Optimal. Leaf size=22 \[ x \log (x) \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]
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Rubi [A] time = 0.0109022, 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 = {6679, 29} \[ x \log (x) \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]
Antiderivative was successfully verified.
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Rule 6679
Rule 29
Rubi steps
\begin{align*} \int \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \, dx &=\left (x \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}\right ) \int \frac{1}{x} \, dx\\ &=x \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \log (x)\\ \end{align*}
Mathematica [A] time = 0.0017707, size = 22, normalized size = 1. \[ x \log (x) \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int \left ( \left ( a \left ( b{x}^{m} \right ) ^{n} \right ) ^{{\frac{1}{mn}}} \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{1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac{1}{m n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83432, size = 53, normalized size = 2.41 \begin{align*} e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \log \left (x\right ) \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 \left (a \left (b x^{m}\right )^{n}\right )^{- \frac{1}{m n}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16167, size = 26, normalized size = 1.18 \begin{align*} e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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