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

Optimal. Leaf size=19 \[ -\frac{1}{5 b x^4 \sqrt{b x^2}} \]

[Out]

-1/(5*b*x^4*Sqrt[b*x^2])

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Rubi [A]  time = 0.00850547, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{1}{5 b x^4 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(b*x^2)^(3/2)),x]

[Out]

-1/(5*b*x^4*Sqrt[b*x^2])

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Rubi in Sympy [A]  time = 2.50083, size = 12, normalized size = 0.63 \[ - \frac{b}{5 \left (b x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b/(5*(b*x**2)**(5/2))

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Mathematica [A]  time = 0.00471399, size = 16, normalized size = 0.84 \[ -\frac{1}{5 x^2 \left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(5*x^2*(b*x^2)^(3/2))

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Maple [A]  time = 0.003, size = 13, normalized size = 0.7 \[ -{\frac{1}{5\,{x}^{2}} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^2)^(3/2),x)

[Out]

-1/5/x^2/(b*x^2)^(3/2)

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Maxima [A]  time = 1.41939, size = 11, normalized size = 0.58 \[ -\frac{1}{5 \, b^{\frac{3}{2}} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5/(b^(3/2)*x^5)

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Fricas [A]  time = 0.215718, size = 20, normalized size = 1.05 \[ -\frac{\sqrt{b x^{2}}}{5 \, b^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*sqrt(b*x^2)/(b^2*x^6)

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Sympy [A]  time = 3.00533, size = 19, normalized size = 1. \[ - \frac{1}{5 b^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(5*b**(3/2)*x**2*(x**2)**(3/2))

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GIAC/XCAS [A]  time = 0.211928, size = 20, normalized size = 1.05 \[ -\frac{1}{5 \, \sqrt{b x^{2}} b x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5/(sqrt(b*x^2)*b*x^4)