Optimal. Leaf size=16 \[ \log (x) x^{-n p} \left (a x^n\right )^p \]
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Rubi [A] time = 0.0031623, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {15, 29} \[ \log (x) x^{-n p} \left (a x^n\right )^p \]
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rubi steps
\begin{align*} \int x^{-1-n p} \left (a x^n\right )^p \, dx &=\left (x^{-n p} \left (a x^n\right )^p\right ) \int \frac{1}{x} \, dx\\ &=x^{-n p} \left (a x^n\right )^p \log (x)\\ \end{align*}
Mathematica [A] time = 0.0050411, size = 16, normalized size = 1. \[ \log (x) x^{-n p} \left (a x^n\right )^p \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{x}^{-np-1} \left ( a{x}^{n} \right ) ^{p}\, 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 \left (a x^{n}\right )^{p} x^{-n p - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75884, size = 16, normalized size = 1. \begin{align*} a^{p} \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 x^{- n p - 1} \left (a x^{n}\right )^{p}\, 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 \left (a x^{n}\right )^{p} x^{-n p - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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