3.1743 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^p}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (-\frac{1}{2},-2 p;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{e \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.119446, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ -\frac{2 \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (-\frac{1}{2},-2 p;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.3821, size = 73, normalized size = 0.9 \[ - \frac{2 \left (\frac{e \left (a + b x\right )}{a e - b d}\right )^{- 2 p} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{e \sqrt{d + e x}} \]

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)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0519624, size = 71, normalized size = 0.88 \[ -\frac{2 \left ((a+b x)^2\right )^p \left (\frac{e (a+b x)}{a e-b d}\right )^{-2 p} \, _2F_1\left (-\frac{1}{2},-2 p;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.106, size = 0, normalized size = 0. \[ \int{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}\, dx \]

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)^(3/2),x)

[Out]

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

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

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)^(3/2),x, algorithm="maxima")

[Out]

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

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

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)^(3/2),x, algorithm="fricas")

[Out]

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

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

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)**(3/2),x)

[Out]

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

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

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)^(3/2),x, algorithm="giac")

[Out]

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