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

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

[Out]

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

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Rubi [A]  time = 0.0207925, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 x^{2-n}}{b (4-3 n) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x/(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 = 3.62614, size = 31, normalized size = 1.11 \[ \frac{2 x^{- \frac{n}{2}} x^{- \frac{3 n}{2} + 2} \sqrt{b x^{n}}}{b^{2} \left (- 3 n + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(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 = 0.0120141, size = 22, normalized size = 0.79 \[ \frac{x^2}{\left (2-\frac{3 n}{2}\right ) \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(b*x^n)^(3/2),x]

[Out]

x^2/((2 - (3*n)/2)*(b*x^n)^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^n)^(3/2),x)

[Out]

-2*x^2/(3*n-4)/(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(x/(b*x^n)^(3/2),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(x/(b*x^n)^(3/2),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(x/(b*x**n)**(3/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^n)^(3/2), x)