\(\int (d+e x)^{-6-2 p} (a+b x+c x^2)^p \, dx\) [792]

Optimal result

Integrand size = 24, antiderivative size = 809 \[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=-\frac {e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac {e \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (18+11 p+2 p^2\right )-2 c e \left (3 a e (2+p)+b d \left (18+11 p+2 p^2\right )\right )\right ) (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}-\frac {e (2 c d-b e) (3+p) \left (b^2 e^2 \left (8+6 p+p^2\right )+2 c^2 d^2 \left (8+7 p+2 p^2\right )-2 c e \left (a e (8+5 p)+b d \left (8+7 p+2 p^2\right )\right )\right ) (d+e x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{4 \left (c d^2-b d e+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac {e (2 c d-b e) (4+p) (d+e x)^{-2 (2+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}+\frac {\left (b^4 e^4 \left (12+7 p+p^2\right )+4 c^4 d^4 \left (15+16 p+4 p^2\right )-8 c^3 d^2 e (5+2 p) (3 a e+b d (3+2 p))-4 b^2 c e^3 (3+p) (3 a e+b d (5+2 p))+12 c^2 e^2 \left (a^2 e^2+2 a b d e (5+2 p)+b^2 d^2 \left (10+9 p+2 p^2\right )\right )\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{4 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right )^4 (1+2 p) (3+2 p) (5+2 p)} \] Output:

-e*(e*x+d)^(-5-2*p)*(c*x^2+b*x+a)^(p+1)/(a*e^2-b*d*e+c*d^2)/(5+2*p)-1/2*e* 
(b^2*e^2*(p^2+7*p+12)+2*c^2*d^2*(2*p^2+11*p+18)-2*c*e*(3*a*e*(2+p)+b*d*(2* 
p^2+11*p+18)))*(e*x+d)^(-3-2*p)*(c*x^2+b*x+a)^(p+1)/(a*e^2-b*d*e+c*d^2)^3/ 
(2+p)/(3+2*p)/(5+2*p)-1/4*e*(-b*e+2*c*d)*(3+p)*(b^2*e^2*(p^2+6*p+8)+2*c^2* 
d^2*(2*p^2+7*p+8)-2*c*e*(a*e*(8+5*p)+b*d*(2*p^2+7*p+8)))*(c*x^2+b*x+a)^(p+ 
1)/(a*e^2-b*d*e+c*d^2)^4/(p+1)/(2+p)/(3+2*p)/(5+2*p)/((e*x+d)^(2*p+2))-1/2 
*e*(-b*e+2*c*d)*(4+p)*(c*x^2+b*x+a)^(p+1)/(a*e^2-b*d*e+c*d^2)^2/(2+p)/(5+2 
*p)/((e*x+d)^(4+2*p))+1/4*(b^4*e^4*(p^2+7*p+12)+4*c^4*d^4*(4*p^2+16*p+15)- 
8*c^3*d^2*e*(5+2*p)*(3*a*e+b*d*(3+2*p))-4*b^2*c*e^3*(3+p)*(3*a*e+b*d*(5+2* 
p))+12*c^2*e^2*(a^2*e^2+2*a*b*d*e*(5+2*p)+b^2*d^2*(2*p^2+9*p+10)))*(b-(-4* 
a*c+b^2)^(1/2)+2*c*x)*(e*x+d)^(-1-2*p)*(c*x^2+b*x+a)^p*hypergeom([-p, -1-2 
*p],[-2*p],-4*c*(-4*a*c+b^2)^(1/2)*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e 
)/(b-(-4*a*c+b^2)^(1/2)+2*c*x))/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)/(a*e^2-b* 
d*e+c*d^2)^4/(1+2*p)/(3+2*p)/(5+2*p)/(((2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)*(2 
*c*x+(-4*a*c+b^2)^(1/2)+b)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)/(b-(-4*a*c+b^2 
)^(1/2)+2*c*x))^p)
 

Mathematica [A] (verified)

Time = 6.56 (sec) , antiderivative size = 1577, normalized size of antiderivative = 1.95 \[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx =\text {Too large to display} \] Input:

Integrate[(d + e*x)^(-6 - 2*p)*(a + b*x + c*x^2)^p,x]
 

Output:

