Optimal. Leaf size=809 \[ -\frac{3 B n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (b c-a d)^4}{4 b d^4}-\frac{B^3 n^3 \log \left (\frac{a+b x}{c+d x}\right ) (b c-a d)^4}{4 b d^4}+\frac{3 B^3 n^3 \log (c+d x) (b c-a d)^4}{2 b d^4}-\frac{9 B^2 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (b c-a d)^4}{2 b d^4}+\frac{7 B^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) (b c-a d)^4}{4 b d^4}-\frac{9 B^3 n^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)^4}{2 b d^4}-\frac{3 B^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)^4}{2 b d^4}-\frac{7 B^3 n^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) (b c-a d)^4}{4 b d^4}+\frac{3 B^3 n^3 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)^4}{2 b d^4}-\frac{9 B n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (b c-a d)^3}{4 b d^3}-\frac{B^3 n^3 x (b c-a d)^3}{4 d^3}-\frac{7 B^2 n^2 (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (b c-a d)^3}{4 b d^3}+\frac{9 b B n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (b c-a d)^2}{8 d^4}+\frac{b B^2 n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (b c-a d)^2}{4 d^4}-\frac{b^2 B n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (b c-a d)}{4 d^4}+\frac{(a+b x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.40354, antiderivative size = 1203, normalized size of antiderivative = 1.49, number of steps used = 56, number of rules used = 13, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.394, Rules used = {6742, 2492, 43, 2514, 2486, 31, 2488, 2411, 2343, 2333, 2315, 2506, 6610} \[ -\frac{3 B^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 ) (b c-a d)^4}{4 b d^4}+\frac{3 B^3 n^3 \log (c+d x) (b c-a d)^4}{2 b d^4}+\frac{11 A B^2 n^2 \log (c+d x) (b c-a d)^4}{4 b d^4}+\frac{3 A^2 B n \log (c+d x) (b c-a d)^4}{4 b d^4}-\frac{11 B^3 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 c-a d)^4}{4 b d^4}-\frac{3 A B^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 ) (b c-a d)^4}{2 b d^4}-\frac{11 B^3 n^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)^4}{4 b d^4}-\frac{3 A B^2 n^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)^4}{2 b d^4}-\frac{3 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,1-\frac{b c-a d}{b (c+d x)}\right ) (b c-a d)^4}{2 b d^4}+\frac{3 B^3 n^3 \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right ) (b c-a d)^4}{2 b d^4}-\frac{3 B^3 n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) (b c-a d)^3}{4 b d^3}-\frac{B^3 n^3 x (b c-a d)^3}{4 d^3}-\frac{5 A B^2 n^2 x (b c-a d)^3}{4 d^3}-\frac{3 A^2 B n x (b c-a d)^3}{4 d^3}-\frac{5 B^3 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) (b c-a d)^3}{4 b d^3}-\frac{3 A B^2 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) (b c-a d)^3}{2 b d^3}+\frac{A B^2 n^2 (a+b x)^2 (b c-a d)^2}{4 b d^2}+\frac{3 A^2 B n (a+b x)^2 (b c-a d)^2}{8 b d^2}+\frac{3 B^3 n (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) (b c-a d)^2}{8 b d^2}+\frac{B^3 n^2 (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) (b c-a d)^2}{4 b d^2}+\frac{3 A B^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) (b c-a d)^2}{4 b d^2}-\frac{A^2 B n (a+b x)^3 (b c-a d)}{4 b d}-\frac{B^3 n (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) (b c-a d)}{4 b d}-\frac{A B^2 n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) (b c-a d)}{2 b d}+\frac{A^3 (a+b x)^4}{4 b}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 2492
Rule 43
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 (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx &=\int \left (A^3 (a+b x)^3+3 A^2 B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 A B^2 (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 (a+b x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac{A^3 (a+b x)^4}{4 b}+\left (3 A^2 B\right ) \int (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+\left (3 A B^2\right ) \int (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+B^3 \int (a+b x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{\left (3 A^2 B (b c-a d) n\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{4 b}-\frac{\left (3 A B^2 (b c-a