3.182 \(\int x^2 \left (a x^n\right )^{-1/n} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{2} x^3 \left (a x^n\right )^{-1/n} \]

[Out]

x^3/(2*(a*x^n)^n^(-1))

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Rubi [A]  time = 0.0079695, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{2} x^3 \left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a*x^n)^n^(-1),x]

[Out]

x^3/(2*(a*x^n)^n^(-1))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ x \left (a x^{n}\right )^{- \frac{1}{n}} \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*x**n)**(-1/n)*Integral(x, x)

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Mathematica [A]  time = 0.00417418, size = 18, normalized size = 1. \[ \frac{1}{2} x^3 \left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a*x^n)^n^(-1),x]

[Out]

x^3/(2*(a*x^n)^n^(-1))

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Maple [A]  time = 0.002, size = 17, normalized size = 0.9 \[{\frac{{x}^{3}}{2\,\sqrt [n]{a{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/((a*x^n)^(1/n)),x)

[Out]

1/2*x^3/((a*x^n)^(1/n))

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Maxima [A]  time = 1.45075, size = 28, normalized size = 1.56 \[ \frac{1}{2} \, a^{-\frac{1}{n}} x^{3}{\left (x^{n}\right )}^{-\frac{1}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^(-1/n)*x^3*(x^n)^(-1/n)

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Fricas [A]  time = 0.226091, size = 16, normalized size = 0.89 \[ \frac{x^{2}}{2 \, a^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2/a^(1/n)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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GIAC/XCAS [A]  time = 0.222881, size = 18, normalized size = 1. \[ \frac{1}{2} \, x^{2} e^{\left (-\frac{{\rm ln}\left (a\right )}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2*e^(-ln(a)/n)