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

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

[Out]

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

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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} \begin {gather*} x \log (x) \left (a x^n\right )^{-1/n} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} x \left (a x^n\right )^{-1/n} \log (x) \end {gather*}

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 = 0.03, size = 29, normalized size = 1.93

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

Verification of antiderivative is not currently implemented for this CAS.

[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))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.35, size = 10, normalized size = 0.67 \begin {gather*} \frac {\log \left (x\right )}{a^{\left (\frac {1}{n}\right )}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a x^{n}\right )^{- \frac {1}{n}}\, dx \end {gather*}

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 = 1.40, size = 10, normalized size = 0.67 \begin {gather*} \frac {\log \left (x\right )}{a^{\left (\frac {1}{n}\right )}} \end {gather*}

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)

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

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)

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