2*x**2 - 324*(3*x**3 - 4*x**2 + 2)**p*b*p* *2*x - 288*(3*x**3 - 4*x**2 + 2)**p*b*p**2 + 972*(3*x**3 - 4*x**2 + 2)**p* b*p*x**2 - 324*(3*x**3 - 4*x**2 + 2)**p*b*p*x - 432*(3*x**3 - 4*x**2 + 2)* *p*b*p + 243*(3*x**3 - 4*x**2 + 2)**p*b*x**2 - 144*(3*x**3 - 4*x**2 + 2)** p*b + 729*(3*x**3 - 4*x**2 + 2)**p*c*p**2*x**3 - 324*(3*x**3 - 4*x**2 + 2) **p*c*p**2*x**2 - 288*(3*x**3 - 4*x**2 + 2)**p*c*p**2*x + 230*(3*x**3 - 4* x**2 + 2)**p*c*p**2 + 729*(3*x**3 - 4*x**2 + 2)**p*c*p*x**3 - 108*(3*x**3 - 4*x**2 + 2)**p*c*p*x**2 - 288*(3*x**3 - 4*x**2 + 2)**p*c*p*x + 102*(3*x* *3 - 4*x**2 + 2)**p*c*p + 162*(3*x**3 - 4*x**2 + 2)**p*c*x**3 - 20*(3*x**3 - 4*x**2 + 2)**p*c + 39366*int((3*x**3 - 4*x**2 + 2)**p/(27*p**2*x**3 - 3 6*p**2*x**2 + 18*p**2 + 27*p*x**3 - 36*p*x**2 + 18*p + 6*x**3 - 8*x**2 + 4 ),x)*a*p**5 + 104976*int((3*x**3 - 4*x**2 + 2)**p/(27*p**2*x**3 - 36*p**2* x**2 + 18*p**2 + 27*p*x**3 - 36*p*x**2 + 18*p + 6*x**3 - 8*x**2 + 4),x)*a* p**4 + 100602*int((3*x**3 - 4*x**2 + 2)**p/(27*p**2*x**3 - 36*p**2*x**2 + 18*p**2 + 27*p*x**3 - 36*p*x**2 + 18*p + 6*x**3 - 8*x**2 + 4),x)*a*p**3 + 40824*int((3*x**3 - 4*x**2 + 2)**p/(27*p**2*x**3 - 36*p**2*x**2 + 18*p**2 + 27*p*x**3 - 36*p*x**2 + 18*p + 6*x**3 - 8*x**2 + 4),x)*a*p**2 + 5832*...