Optimal. Leaf size=151 \[ -\frac{i (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)}-\frac{B i n (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac{B^2 i n^2 (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \]
[Out]
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Rubi [C] time = 2.05542, antiderivative size = 691, normalized size of antiderivative = 4.58, number of steps used = 54, number of rules used = 11, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{B^2 d^2 i n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac{B^2 d^2 i n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac{B d^2 i n \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^3 (b c-a d)}+\frac{B d^2 i n \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^3 (b c-a d)}-\frac{d i \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g^3 (a+b x)}-\frac{B d i n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^3 (a+b x)}-\frac{i (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{B i n (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g^3 (a+b x)^2}+\frac{B^2 d^2 i n^2 \log ^2(a+b x)}{2 b^2 g^3 (b c-a d)}+\frac{B^2 d^2 i n^2 \log ^2(c+d x)}{2 b^2 g^3 (b c-a d)}-\frac{B^2 d^2 i n^2 \log (a+b x)}{2 b^2 g^3 (b c-a d)}+\frac{B^2 d^2 i n^2 \log (c+d x)}{2 b^2 g^3 (b c-a d)}-\frac{B^2 d^2 i n^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac{B^2 d^2 i n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac{B^2 i n^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac{B^2 d i n^2}{2 b^2 g^3 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{(165 c+165 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)^3} \, dx &=\int \left (\frac{165 (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^3 (a+b x)^3}+\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g^3 (a+b x)^2}\right ) \, dx\\ &=\frac{(165 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^3}+\frac{(165 (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)^3} \, dx}{b g^3}\\ &=-\frac{165 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{(330 B 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^3}+\frac{(165 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)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac{165 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{(330 B d (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 (c+d x)} \, dx}{b^2 g^3}+\frac{\left (165 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)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac{165 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{(330 B d (b c-a d) n) \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^3}+\frac{\left (165 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)^3}-\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)^2}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^3}\\ &=-\frac{165 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}-\frac{(165 B 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^3}+\frac{(330 B 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^3}+\frac{\left (165 B 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} \, dx}{b (b c-a d) g^3}-\frac{\left (330 B 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} \, dx}{b (b c-a d) g^3}-\frac{\left (165 B d^3 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 (b c-a d) g^3}+\frac{\left (330 B d^3 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 (b c-a d) g^3}+\frac{(165 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)^3} \, dx}{b g^3}\\ &=-\frac{165 B (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{165 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac{165 B d^2 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 (b c-a d) g^3}-\frac{165 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{165 B d^2 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 (b c-a d) g^3}-\frac{\left (165 B^2 d n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac{\left (330 B^2 d n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}-\frac{\left (165 B^2 d^2 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 (b c-a d) g^3}+\frac{\left (165 B^2 d^2 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 (b c-a d) g^3}+\frac{\left (330 B^2 d^2 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 (b c-a d) g^3}-\frac{\left (330 B^2 d^2 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 (b c-a d) g^3}+\frac{\left (165 B^2 (b c-a d) n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac{165 B (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{165 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac{165 B d^2 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 (b c-a d) g^3}-\frac{165 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{165 B d^2 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 (b c-a d) g^3}-\frac{\left (165 B^2 d^2 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 (b c-a d) g^3}+\frac{\left (165 B^2 d^2 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 (b c-a d) g^3}+\frac{\left (330 B^2 d^2 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 (b c-a d) g^3}-\frac{\left (330 B^2 d^2 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 (b c-a d) g^3}-\frac{\left (165 B^2 d (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac{\left (330 B^2 d (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac{\left (165 B^2 (b c-a d)^2 n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac{165 B (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{165 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac{165 B d^2 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 (b c-a d) g^3}-\frac{165 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{165 B d^2 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 (b c-a d) g^3}-\frac{\left (165 B^2 d^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac{\left (165 B^2 d^2 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac{\left (330 B^2 d^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}-\frac{\left (330 B^2 d^2 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac{\left (165 B^2 d^3 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac{\left (165 B^2 d^3 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac{\left (330 B^2 d^3 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac{\left (330 B^2 d^3 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac{\left (165 B^2 d (b c-a d) 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^3}+\frac{\left (330 B^2 d (b c-a d) 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^3}+\frac{\left (165 B^2 (b c-a d)^2 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^2 g^3}\\ &=-\frac{165 B^2 (b c-a