3.87 \(\int \left (\frac{b}{x^2}\right )^{3/2} \, dx\)

Optimal. Leaf size=17 \[ -\frac{b \sqrt{\frac{b}{x^2}}}{2 x} \]

[Out]

-(b*Sqrt[b/x^2])/(2*x)

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Rubi [A]  time = 0.00790038, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{b \sqrt{\frac{b}{x^2}}}{2 x} \]

Antiderivative was successfully verified.

[In]  Int[(b/x^2)^(3/2),x]

[Out]

-(b*Sqrt[b/x^2])/(2*x)

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Rubi in Sympy [A]  time = 1.32254, size = 14, normalized size = 0.82 \[ - \frac{b \sqrt{\frac{b}{x^{2}}}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*sqrt(b/x**2)/(2*x)

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Mathematica [A]  time = 0.00207925, size = 14, normalized size = 0.82 \[ -\frac{1}{2} x \left (\frac{b}{x^2}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(b/x^2)^(3/2),x]

[Out]

-((b/x^2)^(3/2)*x)/2

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Maple [A]  time = 0.003, size = 11, normalized size = 0.7 \[ -{\frac{x}{2} \left ({\frac{b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b/x^2)^(3/2),x)

[Out]

-1/2*x*(b/x^2)^(3/2)

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Maxima [A]  time = 1.45163, size = 14, normalized size = 0.82 \[ -\frac{1}{2} \, x \left (\frac{b}{x^{2}}\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x*(b/x^2)^(3/2)

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Fricas [A]  time = 0.242181, size = 18, normalized size = 1.06 \[ -\frac{b \sqrt{\frac{b}{x^{2}}}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b*sqrt(b/x^2)/x

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Sympy [A]  time = 1.94604, size = 17, normalized size = 1. \[ - \frac{b^{\frac{3}{2}} x \left (\frac{1}{x^{2}}\right )^{\frac{3}{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b/x**2)**(3/2),x)

[Out]

-b**(3/2)*x*(x**(-2))**(3/2)/2

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GIAC/XCAS [A]  time = 0.230104, size = 14, normalized size = 0.82 \[ -\frac{b^{\frac{3}{2}}{\rm sign}\left (x\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b^(3/2)*sign(x)/x^2