Integrand size = 25, antiderivative size = 571 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^p \, dx =\text {Too large to display} \] Output:
C*(3*x^3-6*x^2+2)^(p+1)/(9*p+9)+3^(-5/2+3/2*p)*(3*B+4*C)*(3*x^3-6*x^2+2)^p *AppellF1(p+1,-p,-1-p,2+p,-1/2*(3*x-2-4*cos(1/6*Pi+1/3*arcsin(1/8)))/(3^(1 /2)*cos(1/3*arcsin(1/8))-3*sin(1/3*arcsin(1/8))),-1/12*(3*x-2-4*cos(1/6*Pi +1/3*arcsin(1/8)))*sec(1/3*arcsin(1/8))*3^(1/2))*(4*cos(1/3*arcsin(1/8)))^ (p+1)*(3*x-2-4*cos(1/6*Pi+1/3*arcsin(1/8)))*(2/3*3^(1/2)*cos(1/3*arcsin(1/ 8))-2*sin(1/3*arcsin(1/8)))^p/(p+1)/((3*x-2-4*sin(1/3*arcsin(1/8)))^p)/((3 *x-2+4*sin(1/3*Pi+1/3*arcsin(1/8)))^p)+3^(-3+1/2*p)*(3*x^3-6*x^2+2)^p*Appe llF1(p+1,-p,-p,2+p,-1/2*(3*x-2-4*cos(1/6*Pi+1/3*arcsin(1/8)))/(3^(1/2)*cos (1/3*arcsin(1/8))-3*sin(1/3*arcsin(1/8))),-1/12*(3*x-2-4*cos(1/6*Pi+1/3*ar csin(1/8)))*sec(1/3*arcsin(1/8))*3^(1/2))*(3*x-2-4*cos(1/6*Pi+1/3*arcsin(1 /8)))*(8*cos(1/3*arcsin(1/8))*(3^(1/2)*cos(1/3*arcsin(1/8))-3*sin(1/3*arcs in(1/8))))^p*(9*A+2*(3*B+4*C)*(1-2*sin(1/3*Pi+1/3*arcsin(1/8))))/(p+1)/((3 *x-2-4*sin(1/3*arcsin(1/8)))^p)/((3*x-2+4*sin(1/3*Pi+1/3*arcsin(1/8)))^p)
\[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^p \, dx=\int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^p \, dx \] Input:
Integrate[(A + B*x + C*x^2)*(2 - 6*x^2 + 3*x^3)^p,x]
Output:
Integrate[(A + B*x + C*x^2)*(2 - 6*x^2 + 3*x^3)^p, x]
Result contains complex when optimal does not.
Time = 8.79 (sec) , antiderivative size = 1729, normalized size of antiderivative = 3.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2526, 27, 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.
| \(\displaystyle \int \left (3 x^3-6 x^2+2\right )^p \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{9} \int 3 (3 A+(3 B+4 C) x) \left (3 x^3-6 x^2+2\right )^pdx+\frac {C \left (3 x^3-6 x^2+2\right )^{p+1}}{9 (p+1)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int (3 A+(3 B+4 C) x) \left (3 x^3-6 x^2+2\right )^pdx+\frac {C \left (3 x^3-6 x^2+2\right )^{p+1}}{9 (p+1)}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {1}{3} \int \left (\frac {1}{9} (27 A+6 (3 B+4 C))+(3 B+4 C) \left (x-\frac {2}{3}\right )\right ) \left (3 \left (x-\frac {2}{3}\right )^3-4 \left (x-\frac {2}{3}\right )+\frac {2}{9}\right )^pd\left (x-\frac {2}{3}\right )+\frac {C \left (3 x^3-6 x^2+2\right )^{p+1}}{9 (p+1)}\) |
| \(\Big \downarrow \) 2486 |
| \(\displaystyle \frac {C \left (3 x^3-6 x^2+2\right )^{p+1}}{9 (p+1)}+\frac {1}{3} \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )^3-4 \left (x-\frac {2}{3}\right )+\frac {2}{9}\right )^p \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^{-p} \int \frac {1}{3} \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^p \left (9 A+6 B+8 C+3 (3 B+4 C) \left (x-\frac {2}{3}\right )\right ) \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^pd\left (x-\frac {2}{3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C \left (3 x^3-6 x^2+2\right )^{p+1}}{9 (p+1)}+\frac {1}{9} \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )^3-4 \left (x-\frac {2}{3}\right )+\frac {2}{9}\right )^p \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^{-p} \int \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^p \left (9 A+6 B+8 C+3 (3 B+4 C) \left (x-\frac {2}{3}\right )\right ) \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^pd\left (x-\frac {2}{3}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {C \left (3 x^3-6 x^2+2\right )^{p+1}}{9 (p+1)}+\frac {1}{9} \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )^3-4 \left (x-\frac {2}{3}\right )+\frac {2}{9}\right )^p \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^{-p} \left (\left (9 A-\frac {\left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) (3 B+4 C)}{\sqrt [3]{1-3 i \sqrt {7}}}+6 B+8 C\right ) \int \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^p \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^pd\left (x-\frac {2}{3}\right )+(3 B+4 C) \int \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{p+1} \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^pd\left (x-\frac {2}{3}\right )\right )\) |
| \(\Big \downarrow \) 1179 |