(e*(d + e*x)^(1 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + 
a*e^2)*(1 - 2*(3 + p))) + (3*c*((e*(d + e*x)^(3 - 2*(3 + p))*(a + b*x + c* 
x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(3 - 2*(3 + p))) + (((c*d*e - e*(b* 
e*(2 + p) - c*d*(3 + 2*p)))*(d + e*x)^(4 - 2*(3 + p))*(a + b*x + c*x^2)^(1 
 + p))/(2*(c*d^2 - b*d*e + a*e^2)*(1 + p)) - ((-2*(a*c*e^2 + c*d*(b*e*(2 + 
 p) - c*d*(3 + 2*p))) + b*(c*d*e + e*(b*e*(2 + p) - c*d*(3 + 2*p))))*(-b + 
 Sqrt[b^2 - 4*a*c] - 2*c*x)*(d + e*x)^(5 - 2*(3 + p))*(a + b*x + c*x^2)^p* 
Hypergeometric2F1[-p, 5 - 2*(3 + p), 6 - 2*(3 + p), (-4*c*Sqrt[b^2 - 4*a*c 
]*(d + e*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 
 2*c*x))])/(2*(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 - b*d*e + a*e^2)* 
(5 - 2*(3 + p))*(((2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(b + Sqrt[b^2 - 4*a* 
c] + 2*c*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 
 2*c*x)))^p))/((c*d^2 - b*d*e + a*e^2)*(3 - 2*(3 + p)))) + ((-3*c*d*e + e* 
(b*e*(4 + p) - c*d*(5 + 2*p)))*((e*(d + e*x)^(2 - 2*(3 + p))*(a + b*x + c* 
x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(2 - 2*(3 + p))) + (2*c*((e*(d + e* 
x)^(4 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(4 
- 2*(3 + p))) + ((2*c*d - b*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*(d + e*x)^ 
(5 - 2*(3 + p))*(a + b*x + c*x^2)^p*Hypergeometric2F1[-p, 5 - 2*(3 + p), 6 
 - 2*(3 + p), (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - b*e - Sqrt[b^2 
- 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))])/(2*(2*c*d - b*e + Sqrt[b...
 

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 842, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1167, 1238, 1238, 1228, 1155}

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.

\(\displaystyle \int (d+e x)^{-2 p-6} \left (a+b x+c x^2\right )^p \, dx\)

\(\Big \downarrow \) 1167

\(\displaystyle -\frac {\int (d+e x)^{-2 p-5} (b e (p+4)-c d (2 p+5)+3 c e x) \left (c x^2+b x+a\right )^pdx}{(2 p+5) \left (a e^2-b d e+c d^2\right )}-\frac {e (d+e x)^{-2 p-5} \left (a+b x+c x^2\right )^{p+1}}{(2 p+5) \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1238

\(\displaystyle -\frac {\frac {e (p+4) (2 c d-b e) (d+e x)^{-2 (p+2)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) \left (a e^2-b d e+c d^2\right )}-\frac {\int (d+e x)^{-2 (p+2)} \left (2 c^2 \left (2 p^2+9 p+10\right ) d^2+b^2 e^2 \left (p^2+7 p+12\right )-2 c e \left (3 a e (p+2)+2 b d \left (p^2+5 p+7\right )\right )-2 c e (2 c d-b e) (p+4) x\right ) \left (c x^2+b x+a\right )^pdx}{2 (p+2) \left (a e^2-b d e+c d^2\right )}}{(2 p+5) \left (a e^2-b d e+c d^2\right )}-\frac {e (d+e x)^{-2 p-5} \left (a+b x+c x^2\right )^{p+1}}{(2 p+5) \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1238

