Integrand size = 41, antiderivative size = 141 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\frac {i (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g}-\frac {(b c-a d) i \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {B (b c-a d) i n \operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {2559, 2541, 2458, 2378, 2370, 2352} \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=-\frac {i (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A-B n\right )}{b^2 g}+\frac {i (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b g}+\frac {B i n (b c-a d) \operatorname {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{b^2 g} \]
[In]
[Out]
Rule 2352
Rule 2370
Rule 2378
Rule 2458
Rule 2541
Rule 2559
Rubi steps \begin{align*} \text {integral}& = \frac {i (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g}+\frac {((b c-a d) i) \int \frac {A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a g+b g x} \, dx}{b} \\ & = \frac {i (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g}-\frac {(b c-a d) i \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {\left (B (b c-a d)^2 i n\right ) \int \frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^2 g} \\ & = \frac {i (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g}-\frac {(b c-a d) i \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {\left (B (b c-a d)^2 i n\right ) \text {Subst}\left (\int \frac {\log \left (\frac {-b c+a d}{d x}\right )}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^3 g} \\ & = \frac {i (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g}-\frac {(b c-a d) i \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}-\frac {\left (B (b c-a d)^2 i n\right ) \text {Subst}\left (\int \frac {\log \left (\frac {(-b c+a d) x}{d}\right )}{\left (\frac {b c-a d}{b}+\frac {d}{b x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{b^3 g} \\ & = \frac {i (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g}-\frac {(b c-a d) i \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}-\frac {\left (B (b c-a d)^2 i n\right ) \text {Subst}\left (\int \frac {\log \left (\frac {(-b c+a d) x}{d}\right )}{\frac {d}{b}+\frac {(b c-a d) x}{b}} \, dx,x,\frac {1}{a+b x}\right )}{b^3 g} \\ & = \frac {i (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g}-\frac {(b c-a d) i \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac {B (b c-a d) i n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.22 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\frac {i \left (B (-b c+a d) n \log ^2(a+b x)+2 \left (A b d x+B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B (-b c+a d) n \log (c+d x)\right )+2 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 B (b c-a d) n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )}{2 b^2 g} \]
[In]
[Out]
\[\int \frac {\left (d i x +c i \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{b g x +a g}d x\]
[In]
[Out]
\[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{b g x + a g} \,d x } \]
[In]
[Out]
\[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\frac {i \left (\int \frac {A c}{a + b x}\, dx + \int \frac {A d x}{a + b x}\, dx + \int \frac {B c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx + \int \frac {B d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx\right )}{g} \]
[In]
[Out]
none
Time = 0.48 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.96 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=A d i {\left (\frac {x}{b g} - \frac {a \log \left (b x + a\right )}{b^{2} g}\right )} - \frac {B c i n \log \left (d x + c\right )}{b g} + \frac {A c i \log \left (b g x + a g\right )}{b g} + \frac {{\left (b c i n - a d i n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{b^{2} g} + \frac {2 \, B b d i x \log \left (e\right ) - {\left (b c i n - a d i n\right )} B \log \left (b x + a\right )^{2} + 2 \, {\left (b c i \log \left (e\right ) + {\left (i n - i \log \left (e\right )\right )} a d\right )} B \log \left (b x + a\right ) + 2 \, {\left (B b d i x + {\left (b c i - a d i\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B b d i x + {\left (b c i - a d i\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, b^{2} g} \]
[In]
[Out]
\[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{b g x + a g} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\int \frac {\left (c\,i+d\,i\,x\right )\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{a\,g+b\,g\,x} \,d x \]
[In]
[Out]