Optimal. Leaf size=261 \[ \frac {2 B d i n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2}-\frac {d i \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g^2}-\frac {2 B i n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b g^2 (a+b x)}-\frac {i (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b g^2 (a+b x)}+\frac {2 B^2 d i n^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g^2}-\frac {2 B^2 i n^2 (c+d x)}{b g^2 (a+b x)} \]
[Out]
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Rubi [B] time = 2.95, antiderivative size = 766, normalized size of antiderivative = 2.93, number of steps used = 40, number of rules used = 20, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ \frac {2 A B d i n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^2}+\frac {2 B^2 d i n \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {2 B^2 d i n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^2}-\frac {2 B^2 d i n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac {2 B^2 d i n^2 \text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{b^2 g^2}-\frac {2 B d i n \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2}-\frac {2 B i n (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2 (a+b x)}+\frac {2 B d i n \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2}+\frac {d i \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g^2}-\frac {i (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g^2 (a+b x)}+\frac {2 A B d i n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {A B d i n \log ^2(a+b x)}{b^2 g^2}-\frac {B^2 d i \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {B^2 d i \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {2 B^2 i n^2 (b c-a d)}{b^2 g^2 (a+b x)}-\frac {2 B^2 d i n^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^2}-\frac {2 B^2 d i n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac {B^2 d i n^2 \log ^2(a+b x)}{b^2 g^2}-\frac {2 B^2 d i n^2 \log (a+b x)}{b^2 g^2}+\frac {B^2 d i n^2 \log ^2(c+d x)}{b^2 g^2}+\frac {2 B^2 d i n^2 \log (c+d x)}{b^2 g^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 44
Rule 2301
Rule 2317
Rule 2344
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2411
Rule 2418
Rule 2488
Rule 2506
Rule 2507
Rule 2524
Rule 2525
Rule 2528
Rule 6610
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int \frac {(164 c+164 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2} \, dx &=\int \left (\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g^2 (a+b x)^2}+\frac {164 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g^2 (a+b x)}\right ) \, dx\\ &=\frac {(164 d) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x} \, dx}{b g^2}+\frac {(164 (b c-a d)) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2} \, dx}{b g^2}\\ &=-\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}+\frac {164 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {(328 B d n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{b^2 g^2}+\frac {(328 B (b c-a d) n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}+\frac {164 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {(328 B d n) \int \frac {(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 )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}+\frac {\left (328 B (b c-a d)^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}+\frac {164 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {(328 B d (b c-a d) n) \int \frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}+\frac {\left (328 B (b c-a d)^2 n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^2}\\ &=-\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}+\frac {164 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {(328 B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b g^2}+\frac {\left (328 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^2 g^2}+\frac {(328 B (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b g^2}-\frac {(328 B d (b c-a d) n) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{b^2 g^2}\\ &=-\frac {328 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}-\frac {328 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}+\frac {164 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}+\frac {328 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}-\frac {(328 A B d (b c-a d) n) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac {\left (328 B^2 d (b c-a d) n\right ) \int \frac {\log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}+\frac {\left (328 B^2 d n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 g^2}-\frac {\left (328 B^2 d n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 g^2}+\frac {\left (328 B^2 (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac {164 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {328 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}-\frac {328 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}+\frac {164 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}+\frac {328 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {\left (164 B^2 d\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b g^2}-\frac {(328 A B d (b c-a d) n) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^3 g^2}+\frac {\left (328 B^2 d n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^2 g^2}-\frac {\left (328 B^2 d n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{b^2 g^2}+\frac {\left (328 B^2 (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac {164 B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {164 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {328 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}-\frac {328 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}+\frac {164 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}+\frac {328 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}-\frac {(328 A B d n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g^2}+\frac {\left (328 A B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b^3 g^2}+\frac {\left (328 B^2 d (b c-a d) n\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}+\frac {\left (328 B^2 d n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b g^2}-\frac {\left (328 B^2 d n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b g^2}-\frac {\left (328 B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 g^2}+\frac {\left (328 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 g^2}+\frac {\left (328 B^2 (b c-a d)^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^2}\\ &=-\frac {328 B^2 (b c-a d) n^2}{b^2 g^2 (a+b x)}-\frac {328 B^2 d n^2 \log (a+b x)}{b^2 g^2}-\frac {164 A B d n \log ^2(a+b x)}{b^2 g^2}-\frac {164 B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {164 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {328 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}-\frac {328 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}+\frac {164 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}+\frac {328 B^2 d n^2 \log (c+d x)}{b^2 g^2}-\frac {328 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac {328 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {328 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {328 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac {328 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g^2}-\frac {(328 A B d n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g^2}+\frac {\left (328 B^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g^2}+\frac {\left (328 B^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 g^2}+\frac {\left (328 B^2 d n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g^2}+\frac {\left (328 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 g^2}-\frac {\left (328 B^2 d (b c-a d) n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac {328 B^2 (b c-a d) n^2}{b^2 g^2 (a+b x)}-\frac {328 B^2 d n^2 \log (a+b x)}{b^2 g^2}-\frac {164 A B d n \log ^2(a+b x)}{b^2 g^2}+\frac {164 B^2 d n^2 \log ^2(a+b x)}{b^2 g^2}-\frac {164 B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {164 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {328 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}-\frac {328 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}+\frac {164 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}+\frac {328 B^2 d n^2 \log (c+d x)}{b^2 g^2}-\frac {328 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac {328 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {164 B^2 d n^2 \log ^2(c+d x)}{b^2 g^2}+\frac {328 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {328 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac {328 A B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^2}+\frac {328 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g^2}+\frac {328 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g^2}+\frac {\left (328 B^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g^2}+\frac {\left (328 B^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 g^2}\\ &=-\frac {328 B^2 (b c-a d) n^2}{b^2 g^2 (a+b x)}-\frac {328 B^2 d n^2 \log (a+b x)}{b^2 g^2}-\frac {164 A B d n \log ^2(a+b x)}{b^2 g^2}+\frac {164 B^2 d n^2 \log ^2(a+b x)}{b^2 g^2}-\frac {164 B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {164 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2 g^2}-\frac {328 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}-\frac {328 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {164 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}+\frac {164 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}+\frac {328 B^2 d n^2 \log (c+d x)}{b^2 g^2}-\frac {328 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac {328 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {164 B^2 d n^2 \log ^2(c+d x)}{b^2 g^2}+\frac {328 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {328 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac {328 A B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^2}-\frac {328 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^2}-\frac {328 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac {328 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g^2}+\frac {328 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g^2}\\ \end {align*}
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Mathematica [B] time = 3.21, size = 1556, normalized size = 5.96 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} d i x + A^{2} c i + {\left (B^{2} d i x + B^{2} c i\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B d i x + A B c i\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, A B c i n {\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} + A^{2} d i {\left (\frac {a}{b^{3} g^{2} x + a b^{2} g^{2}} + \frac {\log \left (b x + a\right )}{b^{2} g^{2}}\right )} - \frac {2 \, A B c i \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{b^{2} g^{2} x + a b g^{2}} - \frac {A^{2} c i}{b^{2} g^{2} x + a b g^{2}} - \frac {{\left ({\left (b c i - a d i\right )} B^{2} - {\left (B^{2} b d i x + B^{2} a d i\right )} \log \left (b x + a\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{b^{3} g^{2} x + a b^{2} g^{2}} - \int -\frac {B^{2} b^{2} c^{2} i \log \relax (e)^{2} + {\left (B^{2} b^{2} d^{2} i \log \relax (e)^{2} + 2 \, A B b^{2} d^{2} i \log \relax (e)\right )} x^{2} + {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + 2 \, {\left (B^{2} b^{2} c d i \log \relax (e)^{2} + A B b^{2} c d i \log \relax (e)\right )} x + 2 \, {\left (B^{2} b^{2} c^{2} i \log \relax (e) + {\left (B^{2} b^{2} d^{2} i \log \relax (e) + A B b^{2} d^{2} i\right )} x^{2} + {\left (2 \, B^{2} b^{2} c d i \log \relax (e) + A B b^{2} c d i\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) + 2 \, {\left ({\left (a b c d i n - a^{2} d^{2} i n - b^{2} c^{2} i \log \relax (e)\right )} B^{2} - {\left (B^{2} b^{2} d^{2} i \log \relax (e) + A B b^{2} d^{2} i\right )} x^{2} - {\left (A B b^{2} c d i + {\left (a b d^{2} i n - {\left (i n - 2 \, i \log \relax (e)\right )} b^{2} c d\right )} B^{2}\right )} x - {\left (B^{2} b^{2} d^{2} i n x^{2} + 2 \, B^{2} a b d^{2} i n x + B^{2} a^{2} d^{2} i n\right )} \log \left (b x + a\right ) - {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b^{4} d g^{2} x^{3} + a^{2} b^{2} c g^{2} + {\left (b^{4} c g^{2} + 2 \, a b^{3} d g^{2}\right )} x^{2} + {\left (2 \, a b^{3} c g^{2} + a^{2} b^{2} d g^{2}\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (c\,i+d\,i\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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