\(\displaystyle -\frac {\frac {e (p+4) (2 c d-b e) (d+e x)^{-2 (p+2)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\int (d+e x)^{-2 p-3} \left (-2 c^3 \left (4 p^3+24 p^2+47 p+30\right ) d^3+2 c^2 e \left (a e \left (10 p^2+43 p+42\right )+b d \left (6 p^3+37 p^2+76 p+54\right )\right ) d+b^3 e^3 \left (p^3+9 p^2+26 p+24\right )-b c e^2 (p+3) \left (2 a e (5 p+8)+b d \left (6 p^2+25 p+28\right )\right )+c e \left (2 c^2 \left (2 p^2+11 p+18\right ) d^2+b^2 e^2 \left (p^2+7 p+12\right )-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )\right ) x\right ) \left (c x^2+b x+a\right )^pdx}{(2 p+3) \left (a e^2-b d e+c d^2\right )}-\frac {e (d+e x)^{-2 p-3} \left (a+b x+c x^2\right )^{p+1} \left (-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )+b^2 e^2 \left (p^2+7 p+12\right )+2 c^2 d^2 \left (2 p^2+11 p+18\right )\right )}{(2 p+3) \left (a e^2-b d e+c d^2\right )}}{2 (p+2) \left (a e^2-b d e+c d^2\right )}}{(2 p+5) \left (a e^2-b d e+c d^2\right )}-\frac {e (d+e x)^{-2 p-5} \left (a+b x+c x^2\right )^{p+1}}{(2 p+5) \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {\frac {e (p+4) (2 c d-b e) (d+e x)^{-2 (p+2)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\frac {e (p+3) (2 c d-b e) (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1} \left (-2 c e \left (a e (5 p+8)+b d \left (2 p^2+7 p+8\right )\right )+b^2 e^2 \left (p^2+6 p+8\right )+2 c^2 d^2 \left (2 p^2+7 p+8\right )\right )}{2 (p+1) \left (a e^2-b d e+c d^2\right )}-\frac {(p+2) \left (12 c^2 e^2 \left (a^2 e^2+2 a b d e (2 p+5)+b^2 d^2 \left (2 p^2+9 p+10\right )\right )-4 b^2 c e^3 (p+3) (3 a e+b d (2 p+5))-8 c^3 d^2 e (2 p+5) (3 a e+b d (2 p+3))+b^4 e^4 \left (p^2+7 p+12\right )+4 c^4 d^4 \left (4 p^2+16 p+15\right )\right ) \int (d+e x)^{-2 (p+1)} \left (c x^2+b x+a\right )^pdx}{2 \left (a e^2-b d e+c d^2\right )}}{(2 p+3) \left (a e^2-b d e+c d^2\right )}-\frac {e (d+e x)^{-2 p-3} \left (a+b x+c x^2\right )^{p+1} \left (-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )+b^2 e^2 \left (p^2+7 p+12\right )+2 c^2 d^2 \left (2 p^2+11 p+18\right )\right )}{(2 p+3) \left (a e^2-b d e+c d^2\right )}}{2 (p+2) \left (a e^2-b d e+c d^2\right )}}{(2 p+5) \left (a e^2-b d e+c d^2\right )}-\frac {e (d+e x)^{-2 p-5} \left (a+b x+c x^2\right )^{p+1}}{(2 p+5) \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1155

\(\displaystyle -\frac {e \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-5}}{\left (c d^2-b e d+a e^2\right ) (2 p+5)}-\frac {\frac {e (2 c d-b e) (p+4) (d+e x)^{-2 (p+2)} \left (c x^2+b x+a\right )^{p+1}}{2 \left (c d^2-b e d+a e^2\right ) (p+2)}-\frac {-\frac {e \left (2 c^2 \left (2 p^2+11 p+18\right ) d^2+b^2 e^2 \left (p^2+7 p+12\right )-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-3}}{\left (c d^2-b e d+a e^2\right ) (2 p+3)}-\frac {\frac {e (2 c d-b e) (p+3) \left (2 c^2 \left (2 p^2+7 p+8\right ) d^2+b^2 e^2 \left (p^2+6 p+8\right )-2 c e \left (a e (5 p+8)+b d \left (2 p^2+7 p+8\right )\right )\right ) (d+e x)^{-2 (p+1)} \left (c x^2+b x+a\right )^{p+1}}{2 \left (c d^2-b e d+a e^2\right ) (p+1)}-\frac {(p+2) \left (4 c^4 \left (4 p^2+16 p+15\right ) d^4-8 c^3 e (2 p+5) (3 a e+b d (2 p+3)) d^2+b^4 e^4 \left (p^2+7 p+12\right )-4 b^2 c e^3 (p+3) (3 a e+b d (2 p+5))+12 c^2 e^2 \left (b^2 \left (2 p^2+9 p+10\right ) d^2+2 a b e (2 p+5) d+a^2 e^2\right )\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right ) \left (\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )^{-p} (d+e x)^{-2 p-1} \left (c x^2+b x+a\right )^p \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )}{2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right ) (2 p+1)}}{\left (c d^2-b e d+a e^2\right ) (2 p+3)}}{2 \left (c d^2-b e d+a e^2\right ) (p+2)}}{\left (c d^2-b e d+a e^2\right ) (2 p+5)}\)

Input:

Int[(d + e*x)^(-6 - 2*p)*(a + b*x + c*x^2)^p,x]
 

Output:

