Optimal. Leaf size=472 \[ \frac{4 B d i^2 n (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2}+\frac{2 B^2 d i^2 n^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b^3 g^2}+\frac{4 B^2 d i^2 n^2 (b c-a d) \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac{d^2 i^2 (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^3 g^2}-\frac{i^2 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g^2 (a+b x)}-\frac{2 B i^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2 (a+b x)}+\frac{2 B d i^2 n (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2}-\frac{2 d i^2 (b c-a d) \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^3 g^2}-\frac{2 B^2 i^2 n^2 (c+d x) (b c-a d)}{b^2 g^2 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 3.76458, antiderivative size = 1309, normalized size of antiderivative = 2.77, number of steps used = 60, number of rules used = 21, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 44, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ -\frac{a B^2 d^2 n^2 \log ^2(a+b x) i^2}{b^3 g^2}+\frac{B^2 d (b c-a d) n^2 \log ^2(a+b x) i^2}{b^3 g^2}-\frac{2 A B d (b c-a d) n \log ^2(a+b x) i^2}{b^3 g^2}-\frac{2 B^2 d (b c-a 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 ) i^2}{b^3 g^2}-\frac{2 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) i^2}{b^3 g^2}+\frac{d^2 x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 i^2}{b^2 g^2}+\frac{2 d (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 )^2 i^2}{b^3 g^2}-\frac{(b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 i^2}{b^3 g^2 (a+b x)}-\frac{B^2 c d n^2 \log ^2(c+d x) i^2}{b^2 g^2}+\frac{B^2 d (b c-a d) n^2 \log ^2(c+d x) i^2}{b^3 g^2}-\frac{2 B^2 d (b c-a d) n^2 \log (a+b x) i^2}{b^3 g^2}+\frac{2 a 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 ) i^2}{b^3 g^2}-\frac{2 B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) i^2}{b^3 g^2}-\frac{2 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) i^2}{b^3 g^2 (a+b x)}+\frac{2 B^2 d (b c-a d) n^2 \log (c+d x) i^2}{b^3 g^2}+\frac{2 B^2 c d n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) i^2}{b^2 g^2}-\frac{2 B^2 d (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) i^2}{b^3 g^2}-\frac{2 B c d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) i^2}{b^2 g^2}+\frac{2 B d (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) i^2}{b^3 g^2}+\frac{2 a B^2 d^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}-\frac{2 B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac{4 A B d (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac{2 a B^2 d^2 n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^2}-\frac{2 B^2 d (b c-a d) n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac{4 A B d (b c-a d) n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac{2 B^2 c d n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) i^2}{b^2 g^2}-\frac{2 B^2 d (b c-a d) n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac{4 B^2 d (b c-a d) n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right ) i^2}{b^3 g^2}+\frac{4 B^2 d (b c-a d) n^2 \text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right ) i^2}{b^3 g^2}-\frac{2 B^2 (b c-a d)^2 n^2 i^2}{b^3 g^2 (a+b x)} \]
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 2525
Rule 44
Rule 6688
Rule 6742
Rule 2411
Rule 2344
Rule 2317
Rule 2507
Rule 2488
Rule 2506
Rule 6610
Rubi steps
\begin{align*} \int \frac{(173 c+173 d x)^2 \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{29929 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}+\frac{29929 (b c-a d)^2 \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)^2}+\frac{59858 d (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)}\right ) \, dx\\ &=\frac{\left (29929 d^2\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2 g^2}+\frac{(59858 d (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} \, dx}{b^2 g^2}+\frac{\left (29929 (b c-a d)^2\right ) \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^2 g^2}\\ &=\frac{29929 d^2 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{29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{59858 d (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 )^2}{b^3 g^2}-\frac{\left (59858 B d^2 n\right ) \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}{b^2 g^2}-\frac{(119716 B d (b c-a 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^3 g^2}+\frac{\left (59858 B (b c-a d)^2 n\right ) \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^3 g^2}\\ &=\frac{29929 d^2 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{29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{59858 d (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 )^2}{b^3 g^2}-\frac{(119716 B d (b c-a 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^3 g^2}-\frac{\left (59858 B d^2 (b c-a d) n\right ) \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}{b^2 g^2}+\frac{\left (59858 B (b c-a d)^3 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^3 g^2}\\ &=\frac{29929 d^2 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{29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{59858 d (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 )^2}{b^3 g^2}-\frac{\left (59858 B d^2 (b c-a d) n\right ) \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}{b^2 g^2}-\frac{\left (119716 B d (b c-a d)^2 n\right ) \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^3 g^2}+\frac{\left (59858 B (b c-a d)^3 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^3 g^2}\\ &=\frac{29929 d^2 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{29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{59858 d (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 )^2}{b^3 g^2}+\frac{\left (59858 a 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^2 