d) n\right ) \int \frac{(a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b}-\frac{\left (3 B^3 (b c-a d) n\right ) \int \frac{(a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{4 b}\\ &=\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{\left (3 A^2 B (b c-a d) n\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{4 b}-\frac{\left (3 A B^2 (b c-a d) n\right ) \int \left (\frac{b (b c-a d)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3}-\frac{b (b c-a d) (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2}+\frac{b (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d}+\frac{(-b c+a d)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 (c+d x)}\right ) \, dx}{2 b}-\frac{\left (3 B^3 (b c-a d) n\right ) \int \left (\frac{b (b c-a d)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3}-\frac{b (b c-a d) (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2}+\frac{b (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d}+\frac{(-b c+a d)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 (c+d x)}\right ) \, dx}{4 b}\\ &=-\frac{3 A^2 B (b c-a d)^3 n x}{4 d^3}+\frac{3 A^2 B (b c-a d)^2 n (a+b x)^2}{8 b d^2}-\frac{A^2 B (b c-a d) n (a+b x)^3}{4 b d}+\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{\left (3 A B^2 (b c-a d) n\right ) \int (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{2 d}-\frac{\left (3 B^3 (b c-a d) n\right ) \int (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{4 d}+\frac{\left (3 A B^2 (b c-a d)^2 n\right ) \int (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{2 d^2}+\frac{\left (3 B^3 (b c-a d)^2 n\right ) \int (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{4 d^2}-\frac{\left (3 A B^2 (b c-a d)^3 n\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{2 d^3}-\frac{\left (3 B^3 (b c-a d)^3 n\right ) \int \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{4 d^3}+\frac{\left (3 A B^2 (b c-a d)^4 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b d^3}+\frac{\left (3 B^3 (b c-a d)^4 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{4 b d^3}\\ &=-\frac{3 A^2 B (b c-a d)^3 n x}{4 d^3}+\frac{3 A^2 B (b c-a d)^2 n (a+b x)^2}{8 b d^2}-\frac{A^2 B (b c-a d) n (a+b x)^3}{4 b d}+\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (b c-a d)^4 n \log (c+d x)}{4 b d^4}-\frac{3 A B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}+\frac{3 A B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{A B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 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 )}{2 b d^4}-\frac{3 B^3 (b c-a d)^3 n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 B^3 (b c-a d)^2 n (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b d^2}-\frac{B^3 (b c-a d) n (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{4 b d^4}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{\left (A B^2 (b c-a d)^2 n^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{2 b d}+\frac{\left (B^3 (b c-a d)^2 n^2\right ) \int \frac{(a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b d}-\frac{\left (3 A B^2 (b c-a d)^3 n^2\right ) \int \frac{a+b x}{c+d x} \, dx}{4 b d^2}-\frac{\left (3 B^3 (b c-a d)^3 n^2\right ) \int \frac{(a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{4 b d^2}+\frac{\left (3 A B^2 (b c-a d)^4 n^2\right ) \int \frac{1}{c+d x} \, dx}{2 b d^3}+\frac{\left (3 B^3 (b c-a d)^4 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b d^3}+\frac{\left (3 A B^2 (b c-a d)^5 n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b d^4}+\frac{\left (3 B^3 (b c-a d)^5 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}{2 b d^4}\\ &=-\frac{3 A^2 B (b c-a d)^3 n x}{4 d^3}+\frac{3 A^2 B (b c-a d)^2 n (a+b x)^2}{8 b d^2}-\frac{A^2 B (b c-a d) n (a+b x)^3}{4 b d}+\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac{3 A B^2 (b c-a d)^4 n^2 \log (c+d x)}{2 b d^4}-\frac{3 A B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}+\frac{3 A B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{A B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 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 )}{2 b d^4}-\frac{3 B^3 (b c-a d)^4 