Optimal. Leaf size=875 \[ -\frac{B^3 h^2 n^3 \log (c+d x) (b c-a d)^3}{b^3 d^3}-\frac{B^2 h^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) (b c-a d)^3}{b^3 d^3}+\frac{B^3 h^2 n^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) (b c-a d)^3}{b^3 d^3}+\frac{B^2 h^2 n^2 (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (b c-a d)^2}{b^3 d^2}-\frac{2 B^2 h (3 b d g-2 b c h-a d h) n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (b c-a d)^2}{b^3 d^3}-\frac{2 B^3 h (3 b d g-2 b c h-a d h) n^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)^2}{b^3 d^3}-\frac{B h^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (b c-a d)}{2 b d^3}-\frac{B h (3 b d g-2 b c h-a d h) n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (b c-a d)}{b^3 d^2}+\frac{B \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (b c-a d)}{b^3 d^3}+\frac{2 B^2 \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)}{b^3 d^3}-\frac{2 B^3 \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) n^3 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)}{b^3 d^3}-\frac{(b g-a h)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 b^3 h}+\frac{(g+h x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 h} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.48124, antiderivative size = 1640, normalized size of antiderivative = 1.87, number of steps used = 53, number of rules used = 13, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.394, Rules used = {6742, 2492, 72, 2514, 2486, 31, 2488, 2411, 2343, 2333, 2315, 2506, 6610} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 2492
Rule 72
Rule 2514
Rule 2486
Rule 31
Rule 2488
Rule 2411
Rule 2343
Rule 2333
Rule 2315
Rule 2506
Rule 6610
Rubi steps
\begin{align*} \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx &=\int \left (A^3 (g+h x)^2+3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac{A^3 (g+h x)^3}{3 h}+\left (3 A^2 B\right ) \int (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+\left (3 A B^2\right ) \int (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+B^3 \int (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac{A^3 (g+h x)^3}{3 h}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac{\left (A^2 B (b c-a d) n\right ) \int \frac{(g+h x)^3}{(a+b x) (c+d x)} \, dx}{h}-\frac{\left (2 A B^2 (b c-a d) n\right ) \int \frac{(g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{h}-\frac{\left (B^3 (b c-a d) n\right ) \int \frac{(g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{h}\\ &=\frac{A^3 (g+h x)^3}{3 h}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac{\left (A^2 B (b c-a d) n\right ) \int \left (\frac{h^2 (3 b d g-b c h-a d h)}{b^2 d^2}+\frac{h^3 x}{b d}+\frac{(b g-a h)^3}{b^2 (b c-a d) (a+b x)}+\frac{(d g-c h)^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{h}-\frac{\left (2 A B^2 (b c-a d) n\right ) \int \left (\frac{h^2 (3 b d g-b c h-a d h) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 d^2}+\frac{h^3 x \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac{(b g-a h)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 (b c-a d) (a+b x)}+\frac{(d g-c h)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{h}-\frac{\left (B^3 (b c-a d) n\right ) \int \left (\frac{h^2 (3 b d g-b c h-a d h) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 d^2}+\frac{h^3 x \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac{(b g-a h)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 (b c-a d) (a+b x)}+\frac{(d g-c h)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{h}\\ &=-\frac{A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}-\frac{A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac{A^3 (g+h x)^3}{3 h}-\frac{A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac{A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac{\left (2 A B^2 (b c-a d) h^2 n\right ) \int x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b d}-\frac{\left (B^3 (b c-a d) h^2 n\right ) \int x \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b d}-\frac{\left (2 A B^2 (b g-a h)^3 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{b^2 h}-\frac{\left (B^3 (b g-a h)^3 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{b^2 h}+\frac{\left (2 A B^2 (d g-c h)^3 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{d^2 h}+\frac{\left (B^3 (d g-c h)^3 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{d^2 h}-\frac{\left (2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b^2 d^2}-\frac{\left (B^3 (b c-a d) h (3 b d g-b c h-a d h) n\right ) \int \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b^2 d^2}\\ &=-\frac{A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}-\frac{A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac{A^3 (g+h x)^3}{3 h}-\frac{A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac{A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac{A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac{2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{2 A B^2 (b g-a h)^3 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 )}{b^3 h}-\frac{2 A B^2 (d g-c h)^3 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 )}{d^3 h}-\frac{B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac{B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (b g-a h)^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 )}{b^3 h}-\frac{B^3 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}+\frac{\left (A B^2 (b c-a d)^2 h^2 n^2\right ) \int \frac{x^2}{(a+b x) (c+d x)} \, dx}{b d}+\frac{\left (B^3 (b c-a d)^2 h^2 n^2\right ) \int \frac{x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{b d}-\frac{\left (2 A