Integrand size = 32, antiderivative size = 217 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+\frac {g}{x}} \, dx=\frac {A x}{f}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {B (b c-a d) n \log (c+d x)}{b d f}+\frac {B g n \log \left (\frac {f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^2}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac {B g n \log \left (\frac {f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^2}+\frac {B g n \operatorname {PolyLog}\left (2,-\frac {b (g+f x)}{a f-b g}\right )}{f^2}-\frac {B g n \operatorname {PolyLog}\left (2,-\frac {d (g+f x)}{c f-d g}\right )}{f^2} \]
[Out]
Time = 0.17 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2608, 2535, 31, 2545, 2441, 2440, 2438} \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+\frac {g}{x}} \, dx=-\frac {g \log (f x+g) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{f^2}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {B n (b c-a d) \log (c+d x)}{b d f}+\frac {B g n \operatorname {PolyLog}\left (2,-\frac {b (g+f x)}{a f-b g}\right )}{f^2}+\frac {B g n \log (f x+g) \log \left (\frac {f (a+b x)}{a f-b g}\right )}{f^2}+\frac {A x}{f}-\frac {B g n \operatorname {PolyLog}\left (2,-\frac {d (g+f x)}{c f-d g}\right )}{f^2}-\frac {B g n \log (f x+g) \log \left (\frac {f (c+d x)}{c f-d g}\right )}{f^2} \]
[In]
[Out]
Rule 31
Rule 2438
Rule 2440
Rule 2441
Rule 2535
Rule 2545
Rule 2608
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{f (g+f x)}\right ) \, dx \\ & = \frac {\int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{f}-\frac {g \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g+f x} \, dx}{f} \\ & = \frac {A x}{f}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}+\frac {B \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{f}+\frac {(b B g n) \int \frac {\log (g+f x)}{a+b x} \, dx}{f^2}-\frac {(B d g n) \int \frac {\log (g+f x)}{c+d x} \, dx}{f^2} \\ & = \frac {A x}{f}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}+\frac {B g n \log \left (\frac {f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^2}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac {B g n \log \left (\frac {f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^2}-\frac {(B (b c-a d) n) \int \frac {1}{c+d x} \, dx}{b f}-\frac {(B g n) \int \frac {\log \left (\frac {f (a+b x)}{a f-b g}\right )}{g+f x} \, dx}{f}+\frac {(B g n) \int \frac {\log \left (\frac {f (c+d x)}{c f-d g}\right )}{g+f x} \, dx}{f} \\ & = \frac {A x}{f}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {B (b c-a d) n \log (c+d x)}{b d f}+\frac {B g n \log \left (\frac {f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^2}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac {B g n \log \left (\frac {f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^2}-\frac {(B g n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a f-b g}\right )}{x} \, dx,x,g+f x\right )}{f^2}+\frac {(B g n) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{c f-d g}\right )}{x} \, dx,x,g+f x\right )}{f^2} \\ & = \frac {A x}{f}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {B (b c-a d) n \log (c+d x)}{b d f}+\frac {B g n \log \left (\frac {f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^2}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac {B g n \log \left (\frac {f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^2}+\frac {B g n \text {Li}_2\left (-\frac {b (g+f x)}{a f-b g}\right )}{f^2}-\frac {B g n \text {Li}_2\left (-\frac {d (g+f x)}{c f-d g}\right )}{f^2} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.85 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+\frac {g}{x}} \, dx=\frac {A f x+\frac {B f (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}-\frac {B (b c-a d) f n \log (c+d x)}{b d}-g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)+B g n \left (\left (\log \left (\frac {f (a+b x)}{a f-b g}\right )-\log \left (\frac {f (c+d x)}{c f-d g}\right )\right ) \log (g+f x)+\operatorname {PolyLog}\left (2,\frac {b (g+f x)}{-a f+b g}\right )-\operatorname {PolyLog}\left (2,\frac {d (g+f x)}{-c f+d g}\right )\right )}{f^2} \]
[In]
[Out]
\[\int \frac {A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{f +\frac {g}{x}}d x\]
[In]
[Out]
\[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+\frac {g}{x}} \, dx=\int { \frac {B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}{f + \frac {g}{x}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+\frac {g}{x}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+\frac {g}{x}} \, dx=\int { \frac {B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}{f + \frac {g}{x}} \,d x } \]
[In]
[Out]
\[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+\frac {g}{x}} \, dx=\int { \frac {B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}{f + \frac {g}{x}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+\frac {g}{x}} \, dx=\int \frac {A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{f+\frac {g}{x}} \,d x \]
[In]
[Out]