3.1931 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^3} \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{b \sqrt{a+\frac{b}{x^2}}} \]

[Out]

1/(b*Sqrt[a + b/x^2])

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Rubi [A]  time = 0.0291111, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{1}{b \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

1/(b*Sqrt[a + b/x^2])

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Rubi in Sympy [A]  time = 2.13514, size = 12, normalized size = 0.8 \[ \frac{1}{b \sqrt{a + \frac{b}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(b*sqrt(a + b/x**2))

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Mathematica [A]  time = 0.0166078, size = 15, normalized size = 1. \[ \frac{1}{b \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

1/(b*Sqrt[a + b/x^2])

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Maple [B]  time = 0.006, size = 28, normalized size = 1.9 \[{\frac{a{x}^{2}+b}{b{x}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(3/2)/x^3,x)

[Out]

(a*x^2+b)/x^2/b/((a*x^2+b)/x^2)^(3/2)

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Maxima [A]  time = 1.43849, size = 18, normalized size = 1.2 \[ \frac{1}{\sqrt{a + \frac{b}{x^{2}}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(sqrt(a + b/x^2)*b)

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Fricas [A]  time = 0.237408, size = 39, normalized size = 2.6 \[ \frac{x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a b x^{2} + b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2*sqrt((a*x^2 + b)/x^2)/(a*b*x^2 + b^2)

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Sympy [A]  time = 6.96624, size = 26, normalized size = 1.73 \[ \begin{cases} \frac{1}{b \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{3}{2}} x^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((1/(b*sqrt(a + b/x**2)), Ne(b, 0)), (-1/(2*a**(3/2)*x**2), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((a + b/x^2)^(3/2)*x^3), x)