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

Optimal. Leaf size=20 \[ \frac{x^{m+1} \left (a x^n\right )^{-1/n}}{m} \]

[Out]

x^(1 + m)/(m*(a*x^n)^n^(-1))

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Rubi [A]  time = 0.0026378, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 30} \[ \frac{x^{m+1} \left (a x^n\right )^{-1/n}}{m} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^(1 + m)/(m*(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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0041813, size = 20, normalized size = 1. \[ \frac{x^{m+1} \left (a x^n\right )^{-1/n}}{m} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^(1 + m)/(m*(a*x^n)^n^(-1))

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Maple [A]  time = 0.002, size = 21, normalized size = 1.1 \begin{align*}{\frac{{x}^{1+m}}{m\sqrt [n]{a{x}^{n}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1+m)/m/((a*x^n)^(1/n))

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Maxima [A]  time = 1.01143, size = 36, normalized size = 1.8 \begin{align*} \frac{x e^{\left (m \log \left (x\right ) - \frac{\log \left (x^{n}\right )}{n}\right )}}{a^{\left (\frac{1}{n}\right )} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.81443, size = 23, normalized size = 1.15 \begin{align*} \frac{x^{m}}{a^{\left (\frac{1}{n}\right )} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.12959, size = 19, normalized size = 0.95 \begin{align*} \frac{x^{m}}{a^{\left (\frac{1}{n}\right )} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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