Optimal. Leaf size=26 \[ \frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{1+p} \]
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Rubi [A]
time = 0.12, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2624}
\begin {gather*} \frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 2624
Rubi steps
\begin {align*} \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx &=\frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{1+p}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 26, normalized size = 1.00 \begin {gather*} \frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{1+p} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (a m \,x^{m}+b n q \ln \left (c \,x^{n}\right )^{-1+q}\right ) \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{p}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 42, normalized size = 1.62 \begin {gather*} \frac {{\left ({\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}\right )} {\left ({\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}\right )}^{p}}{p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 = -0.04 \begin {gather*} \int \frac {\left (a\,m\,x^m+b\,n\,q\,{\ln \left (c\,x^n\right )}^{q-1}\right )\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^p}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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