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

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 = 0.0015194, 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, Rules used = {15, 29} \[ x \log (x) \left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^n)^(-n^(-1)),x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \left (a x^n\right )^{-1/n} \, dx &=\left (x \left (a x^n\right )^{-1/n}\right ) \int \frac{1}{x} \, dx\\ &=x \left (a x^n\right )^{-1/n} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0007678, 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 = 0.017, size = 29, normalized size = 1.9 \begin{align*}{\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}} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a x^{n}\right )^{\left (\frac{1}{n}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a*x^n)^(1/n)), x)

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Fricas [A]  time = 1.7949, size = 22, normalized size = 1.47 \begin{align*} \frac{\log \left (x\right )}{a^{\left (\frac{1}{n}\right )}} \end{align*}

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. \begin{align*} \int \left (a x^{n}\right )^{- \frac{1}{n}}\, dx \end{align*}

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 [A]  time = 1.12788, size = 14, normalized size = 0.93 \begin{align*} \frac{\log \left (x\right )}{a^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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