B^2 (b c-a d) (b g-a h)^3 n^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^3 h}-\frac{\left (2 B^3 (b c-a d) (b g-a h)^3 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}{b^3 h}+\frac{\left (2 A B^2 (b c-a d) (d g-c h)^3 n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{d^3 h}+\frac{\left (2 B^3 (b c-a d) (d g-c h)^3 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}{d^3 h}+\frac{\left (2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2\right ) \int \frac{1}{c+d x} \, dx}{b^3 d^2}+\frac{\left (2 B^3 (b c-a d)^2 h (3 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 )}{c+d x} \, dx}{b^3 d^2}\\ &=-\frac{A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}-\frac{A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac{A^3 (g+h x)^3}{3 h}-\frac{A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac{A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}+\frac{2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac{A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac{2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{2 A B^2 (b g-a h)^3 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 )}{b^3 h}-\frac{2 A B^2 (d g-c h)^3 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 )}{d^3 h}-\frac{2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac{B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac{B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (b g-a h)^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 )}{b^3 h}-\frac{B^3 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac{2 B^3 (b g-a h)^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 )}{b^3 h}-\frac{2 B^3 (d g-c h)^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}{b (c+d x)}\right )}{d^3 h}+\frac{\left (A B^2 (b c-a d)^2 h^2 n^2\right ) \int \left (\frac{1}{b d}+\frac{a^2}{b (b c-a d) (a+b x)}+\frac{c^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{b d}+\frac{\left (B^3 (b c-a d)^2 h^2 n^2\right ) \int \left (\frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac{a^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)}+\frac{c^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d (-b c+a d) (c+d x)}\right ) \, dx}{b d}-\frac{\left (2 A B^2 (b c-a d) (b g-a h)^3 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 )}{b^4 h}+\frac{\left (2 A B^2 (b c-a d) (d g-c h)^3 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 )}{d^4 h}+\frac{\left (2 B^3 (b c-a d) (b g-a h)^3 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}{b^3 h}+\frac{\left (2 B^3 (b c-a d) (d g-c h)^3 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}{d^3 h}+\frac{\left (2 B^3 (b c-a d)^3 h (3 b d g-b c h-a d h) n^3\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{b^3 d^3}\\ &=-\frac{A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac{A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac{A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac{A^3 (g+h x)^3}{3 h}-\frac{A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac{a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac{A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac{A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac{2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac{A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac{2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{2 A B^2 (b g-a h)^3 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 )}{b^3 h}-\frac{2 A B^2 (d g-c h)^3 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 )}{d^3 h}-\frac{2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac{B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac{B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (b g-a h)^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 )}{b^3 h}-\frac{B^3 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac{2 B^3 (b g-a h)^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 )}{b^3 h}-\frac{2 B^3 (d g-c h)^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}{b (c+d x)}\right )}{d^3 h}-\frac{2 B^3 (b g-a h)^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac{2 B^3 (d g-c h)^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{d^3 h}-\frac{\left (B^3 c^2 (b c-a d) h^2 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b d^2}+\frac{\left (a^2 B^3 (b c-a d) h^2 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{b^2 d}+\frac{\left (B^3 (b c-a d)^2 h^2 n^2\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b^2 d^2}+\frac{\left (2 A B^2 (b c-a d) (b g-a h)^3 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 )}{b^4 h}-\frac{\left (2 A B^2 (b c-a d) (d g-c h)^3 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 )}{d^4 h}+\frac{\left (2 B^3 (b c-a d)^3 h (3 b d g-b c h-a d h) n^3\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 )}{b^3 d^4}\\ &=-\frac{A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac{A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac{A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac{A^3 (g+h x)^3}{3 h}-\frac{A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac{a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac{A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac{A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac{2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac{A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac{2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{B^3 (b c-a d)^2 h^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{2 A B^2 (b g-a h)^3 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 )}{b^3 h}-\frac{a^2 B^3 (b c-a d) h^2 n^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d}-\frac{2 