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

Optimal. Leaf size=15 \[ x^2 \left (a x^n\right )^{-1/n} \]

[Out]

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

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Rubi [A]  time = 0.00696795, antiderivative size = 15, 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 \[ x^2 \left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]  Int[x/(a*x^n)^n^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.12268, size = 10, normalized size = 0.67 \[ x^{2} \left (a x^{n}\right )^{- \frac{1}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0022866, size = 15, normalized size = 1. \[ x^2 \left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a*x^n)^n^(-1),x]

[Out]

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

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{\frac{x}{\sqrt [n]{a{x}^{n}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/((a*x^n)^(1/n)),x)

[Out]

int(x/((a*x^n)^(1/n)),x)

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Maxima [A]  time = 1.46423, size = 27, normalized size = 1.8 \[ a^{-\frac{1}{n}} x^{2}{\left (x^{n}\right )}^{-\frac{1}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224897, size = 12, normalized size = 0.8 \[ \frac{x}{a^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/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/((a*x**n)**(1/n)),x)

[Out]

Exception raised: RecursionError

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GIAC/XCAS [A]  time = 0.238496, size = 14, normalized size = 0.93 \[ x e^{\left (-\frac{{\rm ln}\left (a\right )}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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