Optimal. Leaf size=21 \[ x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1971, 29}
\begin {gather*} \log (x) x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 1971
Rubi steps
\begin {align*} \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx &=\left (x^{-n p q} \left (a \left (b x^n\right )^p\right )^q\right ) \int \frac {1}{x} \, dx\\ &=x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int x^{-n p q -1} \left (a \left (b \,x^{n}\right )^{p}\right )^{q}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 14, normalized size = 0.67 \begin {gather*} e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \log \left (x\right ) \end {gather*}
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 \begin {gather*} \int x^{- n p q - 1} \left (a \left (b x^{n}\right )^{p}\right )^{q}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{x^{n\,p\,q+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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