-((e*(d + e*x)^(-5 - 2*p)*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e 
^2)*(5 + 2*p))) - ((e*(2*c*d - b*e)*(4 + p)*(a + b*x + c*x^2)^(1 + p))/(2* 
(c*d^2 - b*d*e + a*e^2)*(2 + p)*(d + e*x)^(2*(2 + p))) - (-((e*(b^2*e^2*(1 
2 + 7*p + p^2) + 2*c^2*d^2*(18 + 11*p + 2*p^2) - 2*c*e*(3*a*e*(2 + p) + b* 
d*(18 + 11*p + 2*p^2)))*(d + e*x)^(-3 - 2*p)*(a + b*x + c*x^2)^(1 + p))/(( 
c*d^2 - b*d*e + a*e^2)*(3 + 2*p))) - ((e*(2*c*d - b*e)*(3 + p)*(b^2*e^2*(8 
 + 6*p + p^2) + 2*c^2*d^2*(8 + 7*p + 2*p^2) - 2*c*e*(a*e*(8 + 5*p) + b*d*( 
8 + 7*p + 2*p^2)))*(a + b*x + c*x^2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)*( 
1 + p)*(d + e*x)^(2*(1 + p))) - ((2 + p)*(b^4*e^4*(12 + 7*p + p^2) + 4*c^4 
*d^4*(15 + 16*p + 4*p^2) - 8*c^3*d^2*e*(5 + 2*p)*(3*a*e + b*d*(3 + 2*p)) - 
 4*b^2*c*e^3*(3 + p)*(3*a*e + b*d*(5 + 2*p)) + 12*c^2*e^2*(a^2*e^2 + 2*a*b 
*d*e*(5 + 2*p) + b^2*d^2*(10 + 9*p + 2*p^2)))*(b - Sqrt[b^2 - 4*a*c] + 2*c 
*x)*(d + e*x)^(-1 - 2*p)*(a + b*x + c*x^2)^p*Hypergeometric2F1[-1 - 2*p, - 
p, -2*p, (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - (b + Sqrt[b^2 - 4*a* 
c])*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))])/(2*(2*c*d - (b - Sqrt[b^2 - 4*a* 
c])*e)*(c*d^2 - b*d*e + a*e^2)*(1 + 2*p)*(((2*c*d - (b - Sqrt[b^2 - 4*a*c] 
)*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e) 
*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))^p))/((c*d^2 - b*d*e + a*e^2)*(3 + 2*p)) 
)/(2*(c*d^2 - b*d*e + a*e^2)*(2 + p)))/((c*d^2 - b*d*e + a*e^2)*(5 + 2*p))
 

Defintions of rubi rules used

rule 1155

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[(-(b - q + 2*c*x))*(d + e*x)^ 
(m + 1)*((a + b*x + c*x^2)^p/((m + 1)*(2*c*d - b*e + e*q)*((2*c*d - b*e + e 
*q)*((b + q + 2*c*x)/((2*c*d - b*e - e*q)*(b - q + 2*c*x))))^p))*Hypergeome 
tric2F1[m + 1, -p, m + 2, -4*c*q*((d + e*x)/((2*c*d - b*e - e*q)*(b - q + 2 
*c*x)))], x]] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[m + 2*p + 2, 0]
 

rule 1167

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d 
^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[ 
(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1238

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && ILtQ[S 
implify[m + 2*p + 3], 0] && NeQ[m, -1]
 

Maple [F]

\[\int \left (e x +d \right )^{-6-2 p} \left (c \,x^{2}+b x +a \right )^{p}d x\]

Input:

int((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x)
 

Output:

int((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x)
 

Fricas [F]

\[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6} \,d x } \] Input:

integrate((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x, algorithm="fricas")
 

Output:

integral((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6), x)
 

Sympy [F(-1)]

Timed out. \[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(-6-2*p)*(c*x**2+b*x+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6} \,d x } \] Input:

integrate((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6), x)
 

Giac [F]

\[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6} \,d x } \] Input:

integrate((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6), x)
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+6}} \,d x \] Input:

int((a + b*x + c*x^2)^p/(d + e*x)^(2*p + 6),x)
 

Output:

int((a + b*x + c*x^2)^p/(d + e*x)^(2*p + 6), x)
 

Reduce [F]

\[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\int \frac {\left (c \,x^{2}+b x +a \right )^{p}}{\left (e x +d \right )^{2 p} d^{6}+6 \left (e x +d \right )^{2 p} d^{5} e x +15 \left (e x +d \right )^{2 p} d^{4} e^{2} x^{2}+20 \left (e x +d \right )^{2 p} d^{3} e^{3} x^{3}+15 \left (e x +d \right )^{2 p} d^{2} e^{4} x^{4}+6 \left (e x +d \right )^{2 p} d \,e^{5} x^{5}+\left (e x +d \right )^{2 p} e^{6} x^{6}}d x \] Input:

int((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x)
 

Output:

int((a + b*x + c*x**2)**p/((d + e*x)**(2*p)*d**6 + 6*(d + e*x)**(2*p)*d**5 
*e*x + 15*(d + e*x)**(2*p)*d**4*e**2*x**2 + 20*(d + e*x)**(2*p)*d**3*e**3* 
x**3 + 15*(d + e*x)**(2*p)*d**2*e**4*x**4 + 6*(d + e*x)**(2*p)*d*e**5*x**5 
 + (d + e*x)**(2*p)*e**6*x**6),x)