Optimal. Leaf size=303 \[ \frac{2 B g n (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i}+\frac{2 B^2 g n^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^2 i}-\frac{2 B^2 g n^2 (b c-a d) \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )}{d^2 i}+\frac{2 B g n (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i}+\frac{g (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^2 i}+\frac{g (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i} \]
[Out]
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Rubi [B] time = 4.03193, antiderivative size = 1156, normalized size of antiderivative = 3.82, number of steps used = 65, number of rules used = 24, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.558, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2500, 2433, 2375, 2317, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ -\frac{B^2 (b c-a d) g n^2 \log ^3(c+d x)}{3 d^2 i}-\frac{b B^2 c g n^2 \log ^2(c+d x)}{d^2 i}-\frac{A B (b c-a d) g n \log ^2(c+d x)}{d^2 i}+\frac{B^2 (b c-a d) g n^2 \log (a+b x) \log ^2(c+d x)}{d^2 i}-\frac{B^2 (b c-a d) g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{d^2 i}+\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{d^2 i}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{d^2 i}+\frac{2 b B^2 c g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 i}+\frac{2 A B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 i}-\frac{2 b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2 i}+\frac{2 B^2 (b c-a d) g n \log (a+b x) \log \left ((c+d x)^{-n}\right ) \log (c+d x)}{d^2 i}-\frac{2 B^2 (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (c+d x)}{d^2 i}-\frac{a B^2 g n^2 \log ^2(a+b x)}{d i}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d i}+\frac{B^2 (b c-a d) g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d^2 i}-\frac{B^2 (b c-a d) g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{d^2 i}+\frac{2 a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d i}-\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac{2 a B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d i}+\frac{2 a B^2 g n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d i}-\frac{2 B^2 (b c-a d) g n \log \left ((a+b x)^n\right ) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d^2 i}+\frac{2 b B^2 c g n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac{2 A B (b c-a d) g n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac{2 B^2 (b c-a d) g n \log \left ((c+d x)^{-n}\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i}-\frac{2 B^2 (b c-a d) g n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac{2 B^2 (b c-a d) g n^2 \text{PolyLog}\left (3,-\frac{d (a+b x)}{b c-a d}\right )}{d^2 i}+\frac{2 B^2 (b c-a d) g n^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2523
Rule 12
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 6688
Rule 6742
Rule 2500
Rule 2433
Rule 2375
Rule 2317
Rule 2374
Rule 6589
Rule 2440
Rule 2434
Rule 2499
Rule 2396
Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{(a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 c+188 d x} \, dx &=\int \left (\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac{(-b c+a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d (c+d x)}\right ) \, dx\\ &=\frac{(b g) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{188 d}-\frac{((b c-a d) g) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x} \, dx}{188 d}\\ &=\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac{(b B g n) \int \frac{(b c-a d) 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}{94 d}+\frac{(B (b c-a d) g n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{94 d^2}\\ &=\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}+\frac{(B (b c-a d) g 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 ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{94 d^2}-\frac{(b B (b c-a d) g n) \int \frac{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}{94 d}\\ &=\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac{(b B (b c-a d) g n) \int \left (-\frac{a \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)}+\frac{c \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{94 d}+\frac{\left (B (b c-a d)^2 g n\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{94 d^2}\\ &=\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}+\frac{(a b B g n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{94 d}-\frac{(b B c g n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{94 d}+\frac{\left (B (b c-a d)^2 g n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{94 d^2}\\ &=\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}+\frac{(b B (b c-a d) g n) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{94 d^2}-\frac{(B (b c-a d) g n) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{c+d x} \, dx}{94 d}+\frac{\left (b B^2 c g 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}{94 d^2}-\frac{\left (a B^2 g 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}{94 d}\\ &=\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}+\frac{(b B (b c-a d) g n) \int \left (\frac{A \log (c+d x)}{a+b x}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x}\right ) \, dx}{94 d^2}-\frac{(B (b c-a d) g n) \int \left (\frac{A \log (c+d x)}{c+d x}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x}\right ) \, dx}{94 d}+\frac{\left (b B^2 c g 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}{94 d^2}-\frac{\left (a B^2 g 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}{94 d}\\ &=\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}+\frac{(A b B (b c-a d) g n) \int \frac{\log (c+d x)}{a+b x} \, dx}{94 d^2}+\frac{\left (b B^2 (b c-a d) g n\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x} \, dx}{94 d^2}-\frac{(A B (b c-a d) g n) \int \frac{\log (c+d x)}{c+d x} \, dx}{94 d}-\frac{\left (B^2 (b c-a d) g n\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{94 d}+\frac{1}{94} \left (a B^2 g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx+\frac{\left (b^2 B^2 c g n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{94 d^2}-\frac{\left (a b B^2 g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{94 d}-\frac{\left (b B^2 c g n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{94 d}\\ &=\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac{A B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{b B^2 c g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}+\frac{a B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{94 d}-\frac{(A B (b c-a d) g n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{94 d^2}+\frac{\left (b B^2 (b c-a d) g n\right ) \int \frac{\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{94 d^2}+\frac{\left (b B^2 (b c-a d) g n\right ) \int \frac{\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{94 d^2}-\frac{(A B (b c-a d) g n) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{94 d}-\frac{\left (b B^2 c g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{94 d^2}-\frac{\left (a B^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{94 d}-\frac{\left (a b B^2 g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{94 d}-\frac{\left (b B^2 c g n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{94 d}+\frac{\left (b B^2 (b c-a d) g n^2\right ) \int \frac{\log ^2(c+d x)}{a+b x} \, dx}{188 d^2}-\frac{\left (B^2 (b c-a d) g n^2\right ) \int \frac{\log ^2(c+d x)}{c+d x} \, dx}{188 d}+\frac{\left (b B^2 (b c-a d) g n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{94 d^2}\\ &=-\frac{a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac{A B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{b B^2 c g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac{A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac{b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac{B^2 (b c-a d) g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}+\frac{a B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{94 d}-\frac{B^2 (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{94 d^2}-\frac{(A B (b c-a d) g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}+\frac{\left (B^2 (b c-a d) g n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^n\right ) \log \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )}{x} \, dx,x,a+b x\right )}{94 d^2}+\frac{\left (B^2 (b c-a d) g n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right ) \log \left (\left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{94 d^2}-\frac{\left (b B^2 c g 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 )}{94 d^2}-\frac{\left (a B^2 g 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 )}{94 d}-\frac{\left (B^2 (b c-a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,c+d x\right )}{188 d^2}-\frac{\left (B^2 (b c-a d) g n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{94 d}-\frac{\left (B^2 (b c-a d) g n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{94 d}\\ &=-\frac{a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac{A B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{b B^2 c g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac{A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac{b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac{B^2 (b c-a d) g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}+\frac{a B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{94 d}+\frac{B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}-\frac{B^2 (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{94 d^2}+\frac{a B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d}+\frac{A B (b c-a d) g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{b B^2 c g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac{\left (B^2 (b c-a d) g\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (x^n\right )}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{188 b d}-\frac{\left (B^2 (b c-a d) g n\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )^{-n}\right )}{-\frac{-b c+a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{94 b d}-\frac{\left (B^2 (b c-a d) g n^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{188 d^2}-\frac{\left (B^2 (b c-a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{d \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}+\frac{\left (B^2 (b c-a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )}{-\frac{-b c+a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{94 b d}-\frac{\left (B^2 (b c-a d) g n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}\\ &=-\frac{a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac{A B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{b B^2 c g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac{A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac{b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac{B^2 (b c-a d) g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n^2 \log ^3(c+d x)}{564 d^2}+\frac{a B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{94 d}-\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{188 d^2}+\frac{B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}-\frac{B^2 (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{94 d^2}+\frac{a B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d}+\frac{A B (b c-a d) g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{b B^2 c g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{B^2 (b c-a d) g n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac{B^2 (b c-a d) g n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac{\left (B^2 (b c-a d) g n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}+\frac{\left (B^2 (b c-a d) g n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^n\right ) \log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{94 d^2}+\frac{\left (B^2 (b c-a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}-\frac{\left (B^2 (b c-a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}\\ &=-\frac{a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac{A B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{b B^2 c g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac{A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac{b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac{B^2 (b c-a d) g n^2 \log (a+b x) \log ^2(c+d x)}{188 d^2}+\frac{B^2 (b c-a d) g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n^2 \log ^3(c+d x)}{564 d^2}+\frac{a B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{94 d}-\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{188 d^2}+\frac{B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}+\frac{B^2 (b c-a d) g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{94 d^2}+\frac{a B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d}-\frac{B^2 (b c-a d) g n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d^2}+\frac{A B (b c-a d) g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{b B^2 c g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{B^2 (b c-a d) g n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac{B^2 (b c-a d) g n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac{B^2 (b c-a d) g n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac{\left (b B^2 (b c-a d) g\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (x^{-n}\right )}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{188 d^3}-\frac{\left (b B^2 (b c-a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{188 d^3}+\frac{\left (B^2 (b c-a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{94 d^2}\\ &=-\frac{a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac{A B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{b B^2 c g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac{A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac{b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac{B^2 (b c-a d) g n^2 \log (a+b x) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n^2 \log ^3(c+d x)}{564 d^2}+\frac{a B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{94 d}-\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{188 d^2}+\frac{B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}+\frac{B^2 (b c-a d) g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac{B^2 (b c-a d) g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{94 d^2}+\frac{a B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d}-\frac{B^2 (b c-a d) g n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d^2}+\frac{A B (b c-a d) g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{b B^2 c g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{B^2 (b c-a d) g n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac{B^2 (b c-a d) g n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{B^2 (b c-a d) g n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d^2}-\frac{B^2 (b c-a d) g n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac{\left (B^2 (b c-a d) g n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}+\frac{\left (B^2 (b c-a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}\\ &=-\frac{a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac{A B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{b B^2 c g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac{A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac{b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac{B^2 (b c-a d) g n^2 \log (a+b x) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n^2 \log ^3(c+d x)}{564 d^2}+\frac{a B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{94 d}-\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{188 d^2}+\frac{B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}+\frac{B^2 (b c-a d) g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac{B^2 (b c-a d) g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{94 d^2}+\frac{a B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d}-\frac{B^2 (b c-a d) g n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d^2}+\frac{A B (b c-a d) g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{b B^2 c g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{B^2 (b c-a d) g n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac{B^2 (b c-a d) g n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{B^2 (b c-a d) g n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d^2}-\frac{B^2 (b c-a d) g n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+2 \frac{\left (B^2 (b c-a d) g n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{94 d^2}\\ &=-\frac{a B^2 g n^2 \log ^2(a+b x)}{188 d}+\frac{a B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{94 d}+\frac{b g x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{188 