3.155 \(\int \frac{1}{x^4 \left (b x^n\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0197548, 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 \[ -\frac{2 x^{-n-3}}{3 b (n+2) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(b*x^n)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.61729, size = 34, normalized size = 1.21 \[ - \frac{2 x^{- \frac{n}{2}} x^{- \frac{3 n}{2} - 3} \sqrt{b x^{n}}}{3 b^{2} \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**n)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0144876, size = 22, normalized size = 0.79 \[ \frac{1}{\left (-\frac{3 n}{2}-3\right ) x^3 \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(b*x^n)^(3/2)),x]

[Out]

1/((-3 - (3*n)/2)*x^3*(b*x^n)^(3/2))

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Maple [A]  time = 0.002, size = 18, normalized size = 0.6 \[ -{\frac{2}{3\,{x}^{3} \left ( 2+n \right ) } \left ( b{x}^{n} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^n)^(3/2),x)

[Out]

-2/3/x^3/(2+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^4),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^4),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [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/x**4/(b*x**n)**(3/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x^{n}\right )^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n)^(3/2)*x^4),x, algorithm="giac")

[Out]

integrate(1/((b*x^n)^(3/2)*x^4), x)