Optimal. Leaf size=629 \[ \frac{3 B^2 n^2 (b c-a d) (-a d h-b c h+2 b d g) \text{PolyLog}\left (2,\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 h n^3 (b c-a d)^2 \text{PolyLog}\left (2,\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right )}{(b g-a h)^2 (d g-c h)^2}-\frac{3 B^3 n^3 (b c-a d) (-a d h-b c h+2 b d g) \text{PolyLog}\left (3,\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{b^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{2 h (b g-a h)^2}+\frac{3 B^2 h n^2 (b c-a d)^2 \log \left (1-\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B h n (a+b x) (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 (g+h x) (b g-a h)^2 (d g-c h)}+\frac{3 B n (b c-a d) (-a d h-b c h+2 b d g) \log \left (1-\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 (b g-a h)^2 (d g-c h)^2}-\frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{2 h (g+h x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 3.7451, antiderivative size = 2207, normalized size of antiderivative = 3.51, number of steps used = 49, number of rules used = 21, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {6742, 2492, 72, 2514, 2488, 2411, 2343, 2333, 2315, 2490, 36, 31, 2494, 2394, 2393, 2391, 2506, 6610, 2503, 2502, 2489} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 2492
Rule 72
Rule 2514
Rule 2488
Rule 2411
Rule 2343
Rule 2333
Rule 2315
Rule 2490
Rule 36
Rule 31
Rule 2494
Rule 2394
Rule 2393
Rule 2391
Rule 2506
Rule 6610
Rule 2503
Rule 2502
Rule 2489
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx &=\int \left (\frac{A^3}{(g+h x)^3}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3}\right ) \, dx\\ &=-\frac{A^3}{2 h (g+h x)^2}+\left (3 A^2 B\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx+\left (3 A B^2\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx+B^3 \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx\\ &=-\frac{A^3}{2 h (g+h x)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{\left (3 A^2 B (b c-a d) n\right ) \int \frac{1}{(a+b x) (c+d x) (g+h x)^2} \, dx}{2 h}+\frac{\left (3 A B^2 (b c-a d) n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x) (g+h x)^2} \, dx}{h}+\frac{\left (3 B^3 (b c-a d) n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x) (g+h x)^2} \, dx}{2 h}\\ &=-\frac{A^3}{2 h (g+h x)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{\left (3 A^2 B (b c-a d) n\right ) \int \left (\frac{b^3}{(b c-a d) (b g-a h)^2 (a+b x)}-\frac{d^3}{(b c-a d) (-d g+c h)^2 (c+d x)}+\frac{h^2}{(b g-a h) (d g-c h) (g+h x)^2}-\frac{h^2 (-2 b d g+b c h+a d h)}{(b g-a h)^2 (d g-c h)^2 (g+h x)}\right ) \, dx}{2 h}+\frac{\left (3 A B^2 (b c-a d) n\right ) \int \left (\frac{b^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (b g-a h)^2 (a+b x)}-\frac{d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (-d g+c h)^2 (c+d x)}+\frac{h^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (d g-c h) (g+h x)^2}-\frac{h^2 (-2 b d g+b c h+a d h) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h)^2 (d g-c h)^2 (g+h x)}\right ) \, dx}{h}+\frac{\left (3 B^3 (b c-a d) n\right ) \int \left (\frac{b^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (b g-a h)^2 (a+b x)}-\frac{d^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (-d g+c h)^2 (c+d x)}+\frac{h^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (d g-c h) (g+h x)^2}-\frac{h^2 (-2 b d g+b c h+a d h) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h)^2 (d g-c h)^2 (g+h x)}\right ) \, dx}{2 h}\\ &=-\frac{A^3}{2 h (g+h x)^2}-\frac{3 A^2 B (b c-a d) n}{2 (b g-a h) (d g-c h) (g+h x)}+\frac{3 A^2 b^2 B n \log (a+b x)}{2 h (b g-a h)^2}-\frac{3 A^2 B d^2 n \log (c+d x)}{2 h (d g-c h)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A^2 B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{2 (b g-a h)^2 (d g-c h)^2}+\frac{\left (3 A b^3 B^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{h (b g-a h)^2}+\frac{\left (3 b^3 B^3 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{2 h (b g-a h)^2}-\frac{\left (3 A B^2 d^3 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{h (d g-c h)^2}-\frac{\left (3 B^3 d^3 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 h (d g-c h)^2}+\frac{\left (3 A B^2 (b c-a d) h n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2} \, dx}{(b g-a h) (d g-c h)}+\frac{\left (3 B^3 (b c-a d) h n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2} \, dx}{2 (b g-a h) (d g-c