A B^2 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 c^2 (b c-a d) h^2 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^3}-\frac{2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac{B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac{B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (b g-a h)^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 )}{b^3 h}-\frac{B^3 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac{2 B^3 (b g-a h)^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 )}{b^3 h}-\frac{2 B^3 (d g-c h)^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}{b (c+d x)}\right )}{d^3 h}-\frac{2 B^3 (b g-a h)^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac{2 B^3 (d g-c h)^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{d^3 h}+\frac{\left (2 A B^2 (b c-a d) (b g-a h)^3 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 )}{b^4 h}-\frac{\left (2 A B^2 (b c-a d) (d g-c h)^3 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 )}{d^4 h}-\frac{\left (B^3 c^2 (b c-a d)^2 h^2 n^3\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{b d^3}+\frac{\left (a^2 B^3 (b c-a d)^2 h^2 n^3\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^3 d}-\frac{\left (B^3 (b c-a d)^3 h^2 n^3\right ) \int \frac{1}{c+d x} \, dx}{b^3 d^2}-\frac{\left (2 B^3 (b c-a d)^3 h (3 b d g-b c h-a d h) n^3\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 )}{b^3 d^4}\\ &=-\frac{A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac{A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac{A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac{A^3 (g+h x)^3}{3 h}-\frac{A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac{a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac{A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac{A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac{2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac{B^3 (b c-a d)^3 h^2 n^3 \log (c+d x)}{b^3 d^3}-\frac{A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac{2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{B^3 (b c-a d)^2 h^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{2 A B^2 (b g-a h)^3 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 )}{b^3 h}-\frac{a^2 B^3 (b c-a d) h^2 n^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d}-\frac{2 A B^2 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 c^2 (b c-a d) h^2 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^3}-\frac{2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac{B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac{B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (b g-a h)^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 )}{b^3 h}-\frac{B^3 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac{2 A B^2 (d g-c h)^3 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 h}-\frac{2 A B^2 (b g-a h)^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 h}-\frac{2 B^3 (b g-a h)^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 )}{b^3 h}-\frac{2 B^3 (d g-c h)^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}{b (c+d x)}\right )}{d^3 h}-\frac{2 B^3 (b g-a h)^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac{2 B^3 (d g-c h)^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{d^3 h}-\frac{\left (B^3 c^2 (b c-a d)^2 h^2 n^3\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 )}{b d^4}+\frac{\left (a^2 B^3 (b c-a d)^2 h^2 n^3\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 )}{b^4 d}-\frac{\left (2 B^3 (b c-a d)^3 h (3 b d g-b c h-a d h) n^3\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 )}{b^3 d^4}\\ &=-\frac{A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac{A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac{A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac{A^3 (g+h x)^3}{3 h}-\frac{A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac{a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac{A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac{A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac{2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac{B^3 (b c-a d)^3 h^2 n^3 \log (c+d x)}{b^3 d^3}-\frac{A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac{2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{B^3 (b c-a d)^2 h^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{2 A B^2 (b g-a h)^3 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 )}{b^3 h}-\frac{a^2 B^3 (b c-a d) h^2 n^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d}-\frac{2 A B^2 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 c^2 (b c-a d) h^2 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^3}-\frac{2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac{B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac{B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (b g-a h)^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 )}{b^3 h}-\frac{B^3 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac{2 A B^2 (d g-c h)^3 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 h}-\frac{2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^3 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}-\frac{2 A B^2 (b g-a h)^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 h}-\frac{2 B^3 (b g-a h)^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 )}{b^3 h}-\frac{2 B^3 (d g-c h)^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}{b (c+d x)}\right )}{d^3 h}-\frac{2 B^3 (b g-a h)^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac{2 B^3 (d g-c h)^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{d^3 h}+\frac{\left (B^3 c^2 (b c-a d)^2 h^2 n^3\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 )}{b d^4}-\frac{\left (a^2 B^3 (b c-a d)^2 h^2 n^3\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 )}{b^4 d}\\ &=-\frac{A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac{A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac{A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac{A^3 (g+h x)^3}{3 h}-\frac{A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac{a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac{A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac{A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac{2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac{B^3 (b c-a d)^3 h^2 n^3 \log (c+d x)}{b^3 d^3}-\frac{A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac{2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{B^3 (b c-a d)^2 h^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{2 A B^2 (b g-a h)^3 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 )}{b^3 h}-\frac{a^2 B^3 (b c-a d) h^2 n^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d}-\frac{2 A B^2 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 c^2 (b c-a d) h^2 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^3}-\frac{2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac{B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac{B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (b g-a h)^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 )}{b^3 h}-\frac{B^3 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac{2 A B^2 (d g-c h)^3 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 h}-\frac{2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^3 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}-\frac{2 A B^2 (b g-a h)^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 h}-\frac{2 B^3 (b g-a h)^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 )}{b^3 h}-\frac{2 B^3 (d g-c h)^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}{b (c+d x)}\right )}{d^3 h}-\frac{2 B^3 (b g-a h)^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac{2 B^3 (d g-c h)^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{d^3 h}+\frac{\left (B^3 c^2 (b c-a d)^2 h^2 n^3\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 )}{b d^4}-\frac{\left (a^2 B^3 (b c-a d)^2 h^2 n^3\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 )}{b^4 d}\\ &=-\frac{A^2 B (b c-a d) h (3 b d g-b c h-a d h) n x}{b^2 d^2}+\frac{A B^2 (b c-a d)^2 h^2 n^2 x}{b^2 d^2}-\frac{A^2 B (b c-a d) h^2 n x^2}{2 b d}+\frac{A^3 (g+h x)^3}{3 h}-\frac{A^2 B (b g-a h)^3 n \log (a+b x)}{b^3 h}+\frac{a^2 A B^2 (b c-a d) h^2 n^2 \log (a+b x)}{b^3 d}+\frac{A^2 B (d g-c h)^3 n \log (c+d x)}{d^3 h}-\frac{A B^2 c^2 (b c-a d) h^2 n^2 \log (c+d x)}{b d^3}+\frac{2 A B^2 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log (c+d x)}{b^3 d^3}-\frac{B^3 (b c-a d)^3 h^2 n^3 \log (c+d x)}{b^3 d^3}-\frac{A B^2 (b c-a d) h^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac{2 A B^2 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{B^3 (b c-a d)^2 h^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A^2 B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{2 A B^2 (b g-a h)^3 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 )}{b^3 h}-\frac{a^2 B^3 (b c-a d) h^2 n^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d}-\frac{2 A B^2 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 c^2 (b c-a d) h^2 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^3}-\frac{2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^3}-\frac{B^3 (b c-a d) h^2 n x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}-\frac{B^3 (b c-a d) h (3 b d g-b c h-a d h) n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b^3 d^2}+\frac{A B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 (b g-a h)^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 )}{b^3 h}-\frac{B^3 (d g-c h)^3 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 )}{d^3 h}+\frac{B^3 (g+h x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac{2 A B^2 (d g-c h)^3 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 h}+\frac{B^3 c^2 (b c-a d) h^2 n^3 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{b d^3}-\frac{2 B^3 (b c-a d)^2 h (3 b d g-b c h-a d h) n^3 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}-\frac{2 A B^2 (b g-a h)^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 h}+\frac{a^2 B^3 (b c-a d) h^2 n^3 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 d}-\frac{2 B^3 (b g-a h)^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 )}{b^3 h}-\frac{2 B^3 (d g-c h)^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}{b (c+d x)}\right )}{d^3 h}-\frac{2 B^3 (b g-a h)^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^3 h}+\frac{2 B^3 (d g-c h)^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{d^3 h}\\ \end{align*}
Mathematica [F] time = 5.6802, size = 0, normalized size = 0. \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 5.47, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{2} \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 (A^{3} h^{2} x^{2} + 2 \, A^{3} g h x + A^{3} g^{2} +{\left (B^{3} h^{2} x^{2} + 2 \, B^{3} g h x + B^{3} g^{2}\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3} + 3 \,{\left (A B^{2} h^{2} x^{2} + 2 \, A B^{2} g h x + A B^{2} g^{2}\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 3 \,{\left (A^{2} B h^{2} x^{2} + 2 \, A^{2} B g h x + A^{2} B g^{2}\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ), 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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