Optimal. Leaf size=16 \[ \log (x) x^{-n p} \left (a x^n\right )^p \]
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Rubi [A] time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \log (x) x^{-n p} \left (a x^n\right )^p \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 6, normalized size = 0.38 \[ a^{p} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a x^{n}\right )^{p} x^{-n p - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int x^{-n p -1} \left (a \,x^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a x^{n}\right )^{p} x^{-n p - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {{\left (a\,x^n\right )}^p}{x^{n\,p+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{- n p - 1} \left (a x^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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