Optimal. Leaf size=86 \[ \frac{e^{-\frac{a (m+1)}{b}} (c x)^{-m-1} (d x)^{m+1} (a+b \log (c x))^p \left (-\frac{(m+1) (a+b \log (c x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) (a+b \log (c x))}{b}\right )}{d (m+1)} \]
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Rubi [A] time = 0.0717138, antiderivative size = 86, 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 = {2310, 2181} \[ \frac{e^{-\frac{a (m+1)}{b}} (c x)^{-m-1} (d x)^{m+1} (a+b \log (c x))^p \left (-\frac{(m+1) (a+b \log (c x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) (a+b \log (c x))}{b}\right )}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2181
Rubi steps
\begin{align*} \int (d x)^m (a+b \log (c x))^p \, dx &=\frac{\left ((c x)^{-1-m} (d x)^{1+m}\right ) \operatorname{Subst}\left (\int e^{(1+m) x} (a+b x)^p \, dx,x,\log (c x)\right )}{d}\\ &=\frac{e^{-\frac{a (1+m)}{b}} (c x)^{-1-m} (d x)^{1+m} \Gamma \left (1+p,-\frac{(1+m) (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac{(1+m) (a+b \log (c x))}{b}\right )^{-p}}{d (1+m)}\\ \end{align*}
Mathematica [A] time = 0.108523, size = 82, normalized size = 0.95 \[ \frac{e^{-\frac{a (m+1)}{b}} (c x)^{-m} (d x)^m (a+b \log (c x))^p \left (-\frac{(m+1) (a+b \log (c x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) (a+b \log (c x))}{b}\right )}{c (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.198, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b\ln \left ( cx \right ) \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 (d x\right )^{m}{\left (b \log \left (c x\right ) + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d x\right )^{m}{\left (b \log \left (c x\right ) + a\right )}^{p}, 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 (d x\right )^{m} \left (a + b \log{\left (c x \right )}\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 (d x\right )^{m}{\left (b \log \left (c x\right ) + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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