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

Optimal. Leaf size=30 \[ \frac{2 x^{3-n}}{3 b (2-n) \sqrt{b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.0239139, antiderivative size = 30, 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 \[ \frac{2 x^{3-n}}{3 b (2-n) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(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 = 3.77391, size = 31, normalized size = 1.03 \[ \frac{2 x^{- \frac{n}{2}} x^{- \frac{3 n}{2} + 3} \sqrt{b x^{n}}}{3 b^{2} \left (- n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(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 = 0.0139887, size = 22, normalized size = 0.73 \[ \frac{x^3}{\left (3-\frac{3 n}{2}\right ) \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(b*x^n)^(3/2),x]

[Out]

x^3/((3 - (3*n)/2)*(b*x^n)^(3/2))

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Maple [A]  time = 0.003, size = 18, normalized size = 0.6 \[ -{\frac{2\,{x}^{3}}{-6+3\,n} \left ( b{x}^{n} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(b*x^n)^(3/2), x)