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

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

[Out]

(x*Log[x])/(a*x^n)^n^(-1)

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Rubi [A]  time = 0.00702907, antiderivative size = 15, 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 \[ x \log (x) \left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^n)^(-n^(-1)),x]

[Out]

(x*Log[x])/(a*x^n)^n^(-1)

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Rubi in Sympy [A]  time = 1.35889, size = 12, normalized size = 0.8 \[ x \left (a x^{n}\right )^{- \frac{1}{n}} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00207765, size = 15, normalized size = 1. \[ x \log (x) \left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^n)^(-n^(-1)),x]

[Out]

(x*Log[x])/(a*x^n)^n^(-1)

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Maple [A]  time = 0.029, size = 29, normalized size = 1.9 \[{\frac{x\ln \left ( a{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n} \left ({{\rm e}^{{\frac{\ln \left ( a{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n}}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((a*x^n)^(1/n)),x)

[Out]

1/n*x*ln(a*exp(n*ln(x)))/exp(1/n*ln(a*exp(n*ln(x))))

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Maxima [A]  time = 1.57134, size = 14, normalized size = 0.93 \[ a^{-\frac{1}{n}} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231388, size = 14, normalized size = 0.93 \[ \frac{\log \left (x\right )}{a^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/a^(1/n)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a*x**n)**(-1/n), x)

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GIAC/XCAS [A]  time = 0.232602, size = 15, normalized size = 1. \[ e^{\left (-\frac{{\rm ln}\left (a\right )}{n}\right )}{\rm ln}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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