h)}+\frac{\left (3 A B^2 (b c-a d) h (2 b d g-b c h-a d h) n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx}{(b g-a h)^2 (d g-c h)^2}+\frac{\left (3 B^3 (b c-a d) h (2 b d g-b c h-a d h) n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx}{2 (b g-a h)^2 (d g-c h)^2}\\ &=-\frac{A^3}{2 h (g+h x)^2}-\frac{3 A^2 B (b c-a d) n}{2 (b g-a h) (d g-c h) (g+h x)}+\frac{3 A^2 b^2 B n \log (a+b x)}{2 h (b g-a h)^2}-\frac{3 A^2 B d^2 n \log (c+d x)}{2 h (d g-c h)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h)^2 (d g-c h) (g+h x)}-\frac{3 A b^2 B^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (b g-a h)^2}+\frac{3 A B^2 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (d g-c h)^2}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 B^3 (b c-a d) h n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b g-a h)^2 (d g-c h) (g+h x)}-\frac{3 b^2 B^3 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (b g-a h)^2}+\frac{3 B^3 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (d g-c h)^2}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A^2 B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{2 (b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{\left (3 B^3 d (b c-a d) (2 b d g-b c h-a d h) n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 (b g-a h)^2 (d g-c h)^2}-\frac{\left (3 B^3 (b c-a d) (2 b d g-b c h-a d h) n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{2 (b g-a h)^2 (d g-c h)}+\frac{\left (3 A b^2 B^2 (b c-a d) n^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{h (b g-a h)^2}+\frac{\left (3 b^2 B^3 (b c-a d) n^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{h (b g-a h)^2}-\frac{\left (3 A B^2 d^2 (b c-a d) n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h (d g-c h)^2}-\frac{\left (3 B^3 d^2 (b c-a d) n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{h (d g-c h)^2}-\frac{\left (3 A B^2 (b c-a d)^2 h n^2\right ) \int \frac{1}{(c+d x) (g+h x)} \, dx}{(b g-a h)^2 (d g-c h)}-\frac{\left (3 B^3 (b c-a d)^2 h n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{(b g-a h)^2 (d g-c h)}-\frac{\left (3 A b B^2 (b c-a d) (2 b d g-b c h-a d h) n^2\right ) \int \frac{\log (g+h x)}{a+b x} \, dx}{(b g-a h)^2 (d g-c h)^2}+\frac{\left (3 A B^2 d (b c-a d) (2 b d g-b c h-a d h) n^2\right ) \int \frac{\log (g+h x)}{c+d x} \, dx}{(b g-a h)^2 (d g-c h)^2}\\ &=-\frac{A^3}{2 h (g+h x)^2}-\frac{3 A^2 B (b c-a d) n}{2 (b g-a h) (d g-c h) (g+h x)}+\frac{3 A^2 b^2 B n \log (a+b x)}{2 h (b g-a h)^2}-\frac{3 A^2 B d^2 n \log (c+d x)}{2 h (d g-c h)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h)^2 (d g-c h) (g+h x)}-\frac{3 A b^2 B^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (b g-a h)^2}+\frac{3 A B^2 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (d g-c h)^2}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 B^3 (b c-a d) h n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b g-a h)^2 (d g-c h) (g+h x)}-\frac{3 b^2 B^3 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (b g-a h)^2}+\frac{3 B^3 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (d g-c h)^2}-\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b g-a h)^2 (d g-c h)^2}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A^2 B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{2 (b g-a h)^2 (d g-c h)^2}-\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d)^2 h n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{2 (b g-a h)^2 (d g-c h)^2}+\frac{3 b^2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{h (b g-a h)^2}+\frac{3 B^3 d^2 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{h (d g-c h)^2}+\frac{\left (3 A b B^2 (b c-a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b c-a d}{d x}\right )}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{h (b g-a h)^2}-\frac{\left (3 A B^2 d (b c-a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b c+a d}{b x}\right )}{x \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )} \, dx,x,c+d x\right )}{h (d g-c h)^2}-\frac{\left (3 A B^2 d (b c-a d)^2 h n^2\right ) \int \frac{1}{c+d x} \, dx}{(b g-a h)^2 (d g-c h)^2}+\frac{\left (3 A B^2 (b c-a d)^2 h^2 n^2\right ) \int \frac{1}{g+h x} \, dx}{(b g-a h)^2 (d g-c h)^2}+\frac{\left (3 B^3 (b c-a d)^2 (2 b d g-b c h-a d h) n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h)^2 (d g-c h)^2}-\frac{\left (3 B^3 (b c-a d)^2 (2 b d g-b c h-a d h) n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h)^2 (d g-c h)^2}+\frac{\left (3 A B^2 (b c-a d) h (2 b d g-b c h-a d h) n^2\right ) \int \frac{\log \left (\frac{h (a+b x)}{-b g+a h}\right )}{g+h x} \, dx}{(b g-a h)^2 (d g-c h)^2}-\frac{\left (3 A B^2 (b c-a d) h (2 b d g-b c h-a d h) n^2\right ) \int \frac{\log \left (\frac{h (c+d x)}{-d g+c h}\right )}{g+h x} \, dx}{(b g-a h)^2 (d g-c h)^2}-\frac{\left (3 b^2 B^3 (b c-a d) n^3\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}{h (b g-a h)^2}-\frac{\left (3 B^3 d^2 (b c-a d) n^3\right ) \int \frac{\text{Li}_2\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h (d g-c h)^2}-\frac{\left (3 B^3 (b c-a d)^3 h n^3\right ) \int \frac{\log \left (-\frac{(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h)^2 (d g-c h)^2}\\ &=-\frac{A^3}{2 h (g+h x)^2}-\frac{3 A^2 B (b c-a d) n}{2 (b g-a h) (d g-c h) (g+h x)}+\frac{3 A^2 b^2 B n \log (a+b x)}{2 h (b g-a h)^2}-\frac{3 A^2 B d^2 n \log (c+d x)}{2 h (d g-c h)^2}-\frac{3 A B^2 (b c-a d)^2 h n^2 \log (c+d x)}{(b g-a h)^2 (d g-c h)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h)^2 (d g-c h) (g+h x)}-\frac{3 A b^2 B^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (b g-a h)^2}+\frac{3 A B^2 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (d g-c h)^2}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 B^3 (b c-a d) h n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b g-a h)^2 (d g-c h) (g+h x)}-\frac{3 b^2 B^3 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (b g-a h)^2}+\frac{3 B^3 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (d g-c h)^2}-\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b g-a h)^2 (d g-c h)^2}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A^2 B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{2 (b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d)^2 h n^2 \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}-\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d)^2 h n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{2 (b g-a h)^2 (d g-c h)^2}+\frac{3 b^2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{h (b g-a h)^2}+\frac{3 B^3 d^2 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{h (d g-c h)^2}-\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 b^2 B^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{h (b g-a h)^2}-\frac{3 B^3 d^2 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{h (d g-c h)^2}-\frac{\left (3 A b B^2 (b c-a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\left (\frac{b c-a d}{b}+\frac{d}{b x}\right ) x} \, dx,x,\frac{1}{a+b x}\right )}{h (b g-a h)^2}+\frac{\left (3 A B^2 d (b c-a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\left (\frac{-b c+a d}{d}+\frac{b}{d x}\right ) x} \, dx,x,\frac{1}{c+d x}\right )}{h (d g-c h)^2}+\frac{\left (3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{(b g-a h)^2 (d g-c h)^2}-\frac{\left (3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{\left (3 B^3 (b c-a d)^3 h n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b g-a h}\right )}{1+\frac{(-b c+a d) x}{b g-a h}} \, dx,x,\frac{g+h x}{c+d x}\right )}{(b g-a h)^3 (d g-c h)^2}+\frac{\left (3 B^3 (b c-a d)^2 (2 b d g-b c h-a d h) n^3\right ) \int \frac{\text{Li}_2\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h)^2 (d g-c h)^2}-\frac{\left (3 B^3 (b c-a d)^2 (2 b d g-b c h-a d h) n^3\right ) \int \frac{\text{Li}_2\left (1+\frac{(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h)^2 (d g-c h)^2}\\ &=-\frac{A^3}{2 h (g+h x)^2}-\frac{3 A^2 B (b c-a d) n}{2 (b g-a h) (d g-c h) (g+h x)}+\frac{3 A^2 b^2 B n \log (a+b x)}{2 h (b g-a h)^2}-\frac{3 A^2 B d^2 n \log (c+d x)}{2 h (d g-c h)^2}-\frac{3 A B^2 (b c-a d)^2 h n^2 \log (c+d x)}{(b g-a h)^2 (d g-c h)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h)^2 (d g-c h) (g+h x)}-\frac{3 A b^2 B^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (b g-a h)^2}+\frac{3 A B^2 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (d g-c h)^2}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 B^3 (b c-a d) h n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b g-a h)^2 (d g-c h) (g+h x)}-\frac{3 b^2 B^3 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (b g-a h)^2}+\frac{3 B^3 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (d g-c h)^2}-\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b g-a h)^2 (d g-c h)^2}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A^2 