| \(\displaystyle \frac {1}{9} \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )^3-4 \left (x-\frac {2}{3}\right )+\frac {2}{9}\right )^p \left (\frac {1}{3} \left (9 A+6 B+8 C-\frac {\left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) (3 B+4 C)}{\sqrt [3]{1-3 i \sqrt {7}}}\right ) \left (\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}+1\right )^{-p} \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^p \int \left (\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}+1\right )^p \left (1-\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^p \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^pd\left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right ) \left (1-\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^{-p}+\frac {1}{3} (3 B+4 C) \left (\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}+1\right )^{-p} \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^p \int \left (\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}+1\right )^p \left (1-\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^p \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{p+1}d\left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right ) \left (1-\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^{-p}\right ) \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^{-p}+\frac {C \left (3 x^3-6 x^2+2\right )^{p+1}}{9 (p+1)}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {1}{9} \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )^3-4 \left (x-\frac {2}{3}\right )+\frac {2}{9}\right )^p \left (\frac {\left (9 A+6 B+8 C-\frac {\left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) (3 B+4 C)}{\sqrt [3]{1-3 i \sqrt {7}}}\right ) \left (\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}+1\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{p+1} \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^p \operatorname {AppellF1}\left (p+1,-p,-p,p+2,-\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )},\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right ) \left (1-\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^{-p}}{3 (p+1)}+\frac {(3 B+4 C) \left (\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}+1\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{p+2} \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^p \operatorname {AppellF1}\left (p+2,-p,-p,p+3,-\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )},\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right ) \left (1-\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^{-p}}{3 (p+2)}\right ) \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^{-p}+\frac {C \left (3 x^3-6 x^2+2\right )^{p+1}}{9 (p+1)}\) |
Input:
Int[(A + B*x + C*x^2)*(2 - 6*x^2 + 3*x^3)^p,x]
Output:
(C*(2 - 6*x^2 + 3*x^3)^(1 + p))/(9*(1 + p)) + ((2/9 - 4*(-2/3 + x) + 3*(-2 /3 + x)^3)^p*(((9*A + 6*B + 8*C - ((4 + (1 - (3*I)*Sqrt[7])^(2/3))*(3*B + 4*C))/(1 - (3*I)*Sqrt[7])^(1/3))*((4 + (1 - (3*I)*Sqrt[7])^(2/3))/(1 - (3* I)*Sqrt[7])^(1/3) + 3*(-2/3 + x))^(1 + p)*(-4 + 16/(1 - (3*I)*Sqrt[7])^(2/ 3) + (1 - (3*I)*Sqrt[7])^(2/3) - (3*(4 + (1 - (3*I)*Sqrt[7])^(2/3))*(-2/3 + x))/(1 - (3*I)*Sqrt[7])^(1/3) + 9*(-2/3 + x)^2)^p*AppellF1[1 + p, -p, -p , 2 + p, (-2*(I + 3*Sqrt[7])*((4 + (1 - (3*I)*Sqrt[7])^(2/3))/(1 - (3*I)*S qrt[7])^(1/3) + 3*(-2/3 + x)))/(Sqrt[-3*(1 - (3*I)*Sqrt[7])^(2/3)]*(I + 3* Sqrt[7] - (4*I)*(1 - (3*I)*Sqrt[7])^(1/3)) - (3*I)*(1 - (3*I)*Sqrt[7])^(2/ 3)*(4 + (1 - (3*I)*Sqrt[7])^(2/3))), (2*(I + 3*Sqrt[7])*((4 + (1 - (3*I)*S qrt[7])^(2/3))/(1 - (3*I)*Sqrt[7])^(1/3) + 3*(-2/3 + x)))/(Sqrt[-3*(1 - (3 *I)*Sqrt[7])^(2/3)]*(I + 3*Sqrt[7] - (4*I)*(1 - (3*I)*Sqrt[7])^(1/3)) + (3 *I)*(1 - (3*I)*Sqrt[7])^(2/3)*(4 + (1 - (3*I)*Sqrt[7])^(2/3)))])/(3*(1 + p )*(1 + (2*(I + 3*Sqrt[7])*((4 + (1 - (3*I)*Sqrt[7])^(2/3))/(1 - (3*I)*Sqrt [7])^(1/3) + 3*(-2/3 + x)))/(Sqrt[-3*(1 - (3*I)*Sqrt[7])^(2/3)]*(I + 3*Sqr t[7] - (4*I)*(1 - (3*I)*Sqrt[7])^(1/3)) - (3*I)*(1 - (3*I)*Sqrt[7])^(2/3)* (4 + (1 - (3*I)*Sqrt[7])^(2/3))))^p*(1 - (2*(I + 3*Sqrt[7])*((4 + (1 - (3* I)*Sqrt[7])^(2/3))/(1 - (3*I)*Sqrt[7])^(1/3) + 3*(-2/3 + x)))/(Sqrt[-3*(1 - (3*I)*Sqrt[7])^(2/3)]*(I + 3*Sqrt[7] - (4*I)*(1 - (3*I)*Sqrt[7])^(1/3)) + (3*I)*(1 - (3*I)*Sqrt[7])^(2/3)*(4 + (1 - (3*I)*Sqrt[7])^(2/3))))^p) ...
