3.2568 \(\int (d+e x)^{-4-2 p} \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=442 \[ \frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right ) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{2 (2 p+1) (2 p+3) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{e (d+e x)^{-2 p-3} \left (a+b x+c x^2\right )^{p+1}}{(2 p+3) \left (a e^2-b d e+c d^2\right )}-\frac{e (p+2) (2 c d-b e) (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+1) (2 p+3) \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 1.01903, antiderivative size = 442, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right ) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{2 (2 p+1) (2 p+3) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{e (d+e x)^{-2 p-3} \left (a+b x+c x^2\right )^{p+1}}{(2 p+3) \left (a e^2-b d e+c d^2\right )}-\frac{e (p+2) (2 c d-b e) (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+1) (2 p+3) \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(-4 - 2*p)*(a + b*x + c*x^2)^p,x]

[Out]

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

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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)**(-4-2*p)*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

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Mathematica [B]  time = 43.335, size = 2072, normalized size = 4.69 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)^(-4 - 2*p)*(a + b*x + c*x^2)^p,x]

[Out]

(2^(1 - 2*p)*((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)^(1 - 2*(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 + Sqr
t[b^2 - 4*a*c])*e))^p*Gamma[-2*(1 + p)]*(Gamma[1 - 2*p]*Gamma[-p]*Hypergeometric
2F1[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))] + 3*p*Gamma[1 - 2*p]*Gamma[-p]*Hyperg
eometric2F1[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))] + 2*p^2*Gamma[1 - 2*p]*Gamma[
-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))] + (2*c*(d + e*x)*Ga
mma[1 - 2*p]*Gamma[-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))])
/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e) + (4*c*p*(d + e*x)*Gamma[1 - 2*p]*Gamma[-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))])/(2*c*d + (-b + Sqrt[
b^2 - 4*a*c])*e) + (4*c^2*(d + e*x)^2*Gamma[1 - 2*p]*Gamma[-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))])/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^2
- (6*c*Sqrt[b^2 - 4*a*c]*(d + e*x)*Gamma[1 - p]*Gamma[-2*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))])/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e
)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) - (8*c*Sqrt[b^2 - 4*a*c]*p*(d + e*x)*Gamma[1
- p]*Gamma[-2*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))])
/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) + (16*c^
2*Sqrt[b^2 - 4*a*c]*p*(d + e*x)^2*Gamma[1 - p]*Gamma[-2*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))])/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)
^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) + (24*c^3*Sqrt[b^2 - 4*a*c]*(d + e*x)^3*Gamm
a[1 - p]*Gamma[-2*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
))])/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^3*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) +
(2*c*Sqrt[b^2 - 4*a*c]*(d + e*x)*Gamma[1 - p]*Gamma[-2*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))])/((2*c*d + (-b + Sqrt[b^2 - 4
*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) - (8*c^2*Sqrt[b^2 - 4*a*c]*(d + e*x)^
2*Gamma[1 - p]*Gamma[-2*p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (4*c*S
qrt[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*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^2*(b + Sqrt[b^2 - 4*a
*c] + 2*c*x)) + (8*c^3*Sqrt[b^2 - 4*a*c]*(d + e*x)^3*Gamma[1 - p]*Gamma[-2*p]*Hy
pergeometricPFQ[{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))])/((2*c*d
+ (-b + Sqrt[b^2 - 4*a*c])*e)^3*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))))/(e*(1 - 2*(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*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[1 - 2*p]*Gamma[-2*p]*Gamma[-p])

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Maple [F]  time = 0.219, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-4-2\,p} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-4-2*p)*(c*x^2+b*x+a)^p,x)

[Out]

int((e*x+d)^(-4-2*p)*(c*x^2+b*x+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 4),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 4),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 4), x)

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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)**(-4-2*p)*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 4),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 4), x)