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

Optimal. Leaf size=19 \[ -\frac{1}{2 b x \sqrt{b x^2}} \]

[Out]

-1/(2*b*x*Sqrt[b*x^2])

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Rubi [A]  time = 0.00802965, antiderivative size = 19, 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 \[ -\frac{1}{2 b x \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2)^(-3/2),x]

[Out]

-1/(2*b*x*Sqrt[b*x^2])

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Rubi in Sympy [A]  time = 1.56596, size = 17, normalized size = 0.89 \[ - \frac{\sqrt{b x^{2}}}{2 b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00151832, size = 14, normalized size = 0.74 \[ -\frac{x}{2 \left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2)^(-3/2),x]

[Out]

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

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Maple [A]  time = 0., size = 11, normalized size = 0.6 \[ -{\frac{x}{2} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44336, size = 11, normalized size = 0.58 \[ -\frac{1}{2 \, b^{\frac{3}{2}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229916, size = 20, normalized size = 1.05 \[ -\frac{\sqrt{b x^{2}}}{2 \, b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.01922, size = 15, normalized size = 0.79 \[ - \frac{x}{2 b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.55815, size = 4, normalized size = 0.21 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x