3.1882 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^5} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{4 a \left (a x^2+b\right )^2} \]

[Out]

-1/(4*a*(b + a*x^2)^2)

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Rubi [A]  time = 0.019997, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{1}{4 a \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^3*x^5),x]

[Out]

-1/(4*a*(b + a*x^2)^2)

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Rubi in Sympy [A]  time = 3.50583, size = 14, normalized size = 0.88 \[ - \frac{1}{4 a \left (a x^{2} + b\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**3/x**5,x)

[Out]

-1/(4*a*(a*x**2 + b)**2)

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Mathematica [A]  time = 0.0046052, size = 16, normalized size = 1. \[ -\frac{1}{4 a \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^3*x^5),x]

[Out]

-1/(4*a*(b + a*x^2)^2)

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Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \[ -{\frac{1}{4\,a \left ( a{x}^{2}+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3/x^5,x)

[Out]

-1/4/a/(a*x^2+b)^2

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Maxima [A]  time = 1.4407, size = 35, normalized size = 2.19 \[ -\frac{1}{4 \,{\left (a^{3} x^{4} + 2 \, a^{2} b x^{2} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^5),x, algorithm="maxima")

[Out]

-1/4/(a^3*x^4 + 2*a^2*b*x^2 + a*b^2)

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Fricas [A]  time = 0.216941, size = 35, normalized size = 2.19 \[ -\frac{1}{4 \,{\left (a^{3} x^{4} + 2 \, a^{2} b x^{2} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^5),x, algorithm="fricas")

[Out]

-1/4/(a^3*x^4 + 2*a^2*b*x^2 + a*b^2)

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Sympy [A]  time = 1.59476, size = 27, normalized size = 1.69 \[ - \frac{1}{4 a^{3} x^{4} + 8 a^{2} b x^{2} + 4 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3/x**5,x)

[Out]

-1/(4*a**3*x**4 + 8*a**2*b*x**2 + 4*a*b**2)

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GIAC/XCAS [A]  time = 0.233716, size = 19, normalized size = 1.19 \[ -\frac{1}{4 \,{\left (a x^{2} + b\right )}^{2} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^5),x, algorithm="giac")

[Out]

-1/4/((a*x^2 + b)^2*a)