3.1865 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^3} \, dx\)

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

[Out]

1/(2*b*(a + b/x^2))

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Rubi [A]  time = 0.0186636, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{1}{2 b \left (a+\frac{b}{x^2}\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^2*x^3),x]

[Out]

1/(2*b*(a + b/x^2))

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Rubi in Sympy [A]  time = 2.13541, size = 10, normalized size = 0.62 \[ \frac{1}{2 b \left (a + \frac{b}{x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*b*(a + b/x**2))

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Mathematica [A]  time = 0.00401547, size = 16, normalized size = 1. \[ -\frac{1}{2 a \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*a*(b + a*x^2))

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Maple [A]  time = 0., size = 15, normalized size = 0.9 \[ -{\frac{1}{ \left ( 2\,a{x}^{2}+2\,b \right ) a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^2/x^3,x)

[Out]

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

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Maxima [A]  time = 1.44247, size = 19, normalized size = 1.19 \[ \frac{1}{2 \,{\left (a + \frac{b}{x^{2}}\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/((a + b/x^2)*b)

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Fricas [A]  time = 0.214873, size = 20, normalized size = 1.25 \[ -\frac{1}{2 \,{\left (a^{2} x^{2} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(a^2*x^2 + a*b)

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Sympy [A]  time = 1.27337, size = 15, normalized size = 0.94 \[ - \frac{1}{2 a^{2} x^{2} + 2 a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a**2*x**2 + 2*a*b)

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GIAC/XCAS [A]  time = 0.220764, size = 19, normalized size = 1.19 \[ -\frac{1}{2 \,{\left (a x^{2} + b\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/((a*x^2 + b)*a)