Integrand size = 40, antiderivative size = 66 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{e}+\frac {b n \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{e} \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2481, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx=\frac {b n \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{e}-\frac {\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e} \]
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Rule 2421
Rule 2481
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{e} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{e}+\frac {(b n) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{e} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{e}+\frac {b n \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{e} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx=\frac {-\left (\left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+b n \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )}{e} \]
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\[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right ) \ln \left (\frac {e \left (g x +f \right )}{-d g +e f}\right )}{e x +d}d x\]
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\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (\frac {{\left (g x + f\right )} e}{e f - d g}\right )}{e x + d} \,d x } \]
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Exception generated. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (\frac {{\left (g x + f\right )} e}{e f - d g}\right )}{e x + d} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (\frac {{\left (g x + f\right )} e}{e f - d g}\right )}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx=\int \frac {\ln \left (-\frac {e\,\left (f+g\,x\right )}{d\,g-e\,f}\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{d+e\,x} \,d x \]
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