3.2.80 \(\int x^m (a x^n)^{-1/n} \, dx\) [180]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^{m+1} \left (a x^n\right )^{-1/n}}{m} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {x^{1+m} \left (a x^n\right )^{-1/n}}{m} \end {gather*}

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 = 0.05, size = 21, normalized size = 1.05

method	result	size
gosper	\(\frac {x^{1+m} \left (a \,x^{n}\right )^{-\frac {1}{n}}}{m}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 27, normalized size = 1.35 \begin {gather*} \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 {gather*}

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 = 0.36, size = 14, normalized size = 0.70 \begin {gather*} \frac {x^{m}}{a^{\left (\frac {1}{n}\right )} m} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x x^{m} \left (a x^{n}\right )^{- \frac {1}{n}}}{m} & \text {for}\: m \neq 0 \\\int \left (a x^{n}\right )^{- \frac {1}{n}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*x**m/(m*(a*x**n)**(1/n)), Ne(m, 0)), (Integral((a*x**n)**(-1/n), x), True))

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Giac [A]
time = 2.06, size = 14, normalized size = 0.70 \begin {gather*} \frac {x^{m}}{a^{\left (\frac {1}{n}\right )} m} \end {gather*}

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)

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Mupad [B]
time = 1.13, size = 20, normalized size = 1.00 \begin {gather*} \frac {x^{m+1}}{m\,{\left (a\,x^n\right )}^{1/n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(m + 1)/(m*(a*x^n)^(1/n))

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