Integrand size = 12, antiderivative size = 63 \[ \int x (a+b \log (c x))^p \, dx=\frac {2^{-1-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c^2} \]
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Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2346, 2212} \[ \int x (a+b \log (c x))^p \, dx=\frac {2^{-p-1} e^{-\frac {2 a}{b}} (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p} \Gamma \left (p+1,-\frac {2 (a+b \log (c x))}{b}\right )}{c^2} \]
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Rule 2212
Rule 2346
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log (c x)\right )}{c^2} \\ & = \frac {2^{-1-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int x (a+b \log (c x))^p \, dx=\frac {2^{-1-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {a+b \log (c x)}{b}\right )^{-p}}{c^2} \]
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\[\int x \left (a +b \ln \left (x c \right )\right )^{p}d x\]
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\[ \int x (a+b \log (c x))^p \, dx=\int { {\left (b \log \left (c x\right ) + a\right )}^{p} x \,d x } \]
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\[ \int x (a+b \log (c x))^p \, dx=\int x \left (a + b \log {\left (c x \right )}\right )^{p}\, dx \]
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none
Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int x (a+b \log (c x))^p \, dx=-\frac {{\left (b \log \left (c x\right ) + a\right )}^{p + 1} e^{\left (-\frac {2 \, a}{b}\right )} E_{-p}\left (-\frac {2 \, {\left (b \log \left (c x\right ) + a\right )}}{b}\right )}{b c^{2}} \]
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\[ \int x (a+b \log (c x))^p \, dx=\int { {\left (b \log \left (c x\right ) + a\right )}^{p} x \,d x } \]
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Timed out. \[ \int x (a+b \log (c x))^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,x\right )\right )}^p \,d x \]
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