3.187 \(\int x^m (a x^n)^{-\frac{1+m}{n}} \, dx\)

Optimal. Leaf size=22 \[ x^{m+1} \log (x) \left (a x^n\right )^{-\frac{m+1}{n}} \]

[Out]

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

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Rubi [A]  time = 0.0034739, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 29} \[ x^{m+1} \log (x) \left (a x^n\right )^{-\frac{m+1}{n}} \]

Antiderivative was successfully verified.

[In]

Int[x^m/(a*x^n)^((1 + m)/n),x]

[Out]

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

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

Mathematica [A]  time = 0.0074418, size = 22, normalized size = 1. \[ x^{m+1} \log (x) \left (a x^n\right )^{-\frac{m+1}{n}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/(a*x^n)^((1 + m)/n),x]

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( \left ( a{x}^{n} \right ) ^{{\frac{1+m}{n}}} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\left (a x^{n}\right )^{\frac{m + 1}{n}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/(a*x^n)^((m + 1)/n), x)

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Fricas [A]  time = 1.7429, size = 30, normalized size = 1.36 \begin{align*} \frac{\log \left (x\right )}{a^{\frac{m + 1}{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/a^((m + 1)/n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \left (a x^{n}\right )^{- \frac{m + 1}{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*(a*x**n)**(-(m + 1)/n), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\left (a x^{n}\right )^{\frac{m + 1}{n}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/(a*x^n)^((m + 1)/n), x)