g^2}-\frac{\left (59858 B c 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{(59858 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} \, dx}{b^2 g^2}+\frac{\left (59858 B d^2 (b c-a d) 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^3 g^2}+\frac{\left (59858 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} \, dx}{b^2 g^2}-\frac{\left (119716 B d (b c-a d)^2 n\right ) \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^3 g^2}\\ &=-\frac{59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac{59858 a 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^3 g^2}-\frac{59858 B d (b c-a 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^3 g^2}+\frac{29929 d^2 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{29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{59858 d (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 )^2}{b^3 g^2}-\frac{59858 B c 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{59858 B d (b c-a 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^3 g^2}-\frac{\left (119716 A B d (b c-a d)^2 n\right ) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^3 g^2}-\frac{\left (119716 B^2 d (b c-a d)^2 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^3 g^2}+\frac{\left (59858 B^2 c 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 (59858 a 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^3 g^2}+\frac{\left (59858 B^2 d (b c-a 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^3 g^2}-\frac{\left (59858 B^2 d (b c-a 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^3 g^2}+\frac{\left (59858 B^2 (b c-a d)^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=-\frac{59858 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac{59858 a 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^3 g^2}-\frac{59858 B d (b c-a 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^3 g^2}+\frac{29929 d^2 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{29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{59858 d (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 )^2}{b^3 g^2}-\frac{59858 B c 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{59858 B d (b c-a 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^3 g^2}+\frac{\left (59858 B^2 d (b c-a d)\right ) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g^2}-\frac{\left (119716 A B d (b c-a d)^2 n\right ) \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^4 g^2}+\frac{\left (59858 B^2 c 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 (59858 a 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^3 g^2}+\frac{\left (59858 B^2 d (b c-a 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^3 g^2}-\frac{\left (59858 B^2 d (b c-a 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^3 g^2}+\frac{\left (59858 B^2 (b c-a d)^3 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=-\frac{59858 B^2 d (b c-a 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^3 g^2}-\frac{59858 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac{59858 a 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^3 g^2}-\frac{59858 B d (b c-a 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^3 g^2}+\frac{29929 d^2 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{29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{59858 d (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 )^2}{b^3 g^2}-\frac{59858 B c 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{59858 B d (b c-a 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^3 g^2}-\frac{(119716 A B d (b c-a d) n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac{\left (119716 A B d^2 (b c-a d) 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^4 g^2}+\frac{\left (119716 B^2 d (b c-a d)^2 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^3 g^2}+\frac{\left (59858 B^2 c d n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b g^2}-\frac{\left (59858 a B^2 d^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 g^2}-\frac{\left (59858 B^2 c d^2 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 g^2}+\frac{\left (59858 a B^2 d^3 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 g^2}+\frac{\left (59858 B^2 d (b c-a d) n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 g^2}-\frac{\left (59858 B^2 d (b c-a d) n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 g^2}-\frac{\left (59858 B^2 d^2 (b c-a d) n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 g^2}+\frac{\left (59858 B^2 d^2 (b c-a d) n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 g^2}+\frac{\left (59858 B^2 (b c-a d)^3 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^3 g^2}\\ &=-\frac{59858 B^2 (b c-a d)^2 n^2}{b^3 g^2 (a+b x)}-\frac{59858 B^2 d (b c-a d) n^2 \log (a+b x)}{b^3 g^2}-\frac{59858 A B d (b c-a d) n \log ^2(a+b x)}{b^3 g^2}-\frac{59858 B^2 d (b c-a 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^3 g^2}-\frac{59858 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac{59858 a 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^3 g^2}-\frac{59858 B d (b c-a 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^3 g^2}+\frac{29929 d^2 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{29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{59858 d (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 )^2}{b^3 g^2}+\frac{59858 B^2 d (b c-a d) n^2 \log (c+d x)}{b^3 g^2}+\frac{59858 B^2 c d n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac{59858 B^2 d (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac{59858 B c 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{59858 B d (b c-a 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^3 g^2}+\frac{119716 A B d (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{59858 a B^2 d^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac{59858 