Integrand size = 16, antiderivative size = 86 \[ \int (d x)^m (a+b \log (c x))^p \, dx=\frac {e^{-\frac {a (1+m)}{b}} (c x)^{-1-m} (d x)^{1+m} \Gamma \left (1+p,-\frac {(1+m) (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {(1+m) (a+b \log (c x))}{b}\right )^{-p}}{d (1+m)} \]
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Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2347, 2212} \[ \int (d x)^m (a+b \log (c x))^p \, dx=\frac {e^{-\frac {a (m+1)}{b}} (c x)^{-m-1} (d x)^{m+1} (a+b \log (c x))^p \left (-\frac {(m+1) (a+b \log (c x))}{b}\right )^{-p} \Gamma \left (p+1,-\frac {(m+1) (a+b \log (c x))}{b}\right )}{d (m+1)} \]
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Rule 2212
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((c x)^{-1-m} (d x)^{1+m}\right ) \text {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*}
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int (d x)^m (a+b \log (c x))^p \, dx=\frac {e^{-\frac {a (1+m)}{b}} (c x)^{-m} (d x)^m \Gamma \left (1+p,-\frac {(1+m) (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (-\frac {(1+m) (a+b \log (c x))}{b}\right )^{-p}}{c (1+m)} \]
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\[\int \left (d x \right )^{m} \left (a +b \ln \left (x c \right )\right )^{p}d x\]
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\[ \int (d x)^m (a+b \log (c x))^p \, dx=\int { \left (d x\right )^{m} {\left (b \log \left (c x\right ) + a\right )}^{p} \,d x } \]
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\[ \int (d x)^m (a+b \log (c x))^p \, dx=\int \left (d x\right )^{m} \left (a + b \log {\left (c x \right )}\right )^{p}\, dx \]
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\[ \int (d x)^m (a+b \log (c x))^p \, dx=\int { \left (d x\right )^{m} {\left (b \log \left (c x\right ) + a\right )}^{p} \,d x } \]
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\[ \int (d x)^m (a+b \log (c x))^p \, dx=\int { \left (d x\right )^{m} {\left (b \log \left (c x\right ) + a\right )}^{p} \,d x } \]
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Timed out. \[ \int (d x)^m (a+b \log (c x))^p \, dx=\int {\left (a+b\,\ln \left (c\,x\right )\right )}^p\,{\left (d\,x\right )}^m \,d x \]
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