Integrand size = 32, antiderivative size = 156 \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\frac {b^2 \log ^{3 q}\left (c x^n\right )}{3 n q}-\frac {2 a b x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (2 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q}}{n}-\frac {2^{-q} a^2 x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \Gamma \left (q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}}{n} \]
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Time = 0.18 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2619, 2347, 2212, 2339, 30} \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=-\frac {a^2 2^{-q} x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \Gamma \left (q,-\frac {2 m \log \left (c x^n\right )}{n}\right )}{n}-\frac {2 a b x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} \Gamma \left (2 q,-\frac {m \log \left (c x^n\right )}{n}\right )}{n}+\frac {b^2 \log ^{3 q}\left (c x^n\right )}{3 n q} \]
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Rule 30
Rule 2212
Rule 2339
Rule 2347
Rule 2619
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 x^{-1+2 m} \log ^{-1+q}\left (c x^n\right )+2 a b x^{-1+m} \log ^{-1+2 q}\left (c x^n\right )+\frac {b^2 \log ^{-1+3 q}\left (c x^n\right )}{x}\right ) \, dx \\ & = a^2 \int x^{-1+2 m} \log ^{-1+q}\left (c x^n\right ) \, dx+(2 a b) \int x^{-1+m} \log ^{-1+2 q}\left (c x^n\right ) \, dx+b^2 \int \frac {\log ^{-1+3 q}\left (c x^n\right )}{x} \, dx \\ & = \frac {b^2 \text {Subst}\left (\int x^{-1+3 q} \, dx,x,\log \left (c x^n\right )\right )}{n}+\frac {\left (a^2 x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}}\right ) \text {Subst}\left (\int e^{\frac {2 m x}{n}} x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n}+\frac {\left (2 a b x^m \left (c x^n\right )^{-\frac {m}{n}}\right ) \text {Subst}\left (\int e^{\frac {m x}{n}} x^{-1+2 q} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {b^2 \log ^{3 q}\left (c x^n\right )}{3 n q}-\frac {2 a b x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (2 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q}}{n}-\frac {2^{-q} a^2 x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \Gamma \left (q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}}{n} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.96 \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\frac {\log ^q\left (c x^n\right ) \left (\frac {b^2 \log ^{2 q}\left (c x^n\right )}{q}-6 a b x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (2 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q}-3\ 2^{-q} a^2 x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \Gamma \left (q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}\right )}{3 n} \]
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\[\int \frac {\ln \left (c \,x^{n}\right )^{-1+q} \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{2}}{x}d x\]
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\[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} \log \left (c x^{n}\right )^{q - 1}}{x} \,d x } \]
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\[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\int \frac {\left (a x^{m} + b \log {\left (c x^{n} \right )}^{q}\right )^{2} \log {\left (c x^{n} \right )}^{q - 1}}{x}\, dx \]
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Exception generated. \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} \log \left (c x^{n}\right )^{q - 1}}{x} \,d x } \]
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Timed out. \[ \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx=\int \frac {{\ln \left (c\,x^n\right )}^{q-1}\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^2}{x} \,d x \]
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