Optimal. Leaf size=56 \[ \frac{e^{-\frac{a}{b}} (a+b \log (c x))^p \left (-\frac{a+b \log (c x)}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c x)}{b}\right )}{c} \]
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Rubi [A] time = 0.0329044, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2299, 2181} \[ \frac{e^{-\frac{a}{b}} (a+b \log (c x))^p \left (-\frac{a+b \log (c x)}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c x)}{b}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 2299
Rule 2181
Rubi steps
\begin{align*} \int (a+b \log (c x))^p \, dx &=\frac{\operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log (c x)\right )}{c}\\ &=\frac{e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log (c x)}{b}\right ) (a+b \log (c x))^p \left (-\frac{a+b \log (c x)}{b}\right )^{-p}}{c}\\ \end{align*}
Mathematica [A] time = 0.0260325, size = 56, normalized size = 1. \[ \frac{e^{-\frac{a}{b}} (a+b \log (c x))^p \left (-\frac{a+b \log (c x)}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c x)}{b}\right )}{c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int \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 [A] time = 1.24848, size = 59, normalized size = 1.05 \begin{align*} -\frac{{\left (b \log \left (c x\right ) + a\right )}^{p + 1} e^{\left (-\frac{a}{b}\right )} E_{-p}\left (-\frac{b \log \left (c x\right ) + a}{b}\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.05313, size = 86, normalized size = 1.54 \begin{align*} \frac{e^{\left (-\frac{b p \log \left (-\frac{1}{b}\right ) + a}{b}\right )} \Gamma \left (p + 1, -\frac{b \log \left (c x\right ) + a}{b}\right )}{c} \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 (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 (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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