B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{2 (b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d)^2 h n^2 \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}-\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d)^2 h n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{2 (b g-a h)^2 (d g-c h)^2}-\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 b^2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{h (b g-a h)^2}+\frac{3 B^3 d^2 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{h (d g-c h)^2}-\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d)^2 h n^3 \text{Li}_2\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 b^2 B^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{h (b g-a h)^2}-\frac{3 B^3 d^2 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{h (d g-c h)^2}+\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}-\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^3 \text{Li}_3\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}-\frac{\left (3 A b B^2 (b c-a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\frac{d}{b}+\frac{(b c-a d) x}{b}} \, dx,x,\frac{1}{a+b x}\right )}{h (b g-a h)^2}+\frac{\left (3 A B^2 d (b c-a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\frac{b}{d}+\frac{(-b c+a d) x}{d}} \, dx,x,\frac{1}{c+d x}\right )}{h (d g-c h)^2}\\ &=-\frac{A^3}{2 h (g+h x)^2}-\frac{3 A^2 B (b c-a d) n}{2 (b g-a h) (d g-c h) (g+h x)}+\frac{3 A^2 b^2 B n \log (a+b x)}{2 h (b g-a h)^2}-\frac{3 A^2 B d^2 n \log (c+d x)}{2 h (d g-c h)^2}-\frac{3 A B^2 (b c-a d)^2 h n^2 \log (c+d x)}{(b g-a h)^2 (d g-c h)^2}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h)^2 (d g-c h) (g+h x)}-\frac{3 A b^2 B^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (b g-a h)^2}+\frac{3 A B^2 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (d g-c h)^2}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 B^3 (b c-a d) h n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b g-a h)^2 (d g-c h) (g+h x)}-\frac{3 b^2 B^3 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (b g-a h)^2}+\frac{3 B^3 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (d g-c h)^2}-\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b g-a h)^2 (d g-c h)^2}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac{3 A^2 B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{2 (b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d)^2 h n^2 \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}-\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d)^2 h n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{2 (b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 d^2 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{h (d g-c h)^2}+\frac{3 A b^2 B^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{h (b g-a h)^2}-\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 A B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 b^2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{h (b g-a h)^2}+\frac{3 B^3 d^2 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{h (d g-c h)^2}-\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d)^2 h n^3 \text{Li}_2\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{3 b^2 B^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{h (b g-a h)^2}-\frac{3 B^3 d^2 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{h (d g-c h)^2}+\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}-\frac{3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^3 \text{Li}_3\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}\\ \end{align*}
Mathematica [F] time = 8.24731, size = 0, normalized size = 0. \[ \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 3.369, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( hx+g \right ) ^{3}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{3}}\, 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*} \text{result too large to display} \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{B^{3} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3} + 3 \, A B^{2} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 3 \, A^{2} B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{3}}{h^{3} x^{3} + 3 \, g h^{2} x^{2} + 3 \, g^{2} h x + g^{3}}, 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(-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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