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

Optimal. Leaf size=19 \[ -\frac{1}{3 b x^2 \sqrt{b x^2}} \]

[Out]

-1/(3*b*x^2*Sqrt[b*x^2])

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Rubi [A]  time = 0.00816629, 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 \[ -\frac{1}{3 b x^2 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(b*x^2)^(3/2)),x]

[Out]

-1/(3*b*x^2*Sqrt[b*x^2])

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Rubi in Sympy [A]  time = 2.28735, size = 12, normalized size = 0.63 \[ - \frac{1}{3 \left (b x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00503237, size = 17, normalized size = 0.89 \[ -\frac{b x^2}{3 \left (b x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 10, normalized size = 0.5 \[ -{\frac{1}{3} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.41441, size = 11, normalized size = 0.58 \[ -\frac{1}{3 \, b^{\frac{3}{2}} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216853, size = 20, normalized size = 1.05 \[ -\frac{\sqrt{b x^{2}}}{3 \, b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(b*x^2)/(b^2*x^4)

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Sympy [A]  time = 2.26645, size = 15, normalized size = 0.79 \[ - \frac{1}{3 b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221404, size = 20, normalized size = 1.05 \[ -\frac{1}{3 \, \sqrt{b x^{2}} b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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