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

Optimal. Leaf size=24 \[ -\frac{2 x^{-n}}{3 b n \sqrt{b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.0134057, antiderivative size = 24, 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 \[ -\frac{2 x^{-n}}{3 b n \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(b*x^n)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.811, size = 14, normalized size = 0.58 \[ - \frac{2}{3 n \left (b x^{n}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00612223, size = 16, normalized size = 0.67 \[ -\frac{2}{3 n \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 13, normalized size = 0.5 \[ -{\frac{2}{3\,n} \left ( b{x}^{n} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^n)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279358, size = 27, normalized size = 1.12 \[ -\frac{2}{3 \, \sqrt{b x^{n}} b n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.45232, size = 26, normalized size = 1.08 \[ \begin{cases} - \frac{2}{3 b^{\frac{3}{2}} n \left (x^{n}\right )^{\frac{3}{2}}} & \text{for}\: n \neq 0 \\\frac{\log{\left (x \right )}}{b^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2/(3*b**(3/2)*n*(x**n)**(3/2)), Ne(n, 0)), (log(x)/b**(3/2), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x^{n}\right )^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n)^(3/2)*x), x)