Optimal. Leaf size=22 \[ x^{m+1} \log (x) \left (a x^n\right )^{-\frac{m+1}{n}} \]
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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.
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Rule 15
Rule 29
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.
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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.
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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.
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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.
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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.
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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.
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