Integrand size = 20, antiderivative size = 114 \[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {e^{-\frac {a+a m q}{b n}} x \left (c x^n\right )^{-\frac {1+m q}{n}} \left (d x^q\right )^m \Gamma \left (1+p,-\frac {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m q} \]
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Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {15, 2347, 2212} \[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {x \left (d x^q\right )^m e^{-\frac {a m q+a}{b n}} \left (c x^n\right )^{-\frac {m q+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m q+1} \]
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Rule 15
Rule 2212
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \left (x^{-m q} \left (d x^q\right )^m\right ) \int x^{m q} \left (a+b \log \left (c x^n\right )\right )^p \, dx \\ & = \frac {\left (x \left (c x^n\right )^{-\frac {1+m q}{n}} \left (d x^q\right )^m\right ) \text {Subst}\left (\int e^{\frac {(1+m q) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {e^{-\frac {a+a m q}{b n}} x \left (c x^n\right )^{-\frac {1+m q}{n}} \left (d x^q\right )^m \Gamma \left (1+p,-\frac {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m q} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\frac {e^{-\frac {(1+m q) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} x^{-m q} \left (d x^q\right )^m \Gamma \left (1+p,-\frac {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m q} \]
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\[\int \left (d \,x^{q}\right )^{m} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]
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\[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { \left (d x^{q}\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} \,d x } \]
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\[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int \left (d x^{q}\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )^{p}\, dx \]
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\[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { \left (d x^{q}\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} \,d x } \]
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\[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int { \left (d x^{q}\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} \,d x } \]
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Timed out. \[ \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx=\int {\left (d\,x^q\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]
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