3.2566 \(\int (d+e x)^{-2-2 p} \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=248 \[ \frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{(2 p+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )} \]

[Out]

((b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(d + e*x)^(-1 - 2*p)*(a + b*x + c*x^2)^p*Hyperg
eometric2F1[-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*c*d - (b - Sqrt[b
^2 - 4*a*c])*e)*(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)

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Rubi [A]  time = 0.357234, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{(2 p+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )} \]

Antiderivative was successfully verified.

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

[Out]

((b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(d + e*x)^(-1 - 2*p)*(a + b*x + c*x^2)^p*Hyperg
eometric2F1[-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*c*d - (b - Sqrt[b
^2 - 4*a*c])*e)*(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)

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Rubi in Sympy [A]  time = 23.502, size = 226, normalized size = 0.91 \[ - \frac{\left (\frac{\left (b + 2 c x + \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right )}{\left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right )}\right )^{- p} \left (d + e x\right )^{- 2 p - 1} \left (a + b x + c x^{2}\right )^{p} \left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} - 2 p - 1, - p \\ - 2 p \end{matrix}\middle |{\frac{4 c \left (d + e x\right ) \sqrt{- 4 a c + b^{2}}}{\left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right )}} \right )}}{\left (2 p + 1\right ) \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-((b + 2*c*x + sqrt(-4*a*c + b**2))*(b*e - 2*c*d - e*sqrt(-4*a*c + b**2))/((b +
2*c*x - sqrt(-4*a*c + b**2))*(b*e - 2*c*d + e*sqrt(-4*a*c + b**2))))**(-p)*(d +
e*x)**(-2*p - 1)*(a + b*x + c*x**2)**p*(b + 2*c*x - sqrt(-4*a*c + b**2))*hyper((
-2*p - 1, -p), (-2*p,), 4*c*(d + e*x)*sqrt(-4*a*c + b**2)/((b + 2*c*x - sqrt(-4*
a*c + b**2))*(b*e - 2*c*d + e*sqrt(-4*a*c + b**2))))/((2*p + 1)*(b*e - 2*c*d - e
*sqrt(-4*a*c + b**2)))

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Mathematica [A]  time = 1.2896, size = 248, normalized size = 1. \[ -\frac{(d+e x)^{-2 p-1} (a+x (b+c x))^p \left (\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}\right )^{1-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}\right )^{-p} \left (1-\frac{2 c (d+e x)}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}\right )^{2 p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d\right ) \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}\right )}{2 e p+e} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.213, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-2-2\,p} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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