3.2569 \(\int (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=577 \[ \frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (2 c d-b e) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right ) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{4 (2 p+1) (2 p+3) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )^3}-\frac{e (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1} \left (-2 c e \left (a e (2 p+3)+b d \left (2 p^2+8 p+9\right )\right )+b^2 e^2 \left (p^2+5 p+6\right )+2 c^2 d^2 \left (2 p^2+8 p+9\right )\right )}{4 (p+1) (p+2) (2 p+3) \left (a e^2-b d e+c d^2\right )^3}-\frac{e (p+3) (2 c d-b e) (d+e x)^{-2 p-3} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) (2 p+3) \left (a e^2-b d e+c d^2\right )^2}-\frac{e (d+e x)^{-2 (p+2)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) \left (a e^2-b d e+c d^2\right )} \]

[Out]

-(e*(2*c*d - b*e)*(3 + p)*(d + e*x)^(-3 - 2*p)*(a + b*x + c*x^2)^(1 + p))/(2*(c*
d^2 - b*d*e + a*e^2)^2*(2 + p)*(3 + 2*p)) - (e*(b^2*e^2*(6 + 5*p + p^2) + 2*c^2*
d^2*(9 + 8*p + 2*p^2) - 2*c*e*(a*e*(3 + 2*p) + b*d*(9 + 8*p + 2*p^2)))*(a + b*x
+ c*x^2)^(1 + p))/(4*(c*d^2 - b*d*e + a*e^2)^3*(1 + p)*(2 + p)*(3 + 2*p)*(d + e*
x)^(2*(1 + p))) - (e*(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))) + ((2*c*d - b*e)*(b^2*e^2*(3 + p) + 2*c^2*d^2*(3 + 2*p
) - 2*c*e*(3*a*e + b*d*(3 + 2*p)))*(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))])/(4*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)^3*(1
+ 2*p)*(3 + 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)

_______________________________________________________________________________________

Rubi [A]  time = 2.40783, antiderivative size = 577, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (2 c d-b e) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right ) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{4 (2 p+1) (2 p+3) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )^3}-\frac{e (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1} \left (-2 c e \left (a e (2 p+3)+b d \left (2 p^2+8 p+9\right )\right )+b^2 e^2 \left (p^2+5 p+6\right )+2 c^2 d^2 \left (2 p^2+8 p+9\right )\right )}{4 (p+1) (p+2) (2 p+3) \left (a e^2-b d e+c d^2\right )^3}-\frac{e (p+3) (2 c d-b e) (d+e x)^{-2 p-3} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) (2 p+3) \left (a e^2-b d e+c d^2\right )^2}-\frac{e (d+e x)^{-2 (p+2)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+2) \left (a e^2-b d e+c d^2\right )} \]

Warning: Unable to verify antiderivative.

[In]  Int[(d + e*x)^(-5 - 2*p)*(a + b*x + c*x^2)^p,x]

[Out]

-(e*(2*c*d - b*e)*(3 + p)*(d + e*x)^(-3 - 2*p)*(a + b*x + c*x^2)^(1 + p))/(2*(c*
d^2 - b*d*e + a*e^2)^2*(2 + p)*(3 + 2*p)) - (e*(b^2*e^2*(6 + 5*p + p^2) + 2*c^2*
d^2*(9 + 8*p + 2*p^2) - 2*c*e*(a*e*(3 + 2*p) + b*d*(9 + 8*p + 2*p^2)))*(a + b*x
+ c*x^2)^(1 + p))/(4*(c*d^2 - b*d*e + a*e^2)^3*(1 + p)*(2 + p)*(3 + 2*p)*(d + e*
x)^(2*(1 + p))) - (e*(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))) + ((2*c*d - b*e)*(b^2*e^2*(3 + p) + 2*c^2*d^2*(3 + 2*p
) - 2*c*e*(3*a*e + b*d*(3 + 2*p)))*(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))])/(4*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)^3*(1
+ 2*p)*(3 + 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)

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(-5-2*p)*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [B]  time = 64.7326, size = 3457, normalized size = 5.99 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)^(-5 - 2*p)*(a + b*x + c*x^2)^p,x]

[Out]

