3.812 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (-\frac{1}{4},-2 p;\frac{3}{4};-\frac{b x^2}{a}\right )}{d \sqrt{d x}} \]

[Out]

(-2*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*Hypergeometric2F1[-1/4, -2*p, 3/4, -((b*x^2)/a
)])/(d*Sqrt[d*x]*(1 + (b*x^2)/a)^(2*p))

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Rubi [A]  time = 0.0630066, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{2 \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (-\frac{1}{4},-2 p;\frac{3}{4};-\frac{b x^2}{a}\right )}{d \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/(d*x)^(3/2),x]

[Out]

(-2*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*Hypergeometric2F1[-1/4, -2*p, 3/4, -((b*x^2)/a
)])/(d*Sqrt[d*x]*(1 + (b*x^2)/a)^(2*p))

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Rubi in Sympy [A]  time = 17.2758, size = 61, normalized size = 0.94 \[ - \frac{2 \left (1 + \frac{b x^{2}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{d \sqrt{d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**p/(d*x)**(3/2),x)

[Out]

-2*(1 + b*x**2/a)**(-2*p)*(a**2 + 2*a*b*x**2 + b**2*x**4)**p*hyper((-2*p, -1/4),
(3/4,), -b*x**2/a)/(d*sqrt(d*x))

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Mathematica [A]  time = 0.0261215, size = 54, normalized size = 0.83 \[ -\frac{2 x \left (\left (a+b x^2\right )^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-2 p} \, _2F_1\left (-\frac{1}{4},-2 p;\frac{3}{4};-\frac{b x^2}{a}\right )}{(d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/(d*x)^(3/2),x]

[Out]

(-2*x*((a + b*x^2)^2)^p*Hypergeometric2F1[-1/4, -2*p, 3/4, -((b*x^2)/a)])/((d*x)
^(3/2)*(1 + (b*x^2)/a)^(2*p))

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \[ \int{ \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p} \left ( dx \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(3/2),x)

[Out]

int((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\left (d x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/(d*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/(d*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/(d*x)^(3/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)^p/(sqrt(d*x)*d*x), 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((b**2*x**4+2*a*b*x**2+a**2)**p/(d*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\left (d x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/(d*x)^(3/2),x, algorithm="giac")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/(d*x)^(3/2), x)