B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{119716 B^2 d (b c-a 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^3 g^2}-\frac{(119716 A B d (b c-a 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^3 g^2}-\frac{\left (59858 B^2 c d n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 g^2}-\frac{\left (59858 a B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}-\frac{\left (59858 a 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^2 g^2}-\frac{\left (59858 B^2 c 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 (59858 B^2 d (b c-a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac{\left (59858 B^2 d (b c-a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 g^2}+\frac{\left (59858 B^2 d (b c-a d) n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^2}+\frac{\left (59858 B^2 d^2 (b c-a d) n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 g^2}-\frac{\left (119716 B^2 d (b c-a d)^2 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^3 g^2}\\ &=-\frac{59858 B^2 (b c-a d)^2 n^2}{b^3 g^2 (a+b x)}-\frac{59858 B^2 d (b c-a d) n^2 \log (a+b x)}{b^3 g^2}-\frac{59858 A B d (b c-a d) n \log ^2(a+b x)}{b^3 g^2}-\frac{29929 a B^2 d^2 n^2 \log ^2(a+b x)}{b^3 g^2}+\frac{29929 B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g^2}-\frac{59858 B^2 d (b c-a 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^3 g^2}-\frac{59858 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac{59858 a 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^3 g^2}-\frac{59858 B d (b c-a 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^3 g^2}+\frac{29929 d^2 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{29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{59858 d (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 )^2}{b^3 g^2}+\frac{59858 B^2 d (b c-a d) n^2 \log (c+d x)}{b^3 g^2}+\frac{59858 B^2 c d n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac{59858 B^2 d (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac{59858 B c 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{59858 B d (b c-a 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^3 g^2}-\frac{29929 B^2 c d n^2 \log ^2(c+d x)}{b^2 g^2}+\frac{29929 B^2 d (b c-a d) n^2 \log ^2(c+d x)}{b^3 g^2}+\frac{119716 A B d (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{59858 a B^2 d^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac{59858 B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{119716 A B d (b c-a d) n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac{119716 B^2 d (b c-a 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^3 g^2}+\frac{119716 B^2 d (b c-a d) n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^3 g^2}-\frac{\left (59858 B^2 c 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{\left (59858 a 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^3 g^2}+\frac{\left (59858 B^2 d (b c-a 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^3 g^2}+\frac{\left (59858 B^2 d (b c-a 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^3 g^2}\\ &=-\frac{59858 B^2 (b c-a d)^2 n^2}{b^3 g^2 (a+b x)}-\frac{59858 B^2 d (b c-a d) n^2 \log (a+b x)}{b^3 g^2}-\frac{59858 A B d (b c-a d) n \log ^2(a+b x)}{b^3 g^2}-\frac{29929 a B^2 d^2 n^2 \log ^2(a+b x)}{b^3 g^2}+\frac{29929 B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g^2}-\frac{59858 B^2 d (b c-a 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^3 g^2}-\frac{59858 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac{59858 a 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^3 g^2}-\frac{59858 B d (b c-a 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^3 g^2}+\frac{29929 d^2 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{29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac{59858 d (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 )^2}{b^3 g^2}+\frac{59858 B^2 d (b c-a d) n^2 \log (c+d x)}{b^3 g^2}+\frac{59858 B^2 c d n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac{59858 B^2 d (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac{59858 B c 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{59858 B d (b c-a 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^3 g^2}-\frac{29929 B^2 c d n^2 \log ^2(c+d x)}{b^2 g^2}+\frac{29929 B^2 d (b c-a d) n^2 \log ^2(c+d x)}{b^3 g^2}+\frac{119716 A B d (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{59858 a B^2 d^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac{59858 B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{119716 A B d (b c-a d) n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac{59858 a B^2 d^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^2}-\frac{59858 B^2 d (b c-a d) n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac{59858 B^2 c d n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac{59858 B^2 d (b c-a d) n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{119716 B^2 d (b c-a 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^3 g^2}+\frac{119716 B^2 d (b c-a d) n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^3 g^2}\\ \end{align*}
Mathematica [B] time = 15.5764, size = 2834, normalized size = 6. \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.714, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{ \left ( bgx+ag \right ) ^{2}} \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A^{2} d^{2} i^{2} x^{2} + 2 \, A^{2} c d i^{2} x + A^{2} c^{2} i^{2} +{\left (B^{2} d^{2} i^{2} x^{2} + 2 \, B^{2} c d i^{2} x + B^{2} c^{2} i^{2}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B d^{2} i^{2} x^{2} + 2 \, A B c d i^{2} x + A B c^{2} i^{2}\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}^{2}{\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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]