-((((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/c)^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/c)^p*(
d + e*x)^(-4 - 2*p)*(a + b*x + c*x^2)^p*(1 - (2*c*(d + e*x))/(2*c*d + (-b + Sqrt
[b^2 - 4*a*c])*e))^(1 + p)*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])
*e))^p*(-3*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^4*p*(b + Sqrt[b^2 - 4*a*c] + 2*c
*x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 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))] - 11*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^4*p^2*(b + Sqrt[b^2 - 4*a
*c] + 2*c*x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 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))] - 12*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^4*p^3*(b + Sqrt[
b^2 - 4*a*c] + 2*c*x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 -
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))] - 4*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^4*p^4*(b
+ Sqrt[b^2 - 4*a*c] + 2*c*x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -
p, -3 - 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))] - 12*c^2*(2*c*d + (-b + Sqrt[b^2 - 4*a*c]
)*e)^2*p*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(d + e*x)^2*Gamma[-p]*Gamma[-2*(1 + p)]
*Hypergeometric2F1[1, -p, -1 - 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))] - 24*c^2*(2*c*d +
(-b + Sqrt[b^2 - 4*a*c])*e)^2*p^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(d + e*x)^2*Ga
mma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -1 - 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))] - 24*c^3*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*p*(b + Sqrt[b^2 - 4*a*c] +
2*c*x)*(d + e*x)^3*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -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))] - 6*c*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^3*p*(b + Sqrt
[b^2 - 4*a*c] + 2*c*x)*(d + e*x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1
, -p, -2*(1 + 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))] - 18*c*(2*c*d + (-b + Sqrt[b^2 - 4*a
*c])*e)^3*p^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(d + e*x)*Gamma[-p]*Gamma[-2*(1 +
p)]*Hypergeometric2F1[1, -p, -2*(1 + 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))] - 12*c*(2*c*d
+ (-b + Sqrt[b^2 - 4*a*c])*e)^3*p^3*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(d + e*x)*G
amma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -2*(1 + 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))] - 36*c^2*Sqrt[b^2 - 4*a*c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^2*p*(d
+ e*x)^2*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -1 - 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))] - 96*c^2*Sqrt[b^2 - 4*a*c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*
c])*e)^2*p^2*(d + e*x)^2*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p
, -1 - 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))] - 48*c^2*Sqrt[b^2 - 4*a*c]*(2*c*d + (-b +
Sqrt[b^2 - 4*a*c])*e)^2*p^3*(d + e*x)^2*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeomet
ric2F1[2, 1 - p, -1 - 2*p, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sqr
t[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] - 264*c^4*Sqrt[b^2 - 4*a*c]
*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 1 - 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))] - 176*c^4*Sqrt[b^2 - 4*a*c]*p*(d + e*x)^4*Gamma[-3 - 2*
p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 1 - 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))] -
216*c^3*Sqrt[b^2 - 4*a*c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*p*(d + e*x)^3*Ga
mma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - 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))] - 144*c^3*Sqrt[b^2 - 4*a*c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*p^2*(d +
e*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - 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))] - 144*c^4*Sqrt[b^2 - 4*a*c]*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1
- p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 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))] -
96*c^4*Sqrt[b^2 - 4*a*c]*p*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeomet
ricPFQ[{2, 2, 1 - p}, {1, 1 - 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))] - 72*c^3*Sqrt[b^2
- 4*a*c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*p*(d + e*x)^3*Gamma[-3 - 2*p]*Gamm
a[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, -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))] -
48*c^3*Sqrt[b^2 - 4*a*c]*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*p^2*(d + e*x)^3*G
amma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, -2*p}, (4*c*Sqr
t[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))] - 24*c^4*Sqrt[b^2 - 4*a*c]*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1
- p]*HypergeometricPFQ[{2, 2, 2, 1 - p}, {1, 1, 1 - 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))] - 16*c^4*Sqrt[b^2 - 4*a*c]*p*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*Hyper
geometricPFQ[{2, 2, 2, 1 - p}, {1, 1, 1 - 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*p)*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^4*(-4 - 2*p)*p*(1 + p)*(1 + 2*p)*(3
+ 2*p)*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x)^p*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c)
+ x)^p*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*((-(b*e) - Sqrt[b^2 - 4*a*c]*e - 2*c*e*x)
/(2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e))^p*((-(b*e) + Sqrt[b^2 - 4*a*c]*e - 2*c*e*x
)/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e))^p*Gamma[-p]*Gamma[-2*(1 + p)]))

_______________________________________________________________________________________

Maple [F]  time = 0.225, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-5-2\,p} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-5-2*p)*(c*x^2+b*x+a)^p,x)

[Out]

int((e*x+d)^(-5-2*p)*(c*x^2+b*x+a)^p,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 5),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 5), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 5),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 5), x)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(-5-2*p)*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 5),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 5), x)