d}+\frac{A B (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{b B^2 c g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{94 d^2}+\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{188 d^2}-\frac{b B c g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{94 d^2}-\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{188 d^2}-\frac{A B (b c-a d) g n \log ^2(c+d x)}{188 d^2}-\frac{b B^2 c g n^2 \log ^2(c+d x)}{188 d^2}+\frac{B^2 (b c-a d) g n^2 \log (a+b x) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{188 d^2}-\frac{B^2 (b c-a d) g n^2 \log ^3(c+d x)}{564 d^2}+\frac{a B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{94 d}-\frac{B^2 (b c-a d) g \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{188 d^2}+\frac{B^2 (b c-a d) g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{94 d^2}+\frac{B^2 (b c-a d) g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac{B^2 (b c-a d) g \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{188 d^2}-\frac{B^2 (b c-a d) g n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{94 d^2}+\frac{a B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d}-\frac{B^2 (b c-a d) g n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d^2}+\frac{A B (b c-a d) g n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{b B^2 c g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{B^2 (b c-a d) g n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}-\frac{B^2 (b c-a d) g n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}+\frac{B^2 (b c-a d) g n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{94 d^2}+\frac{B^2 (b c-a d) g n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{94 d^2}\\ \end{align*}
Mathematica [B] time = 0.667736, size = 802, normalized size = 2.65 \[ -\frac{g \left (-B^2 \left (d (a+b x) \log ^2\left (\frac{a+b x}{c+d x}\right )+b c \log \left (\frac{b c-a d}{b c+b d x}\right ) \log ^2\left (\frac{a+b x}{c+d x}\right )-(b c-a d) \left (\log \left (\frac{b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-2 \log \left (\frac{a+b x}{c+d x}\right )+\log \left (\frac{b c-a d}{b c+b d x}\right )\right )-2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )+2 b c \left (\log \left (\frac{a+b x}{c+d x}\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )-\text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )\right )\right ) n^2+a B^2 d \left (\log \left (\frac{b c-a d}{b c+b d x}\right ) \log ^2\left (\frac{a+b x}{c+d x}\right )+2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \log \left (\frac{a+b x}{c+d x}\right )-2 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )\right ) n^2+a B d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )\right ) \left (\log ^2\left (\frac{c}{d}+x\right )+2 \left (\log \left (\frac{a}{b}+x\right )-\log \left (\frac{c}{d}+x\right )-\log \left (\frac{a+b x}{c+d x}\right )\right ) \log (c+d x)-2 \left (\log \left (\frac{a}{b}+x\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )+\text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )\right ) n+B \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )\right ) \left (-b c \log ^2\left (\frac{c}{d}+x\right )-2 d (a+b x) \left (\log \left (\frac{a}{b}+x\right )-1\right )+2 b (c+d x) \left (\log \left (\frac{c}{d}+x\right )-1\right )+2 b \left (\log \left (\frac{a}{b}+x\right )-\log \left (\frac{c}{d}+x\right )-\log \left (\frac{a+b x}{c+d x}\right )\right ) (d x-c \log (c+d x))+2 b c \left (\log \left (\frac{a}{b}+x\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )+\text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )\right ) n-b d x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )\right )^2+(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac{a+b x}{c+d x}\right )\right )^2 \log (c+d x)\right )}{d^2 i} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.52, size = 0, normalized size = 0. \begin{align*} \int{\frac{bgx+ag}{dix+ci} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} A^{2} b g{\left (\frac{x}{d i} - \frac{c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac{A^{2} a g \log \left (d i x + c i\right )}{d i} + \frac{{\left (B^{2} b d g x -{\left (b c g - a d g\right )} B^{2} \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{d^{2} i} - \int -\frac{B^{2} a d g \log \left (e\right )^{2} + 2 \, A B a d g \log \left (e\right ) +{\left (B^{2} b d g x + B^{2} a d g\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} +{\left (B^{2} b d g \log \left (e\right )^{2} + 2 \, A B b d g \log \left (e\right )\right )} x + 2 \,{\left (B^{2} a d g \log \left (e\right ) + A B a d g +{\left (B^{2} b d g \log \left (e\right ) + A B b d g\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \,{\left (B^{2} a d g \log \left (e\right ) + A B a d g -{\left (b c g n - a d g n\right )} B^{2} \log \left (d x + c\right ) +{\left ({\left (g n + g \log \left (e\right )\right )} B^{2} b d + A B b d g\right )} x +{\left (B^{2} b d g x + B^{2} a d g\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{d^{2} i x + c d i}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A^{2} b g x + A^{2} a g +{\left (B^{2} b g x + B^{2} a g\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B b g x + A B a g\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{d i x + c i}, x\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 (b g x + a g\right )}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{d i x + c i}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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