d) n^2}{4 b^2 g^3 (a+b x)^2}-\frac{165 B^2 d n^2}{2 b^2 g^3 (a+b x)}-\frac{165 B^2 d^2 n^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac{165 B (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{165 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac{165 B d^2 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 (b c-a d) g^3}-\frac{165 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{165 B^2 d^2 n^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac{165 B^2 d^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac{165 B d^2 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 (b c-a d) g^3}-\frac{165 B^2 d^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac{\left (165 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}-\frac{\left (165 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}+\frac{\left (330 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}+\frac{\left (330 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}-\frac{\left (165 B^2 d^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}+\frac{\left (330 B^2 d^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac{\left (165 B^2 d^3 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac{\left (330 B^2 d^3 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}\\ &=-\frac{165 B^2 (b c-a d) n^2}{4 b^2 g^3 (a+b x)^2}-\frac{165 B^2 d n^2}{2 b^2 g^3 (a+b x)}-\frac{165 B^2 d^2 n^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac{165 B^2 d^2 n^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac{165 B (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{165 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac{165 B d^2 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 (b c-a d) g^3}-\frac{165 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{165 B^2 d^2 n^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac{165 B^2 d^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac{165 B d^2 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 (b c-a d) g^3}+\frac{165 B^2 d^2 n^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac{165 B^2 d^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac{\left (165 B^2 d^2 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 (b c-a d) g^3}-\frac{\left (165 B^2 d^2 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 (b c-a d) g^3}+\frac{\left (330 B^2 d^2 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 (b c-a d) g^3}+\frac{\left (330 B^2 d^2 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 (b c-a d) g^3}\\ &=-\frac{165 B^2 (b c-a d) n^2}{4 b^2 g^3 (a+b x)^2}-\frac{165 B^2 d n^2}{2 b^2 g^3 (a+b x)}-\frac{165 B^2 d^2 n^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac{165 B^2 d^2 n^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac{165 B (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{165 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac{165 B d^2 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 (b c-a d) g^3}-\frac{165 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{165 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{165 B^2 d^2 n^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac{165 B^2 d^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac{165 B d^2 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 (b c-a d) g^3}+\frac{165 B^2 d^2 n^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac{165 B^2 d^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac{165 B^2 d^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac{165 B^2 d^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}\\ \end{align*}
Mathematica [C] time = 0.927759, size = 801, normalized size = 5.3 \[ -\frac{i \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2-4 d (a d-b c) (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+4 B d n (a+b x) \left (2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+2 d (a+b x) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-2 d (a+b x) \log (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+2 B n (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d n (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+B d n (a+b x) \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )+B n \left (2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2+4 d (a d-b c) (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-4 B d n (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B n \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 n (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-2 B d^2 n (a+b x)^2 \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )\right )}{4 b^2 (b c-a d) g^3 (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.527, size = 0, normalized size = 0. \begin{align*} \int{\frac{dix+ci}{ \left ( bgx+ag \right ) ^{3}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.75431, size = 2723, normalized size = 18.03 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.533099, size = 1214, normalized size = 8.04 \begin{align*} -\frac{{\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} i n^{2} + 2 \,{\left (A B b^{2} c^{2} - A B a^{2} d^{2}\right )} i n + 2 \,{\left (2 \,{\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} i x +{\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} i\right )} \log \left (e\right )^{2} + 2 \,{\left (B^{2} b^{2} d^{2} i n^{2} x^{2} + 2 \, B^{2} b^{2} c d i n^{2} x + B^{2} b^{2} c^{2} i n^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )^{2} + 2 \,{\left (A^{2} b^{2} c^{2} - A^{2} a^{2} d^{2}\right )} i + 2 \,{\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} i n^{2} + 2 \,{\left (A B b^{2} c d - A B a b d^{2}\right )} i n + 2 \,{\left (A^{2} b^{2} c d - A^{2} a b d^{2}\right )} i\right )} x + 2 \,{\left ({\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} i n + 2 \,{\left (A B b^{2} c^{2} - A B a^{2} d^{2}\right )} i + 2 \,{\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} i n + 2 \,{\left (A B b^{2} c d - A B a b d^{2}\right )} i\right )} x + 2 \,{\left (B^{2} b^{2} d^{2} i n x^{2} + 2 \, B^{2} b^{2} c d i n x + B^{2} b^{2} c^{2} i n\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 2 \,{\left (B^{2} b^{2} c^{2} i n^{2} + 2 \, A B b^{2} c^{2} i n +{\left (B^{2} b^{2} d^{2} i n^{2} + 2 \, A B b^{2} d^{2} i n\right )} x^{2} + 2 \,{\left (B^{2} b^{2} c d i n^{2} + 2 \, A B b^{2} c d i n\right )} x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{4 \,{\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \,{\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x +{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}^{2}}{{\left (b g x + a g\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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