3.1739 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^p}{(d+e x)^3} \, dx\)

Optimal. Leaf size=74 \[ \frac{b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (3,2 p+1;2 (p+1);-\frac{e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)^3} \]

[Out]

(b^2*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[3, 1 + 2*p, 2*(1 +
p), -((e*(a + b*x))/(b*d - a*e))])/((b*d - a*e)^3*(1 + 2*p))

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Rubi [A]  time = 0.0901418, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (3,2 p+1;2 (p+1);-\frac{e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^p/(d + e*x)^3,x]

[Out]

(b^2*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[3, 1 + 2*p, 2*(1 +
p), -((e*(a + b*x))/(b*d - a*e))])/((b*d - a*e)^3*(1 + 2*p))

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Rubi in Sympy [A]  time = 18.254, size = 82, normalized size = 1.11 \[ - \frac{b \left (a b + b^{2} x\right )^{- 2 p} \left (a b + b^{2} x\right )^{2 p + 1} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} 3, 2 p + 1 \\ 2 p + 2 \end{matrix}\middle |{\frac{e \left (a + b x\right )}{a e - b d}} \right )}}{\left (2 p + 1\right ) \left (a e - b d\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**p/(e*x+d)**3,x)

[Out]

-b*(a*b + b**2*x)**(-2*p)*(a*b + b**2*x)**(2*p + 1)*(a**2 + 2*a*b*x + b**2*x**2)
**p*hyper((3, 2*p + 1), (2*p + 2,), e*(a + b*x)/(a*e - b*d))/((2*p + 1)*(a*e - b
*d)**3)

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Mathematica [A]  time = 0.0664531, size = 0, normalized size = 0. \[ \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^p}{(d+e x)^3} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p/(d + e*x)^3,x]

[Out]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p/(d + e*x)^3, x]

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Maple [F]  time = 0.226, size = 0, normalized size = 0. \[ \int{\frac{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^p/(e*x+d)^3,x)

[Out]

int((b^2*x^2+2*a*b*x+a^2)^p/(e*x+d)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^p/(e*x + d)^3,x, algorithm="maxima")

[Out]

integrate((b^2*x^2 + 2*a*b*x + a^2)^p/(e*x + d)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^p/(e*x + d)^3,x, algorithm="fricas")

[Out]

integral((b^2*x^2 + 2*a*b*x + a^2)^p/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3),
x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{p}}{\left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**p/(e*x+d)**3,x)

[Out]

Integral(((a + b*x)**2)**p/(d + e*x)**3, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^p/(e*x + d)^3,x, algorithm="giac")

[Out]

integrate((b^2*x^2 + 2*a*b*x + a^2)^p/(e*x + d)^3, x)