\(\int (a x^n)^{-1/n} \, dx\) [184]

Optimal result

Integrand size = 11, antiderivative size = 15 \[ \int \left (a x^n\right )^{-1/n} \, dx=x \left (a x^n\right )^{-1/n} \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 29} \[ \int \left (a x^n\right )^{-1/n} \, dx=x \log (x) \left (a x^n\right )^{-1/n} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (a x^n\right )^{-1/n} \, dx=x \left (a x^n\right )^{-1/n} \log (x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.93

method	result	size
norman	\(\frac {x \ln \left (a \,{\mathrm e}^{n \ln \left (x \right )}\right ) {\mathrm e}^{-\frac {\ln \left (a \,{\mathrm e}^{n \ln \left (x \right )}\right )}{n}}}{n}\)	\(29\)

[In]

int(1/((a*x^n)^(1/n)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \left (a x^n\right )^{-1/n} \, dx=\frac {\log \left (x\right )}{a^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \left (a x^n\right )^{-1/n} \, dx=x \left (a x^{n}\right )^{- \frac {1}{n}} \log {\left (x \right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (a x^n\right )^{-1/n} \, dx=\int { \frac {1}{\left (a x^{n}\right )^{\left (\frac {1}{n}\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/((a*x^n)^(1/n)), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \left (a x^n\right )^{-1/n} \, dx=\frac {\log \left (x\right )}{a^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \left (a x^n\right )^{-1/n} \, dx=\int \frac {1}{{\left (a\,x^n\right )}^{1/n}} \,d x \]

[In]

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

[Out]

int(1/(a*x^n)^(1/n), x)