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]
\[\int \left (C \,x^{2}+B x +A \right ) \left (3 x^{3}-6 x^{2}+2\right )^{p}d x\]
Input:
int((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^p,x)
Output:
int((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^p,x)
\[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 6 \, x^{2} + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^p,x, algorithm="fricas")
Output:
integral((C*x^2 + B*x + A)*(3*x^3 - 6*x^2 + 2)^p, x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^p \, dx=\text {Timed out} \] Input:
integrate((C*x**2+B*x+A)*(3*x**3-6*x**2+2)**p,x)
Output:
Timed out
\[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 6 \, x^{2} + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^p,x, algorithm="maxima")
Output:
integrate((C*x^2 + B*x + A)*(3*x^3 - 6*x^2 + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 6 \, x^{2} + 2\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^p,x, algorithm="giac")
Output:
integrate((C*x^2 + B*x + A)*(3*x^3 - 6*x^2 + 2)^p, x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^p \, dx=\int \left (C\,x^2+B\,x+A\right )\,{\left (3\,x^3-6\,x^2+2\right )}^p \,d x \] Input:
int((A + B*x + C*x^2)*(3*x^3 - 6*x^2 + 2)^p,x)
Output:
int((A + B*x + C*x^2)*(3*x^3 - 6*x^2 + 2)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^p \, dx=\text {too large to display} \] Input:
int((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^p,x)
Output:
(27*(3*x**3 - 6*x**2 + 2)**p*a*p**2*x - 18*(3*x**3 - 6*x**2 + 2)**p*a*p**2 + 45*(3*x**3 - 6*x**2 + 2)**p*a*p*x - 30*(3*x**3 - 6*x**2 + 2)**p*a*p + 1 8*(3*x**3 - 6*x**2 + 2)**p*a*x - 12*(3*x**3 - 6*x**2 + 2)**p*a + 27*(3*x** 3 - 6*x**2 + 2)**p*b*p**2*x**2 - 18*(3*x**3 - 6*x**2 + 2)**p*b*p**2*x - 24 *(3*x**3 - 6*x**2 + 2)**p*b*p**2 + 36*(3*x**3 - 6*x**2 + 2)**p*b*p*x**2 - 18*(3*x**3 - 6*x**2 + 2)**p*b*p*x - 36*(3*x**3 - 6*x**2 + 2)**p*b*p + 9*(3 *x**3 - 6*x**2 + 2)**p*b*x**2 - 12*(3*x**3 - 6*x**2 + 2)**p*b + 27*(3*x**3 - 6*x**2 + 2)**p*c*p**2*x**3 - 18*(3*x**3 - 6*x**2 + 2)**p*c*p**2*x**2 - 24*(3*x**3 - 6*x**2 + 2)**p*c*p**2*x - 14*(3*x**3 - 6*x**2 + 2)**p*c*p**2 + 27*(3*x**3 - 6*x**2 + 2)**p*c*p*x**3 - 6*(3*x**3 - 6*x**2 + 2)**p*c*p*x* *2 - 24*(3*x**3 - 6*x**2 + 2)**p*c*p*x - 30*(3*x**3 - 6*x**2 + 2)**p*c*p + 6*(3*x**3 - 6*x**2 + 2)**p*c*x**3 - 12*(3*x**3 - 6*x**2 + 2)**p*c + 1458* int((3*x**3 - 6*x**2 + 2)**p/(27*p**2*x**3 - 54*p**2*x**2 + 18*p**2 + 27*p *x**3 - 54*p*x**2 + 18*p + 6*x**3 - 12*x**2 + 4),x)*a*p**5 + 3888*int((3*x **3 - 6*x**2 + 2)**p/(27*p**2*x**3 - 54*p**2*x**2 + 18*p**2 + 27*p*x**3 - 54*p*x**2 + 18*p + 6*x**3 - 12*x**2 + 4),x)*a*p**4 + 3726*int((3*x**3 - 6* x**2 + 2)**p/(27*p**2*x**3 - 54*p**2*x**2 + 18*p**2 + 27*p*x**3 - 54*p*x** 2 + 18*p + 6*x**3 - 12*x**2 + 4),x)*a*p**3 + 1512*int((3*x**3 - 6*x**2 + 2 )**p/(27*p**2*x**3 - 54*p**2*x**2 + 18*p**2 + 27*p*x**3 - 54*p*x**2 + 18*p + 6*x**3 - 12*x**2 + 4),x)*a*p**2 + 216*int((3*x**3 - 6*x**2 + 2)**p/(...