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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = 0.00, size = 15, normalized size = 1.00 \[ x \log (x) \left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.85, size = 10, normalized size = 0.67 \[ \frac {\log \relax (x)}{a^{\left (\frac {1}{n}\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 10, normalized size = 0.67 \[ \frac {\log \relax (x)}{a^{\left (\frac {1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.02, size = 29, normalized size = 1.93 \[ \frac {x \,{\mathrm e}^{-\frac {\ln \left (a \,{\mathrm e}^{n \ln \relax (x )}\right )}{n}} \ln \left (a \,{\mathrm e}^{n \ln \relax (x )}\right )}{n} \]

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))))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a x^{n}\right )^{\left (\frac {1}{n}\right )}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{{\left (a\,x^n\right )}^{1/n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*x^n)^(1/n), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a x^{n}\right )^{- \frac {1}{n}}\, dx \]

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)

________________________________________________________________________________________