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

Optimal. Leaf size=28 \[ -\frac{2 x^{-n-2}}{b (3 n+4) \sqrt{b x^n}} \]

[Out]

(-2*x^(-2 - n))/(b*(4 + 3*n)*Sqrt[b*x^n])

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Rubi [A]  time = 0.0193791, antiderivative size = 28, 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 \[ -\frac{2 x^{-n-2}}{b (3 n+4) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(b*x^n)^(3/2)),x]

[Out]

(-2*x^(-2 - n))/(b*(4 + 3*n)*Sqrt[b*x^n])

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Rubi in Sympy [A]  time = 3.76249, size = 34, normalized size = 1.21 \[ - \frac{2 x^{- \frac{n}{2}} x^{- \frac{3 n}{2} - 2} \sqrt{b x^{n}}}{b^{2} \left (3 n + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**(-n/2)*x**(-3*n/2 - 2)*sqrt(b*x**n)/(b**2*(3*n + 4))

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Mathematica [A]  time = 0.0175594, size = 22, normalized size = 0.79 \[ \frac{1}{\left (-\frac{3 n}{2}-2\right ) x^2 \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

1/((-2 - (3*n)/2)*x^2*(b*x^n)^(3/2))

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Maple [A]  time = 0.003, size = 20, normalized size = 0.7 \[ -2\,{\frac{1}{{x}^{2} \left ( 4+3\,n \right ) \left ( b{x}^{n} \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^n)^(3/2),x)

[Out]

-2/x^2/(4+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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n)^(3/2)*x^3), x)