Integrand size = 32, antiderivative size = 32 \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{b n (1+p) q}-\frac {a m \text {Int}\left (x^{-1+m} \left (a x^m+b \log ^q\left (c x^n\right )\right )^p,x\right )}{b n q} \]
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Not integrable
Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{b n (1+p) q}-\frac {(a m) \int x^{-1+m} \left (a x^m+b \log ^q\left (c x^n\right )\right )^p \, dx}{b n q} \\ \end{align*}
Not integrable
Time = 0.90 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00
\[\int \frac {\ln \left (c \,x^{n}\right )^{-1+q} \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{p}}{x}d x\]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{p} \log \left (c x^{n}\right )^{q - 1}}{x} \,d x } \]
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Timed out. \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 1.67 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int \frac {{\ln \left (c\,x^n\right )}^{q-1}\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^p}{x} \,d x \]
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