3.1916 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\sqrt{a+\frac{b}{x^2}}}{b} \]

[Out]

-(Sqrt[a + b/x^2]/b)

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Rubi [A]  time = 0.0289002, antiderivative size = 16, 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 \[ -\frac{\sqrt{a+\frac{b}{x^2}}}{b} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^2]*x^3),x]

[Out]

-(Sqrt[a + b/x^2]/b)

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Rubi in Sympy [A]  time = 2.11404, size = 12, normalized size = 0.75 \[ - \frac{\sqrt{a + \frac{b}{x^{2}}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b/x**2)/b

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Mathematica [A]  time = 0.0259794, size = 16, normalized size = 1. \[ -\frac{\sqrt{a+\frac{b}{x^2}}}{b} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b/x^2]/b)

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Maple [A]  time = 0.007, size = 29, normalized size = 1.8 \[ -{\frac{a{x}^{2}+b}{b{x}^{2}}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(1/2)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.43617, size = 19, normalized size = 1.19 \[ -\frac{\sqrt{a + \frac{b}{x^{2}}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b/x^2)/b

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Fricas [A]  time = 0.226257, size = 24, normalized size = 1.5 \[ -\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt((a*x^2 + b)/x^2)/b

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Sympy [A]  time = 4.08182, size = 26, normalized size = 1.62 \[ \begin{cases} - \frac{\sqrt{a + \frac{b}{x^{2}}}}{b} & \text{for}\: b \neq 0 \\- \frac{1}{2 \sqrt{a} x^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221484, size = 19, normalized size = 1.19 \[ -\frac{\sqrt{a + \frac{b}{x^{2}}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b/x^2)/b