Integrand size = 43, antiderivative size = 532 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \, dx=\frac {C (d+e x) \left (a+b x+c x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p}{3 c e (1+p)}+\frac {\left (C e (b d-a e)+c \left (2 C d^2-3 e (B d-A e)\right )\right ) (d+e x) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (1+p,-p,-p,2+p,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c e^3 (1+p)}-\frac {(2 c C d-3 B c e+2 b C e) (d+e x)^2 \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (2+p,-p,-p,3+p,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c e^3 (2+p)} \] Output:
1/3*C*(e*x+d)*(c*x^2+b*x+a)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^p/c/e/ (p+1)+1/3*(C*e*(-a*e+b*d)+c*(2*C*d^2-3*e*(-A*e+B*d)))*(e*x+d)*(a*d+(a*e+b* d)*x+(b*e+c*d)*x^2+c*e*x^3)^p*AppellF1(p+1,-p,-p,2+p,2*c*(e*x+d)/(2*c*d-(b -(-4*a*c+b^2)^(1/2))*e),2*c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))/c/e^ 3/(p+1)/((1-2*c*(e*x+d)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e))^p)/((1-2*c*(e*x+ d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^p)-1/3*(-3*B*c*e+2*C*b*e+2*C*c*d)*(e* x+d)^2*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^p*AppellF1(2+p,-p,-p,3+p,2* c*(e*x+d)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e),2*c*(e*x+d)/(2*c*d-(b+(-4*a*c+b ^2)^(1/2))*e))/c/e^3/(2+p)/((1-2*c*(e*x+d)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e ))^p)/((1-2*c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^p)
\[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \, dx=\int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \, dx \] Input:
Integrate[(A + B*x + C*x^2)*(a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x ^3)^p,x]
Output:
Integrate[(A + B*x + C*x^2)*(a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x ^3)^p, x]
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.
| \(\displaystyle \int \left (A+B x+C x^2\right ) \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^p \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {\int -\left ((b C d-3 A c e+a C e+(2 c C d-3 B c e+2 b C e) x) \left (c e x^3+(c d+b e) x^2+(b d+a e) x+a d\right )^p\right )dx}{3 c e}+\frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int (b C d-3 A c e+a C e+(2 c C d-3 B c e+2 b C e) x) \left (c e x^3+(c d+b e) x^2+(b d+a e) x+a d\right )^pdx}{3 c e}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3 c e (b C d-3 A c e+a C e)-(c d+b e) (2 c C d-3 B c e+2 b C e)}{3 c e}+(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c e \left (\frac {c d+b e}{3 c e}+x\right )^3+\frac {\left (3 c e (b d+a e)-(c d+b e)^2\right ) \left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}+\frac {(2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right )}{27 c^2 e^2}\right )^pd\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {\left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (c e \left (\frac {c d+b e}{3 c e}+x\right )^3+\frac {\left (3 c e (b d+a e)-(c d+b e)^2\right ) \left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}+\frac {(2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right )}{27 c^2 e^2}\right )^p}{3 c e}+(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (c e \left (\frac {c d+b e}{3 c e}+x\right )^3+\frac {\left (3 c e (b d+a e)-(c d+b e)^2\right ) \left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}+\frac {(2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right )}{27 c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{p+1}}{3 c e (p+1)}-\frac {\int \left (\frac {3^{-3 p-1} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p}{c e}+27^{-p} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^p\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
Input:
Int[(A + B*x + C*x^2)*(a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x^3)^p, x]
Output:
$Aborted
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]
\[\int \left (C \,x^{2}+B x +A \right ) \left (a d +\left (a e +b d \right ) x +\left (e b +c d \right ) x^{2}+c e \,x^{3}\right )^{p}d x\]
Input:
int((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^p,x)
Output:
int((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^p,x)
\[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^p,x, algor ithm="fricas")
Output:
integral((C*x^2 + B*x + A)*(c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)* x)^p, x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \, dx=\text {Timed out} \] Input:
integrate((C*x**2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x**2+c*e*x**3)**p,x)
Output:
Timed out
\[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^p,x, algor ithm="maxima")
Output:
integrate((C*x^2 + B*x + A)*(c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e) *x)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^p,x, algor ithm="giac")
Output:
integrate((C*x^2 + B*x + A)*(c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e) *x)^p, x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \, dx=\int \left (C\,x^2+B\,x+A\right )\,{\left (c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+\left (a\,e+b\,d\right )\,x+a\,d\right )}^p \,d x \] Input:
int((A + B*x + C*x^2)*(a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3)^p, x)
Output:
int((A + B*x + C*x^2)*(a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3)^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^p \, dx=\text {too large to display} \] Input:
int((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^p,x)
Output:
(12*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a**2*c*e**3* p**2 + 20*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a**2*c *e**3*p + 8*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a**2 *c*e**3 - (a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a*b**2 *e**3*p**2 - 2*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a *b**2*e**3*p - (a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a *b**2*e**3 + 34*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p* a*b*c*d*e**2*p**2 + 42*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x* *3)**p*a*b*c*d*e**2*p + 12*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c* e*x**3)**p*a*b*c*d*e**2 + 30*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a*b*c*e**3*p**2*x + 38*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d* x**2 + c*e*x**3)**p*a*b*c*e**3*p*x + 12*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a*b*c*e**3*x + 2*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a*c**2*d**2*e*p**2 - 2*(a*d + a*e*x + b*d*x + b*e *x**2 + c*d*x**2 + c*e*x**3)**p*a*c**2*d**2*e*p + 30*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a*c**2*d*e**2*p**2*x + 38*(a*d + a*e* x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a*c**2*d*e**2*p*x + 12*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*a*c**2*d*e**2*x - (a *d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*b**3*d*e**2*p**2 - 2*(a*d + a*e*x + b*d*x + b*e*x**2 + c*d*x**2 + c*e*x**3)**p*b**3*d*e**...