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 )}{2 b d^4}-\frac{3 B^3 (b c-a d)^3 n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 B^3 (b c-a d)^2 n (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b d^2}-\frac{B^3 (b c-a d) n (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{4 b d^4}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{2 b d^4}+\frac{\left (A B^2 (b c-a d)^2 n^2\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{2 b d}+\frac{\left (B^3 (b c-a d)^2 n^2\right ) \int \left (-\frac{b (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2}+\frac{b (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d}+\frac{(-b c+a d)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2 (c+d x)}\right ) \, dx}{2 b d}-\frac{\left (3 A B^2 (b c-a d)^3 n^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{4 b d^2}-\frac{\left (3 B^3 (b c-a d)^3 n^2\right ) \int \left (\frac{b \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d}+\frac{(-b c+a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d (c+d x)}\right ) \, dx}{4 b d^2}+\frac{\left (3 A B^2 (b c-a d)^5 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 )}{2 b d^5}+\frac{\left (3 B^3 (b c-a d)^5 n^3\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b d^4}+\frac{\left (3 B^3 (b c-a d)^5 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}{2 b d^4}\\ &=-\frac{3 A^2 B (b c-a d)^3 n x}{4 d^3}-\frac{5 A B^2 (b c-a d)^3 n^2 x}{4 d^3}+\frac{3 A^2 B (b c-a d)^2 n (a+b x)^2}{8 b d^2}+\frac{A B^2 (b c-a d)^2 n^2 (a+b x)^2}{4 b d^2}-\frac{A^2 B (b c-a d) n (a+b x)^3}{4 b d}+\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac{11 A B^2 (b c-a d)^4 n^2 \log (c+d x)}{4 b d^4}-\frac{3 A B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}+\frac{3 A B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{A B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 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 )}{2 b d^4}-\frac{3 B^3 (b c-a d)^4 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 )}{2 b d^4}-\frac{3 B^3 (b c-a d)^3 n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 B^3 (b c-a d)^2 n (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b d^2}-\frac{B^3 (b c-a d) n (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{4 b d^4}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{2 b d^4}+\frac{3 B^3 (b c-a d)^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b d^4}+\frac{\left (B^3 (b c-a d)^2 n^2\right ) \int (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{2 d^2}-\frac{\left (B^3 (b c-a d)^3 n^2\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{2 d^3}-\frac{\left (3 B^3 (b c-a d)^3 n^2\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{4 d^3}+\frac{\left (B^3 (b c-a d)^4 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b d^3}+\frac{\left (3 B^3 (b c-a d)^4 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{4 b d^3}-\frac{\left (3 A B^2 (b c-a d)^5 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 )}{2 b d^5}+\frac{\left (3 B^3 (b c-a d)^5 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 )}{2 b d^5}\\ &=-\frac{3 A^2 B (b c-a d)^3 n x}{4 d^3}-\frac{5 A B^2 (b c-a d)^3 n^2 x}{4 d^3}+\frac{3 A^2 B (b c-a d)^2 n (a+b x)^2}{8 b d^2}+\frac{A B^2 (b c-a d)^2 n^2 (a+b x)^2}{4 b d^2}-\frac{A^2 B (b c-a d) n (a+b x)^3}{4 b d}+\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac{11 A B^2 (b c-a d)^4 n^2 \log (c+d x)}{4 b d^4}-\frac{3 A B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}-\frac{5 B^3 (b c-a d)^3 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 A B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}+\frac{B^3 (b c-a d)^2 n^2 (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{A B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 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 )}{2 b d^4}-\frac{11 B^3 (b c-a d)^4 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 )}{4 b d^4}-\frac{3 B^3 (b c-a d)^3 n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 B^3 (b c-a d)^2 n (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b d^2}-\frac{B^3 (b c-a d) n (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{4 b d^4}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{2 b d^4}+\frac{3 B^3 (b c-a d)^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b d^4}-\frac{\left (3 A B^2 (b c-a d)^5 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 )}{2 b d^5}-\frac{\left (B^3 (b c-a d)^3 n^3\right ) \int \frac{a+b x}{c+d x} \, dx}{4 b d^2}+\frac{\left (B^3 (b c-a d)^4 n^3\right ) \int \frac{1}{c+d x} \, dx}{2 b d^3}+\frac{\left (3 B^3 (b c-a d)^4 n^3\right ) \int \frac{1}{c+d x} \, dx}{4 b d^3}-\frac{\left (3 B^3 (b c-a d)^5 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 )}{2 b d^5}+\frac{\left (B^3 (b c-a d)^5 n^3\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b d^4}+\frac{\left (3 B^3 (b c-a d)^5 n^3\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{4 b d^4}\\ &=-\frac{3 A^2 B (b c-a d)^3 n x}{4 d^3}-\frac{5 A B^2 (b c-a d)^3 n^2 x}{4 d^3}+\frac{3 A^2 B (b c-a d)^2 n (a+b x)^2}{8 b d^2}+\frac{A B^2 (b c-a d)^2 n^2 (a+b x)^2}{4 b d^2}-\frac{A^2 B (b c-a d) n (a+b x)^3}{4 b d}+\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac{11 A B^2 (b c-a d)^4 n^2 \log (c+d x)}{4 b d^4}+\frac{5 B^3 (b c-a d)^4 n^3 \log (c+d x)}{4 b d^4}-\frac{3 A B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}-\frac{5 B^3 (b c-a d)^3 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 A B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}+\frac{B^3 (b c-a d)^2 n^2 (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{A B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 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 )}{2 b d^4}-\frac{11 B^3 (b c-a d)^4 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 )}{4 b d^4}-\frac{3 B^3 (b c-a d)^3 n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 B^3 (b c-a d)^2 n (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b d^2}-\frac{B^3 (b c-a d) n (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{4 b d^4}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b d^4}-\frac{3 B^3 (b c-a d)^4 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 )}{2 b d^4}+\frac{3 B^3 (b c-a d)^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b d^4}-\frac{\left (B^3 (b c-a d)^3 n^3\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{4 b d^2}+\frac{\left (B^3 (b c-a d)^5 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 )}{2 b d^5}+\frac{\left (3 B^3 (b c-a d)^5 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 )}{4 b d^5}-\frac{\left (3 B^3 (b c-a d)^5 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 )}{2 b d^5}\\ &=-\frac{3 A^2 B (b c-a d)^3 n x}{4 d^3}-\frac{5 A B^2 (b c-a d)^3 n^2 x}{4 d^3}-\frac{B^3 (b c-a d)^3 n^3 x}{4 d^3}+\frac{3 A^2 B (b c-a d)^2 n (a+b x)^2}{8 b d^2}+\frac{A B^2 (b c-a d)^2 n^2 (a+b x)^2}{4 b d^2}-\frac{A^2 B (b c-a d) n (a+b x)^3}{4 b d}+\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac{11 A B^2 (b c-a d)^4 n^2 \log (c+d x)}{4 b d^4}+\frac{3 B^3 (b c-a d)^4 n^3 \log (c+d x)}{2 b d^4}-\frac{3 A B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}-\frac{5 B^3 (b c-a d)^3 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 A B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}+\frac{B^3 (b c-a d)^2 n^2 (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{A B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 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 )}{2 b d^4}-\frac{11 B^3 (b c-a d)^4 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 )}{4 b d^4}-\frac{3 B^3 (b c-a d)^3 n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 B^3 (b c-a d)^2 n (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b d^2}-\frac{B^3 (b c-a d) n (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{4 b d^4}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b d^4}-\frac{3 B^3 (b c-a d)^4 n^3 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b d^4}-\frac{3 B^3 (b c-a d)^4 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 )}{2 b d^4}+\frac{3 B^3 (b c-a d)^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b d^4}-\frac{\left (B^3 (b c-a d)^5 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 )}{2 b d^5}-\frac{\left (3 B^3 (b c-a d)^5 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 )}{4 b d^5}\\ &=-\frac{3 A^2 B (b c-a d)^3 n x}{4 d^3}-\frac{5 A B^2 (b c-a d)^3 n^2 x}{4 d^3}-\frac{B^3 (b c-a d)^3 n^3 x}{4 d^3}+\frac{3 A^2 B (b c-a d)^2 n (a+b x)^2}{8 b d^2}+\frac{A B^2 (b c-a d)^2 n^2 (a+b x)^2}{4 b d^2}-\frac{A^2 B (b c-a d) n (a+b x)^3}{4 b d}+\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac{11 A B^2 (b c-a d)^4 n^2 \log (c+d x)}{4 b d^4}+\frac{3 B^3 (b c-a d)^4 n^3 \log (c+d x)}{2 b d^4}-\frac{3 A B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}-\frac{5 B^3 (b c-a d)^3 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 A B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}+\frac{B^3 (b c-a d)^2 n^2 (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{A B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 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 )}{2 b d^4}-\frac{11 B^3 (b c-a d)^4 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 )}{4 b d^4}-\frac{3 B^3 (b c-a d)^3 n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 B^3 (b c-a d)^2 n (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b d^2}-\frac{B^3 (b c-a d) n (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{4 b d^4}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b d^4}-\frac{3 B^3 (b c-a d)^4 n^3 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b d^4}-\frac{3 B^3 (b c-a d)^4 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 )}{2 b d^4}+\frac{3 B^3 (b c-a d)^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b d^4}-\frac{\left (B^3 (b c-a d)^5 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 )}{2 b d^5}-\frac{\left (3 B^3 (b c-a d)^5 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 )}{4 b d^5}\\ &=-\frac{3 A^2 B (b c-a d)^3 n x}{4 d^3}-\frac{5 A B^2 (b c-a d)^3 n^2 x}{4 d^3}-\frac{B^3 (b c-a d)^3 n^3 x}{4 d^3}+\frac{3 A^2 B (b c-a d)^2 n (a+b x)^2}{8 b d^2}+\frac{A B^2 (b c-a d)^2 n^2 (a+b x)^2}{4 b d^2}-\frac{A^2 B (b c-a d) n (a+b x)^3}{4 b d}+\frac{A^3 (a+b x)^4}{4 b}+\frac{3 A^2 B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac{11 A B^2 (b c-a d)^4 n^2 \log (c+d x)}{4 b d^4}+\frac{3 B^3 (b c-a d)^4 n^3 \log (c+d x)}{2 b d^4}-\frac{3 A B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}-\frac{5 B^3 (b c-a d)^3 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 A B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}+\frac{B^3 (b c-a d)^2 n^2 (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{A B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d}+\frac{3 A^2 B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 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 )}{2 b d^4}-\frac{11 B^3 (b c-a d)^4 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 )}{4 b d^4}-\frac{3 B^3 (b c-a d)^3 n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^3}+\frac{3 B^3 (b c-a d)^2 n (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b d^2}-\frac{B^3 (b c-a d) n (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d}+\frac{3 A B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 B^3 (b c-a d)^4 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 )}{4 b d^4}+\frac{B^3 (a+b x)^4 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{3 A B^2 (b c-a d)^4 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b d^4}-\frac{11 B^3 (b c-a d)^4 n^3 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{4 b d^4}-\frac{3 B^3 (b c-a d)^4 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 )}{2 b d^4}+\frac{3 B^3 (b c-a d)^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b d^4}\\ \end{align*}
Mathematica [B] time = 9.69639, size = 9054, normalized size = 11.19 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 5.133, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \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 (A^{3} b^{3} x^{3} + 3 \, A^{3} a b^{2} x^{2} + 3 \, A^{3} a^{2} b x + A^{3} a^{3} +{\left (B^{3} b^{3} x^{3} + 3 \, B^{3} a b^{2} x^{2} + 3 \, B^{3} a^{2} b x + B^{3} a^{3}\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3} + 3 \,{\left (A B^{2} b^{3} x^{3} + 3 \, A B^{2} a b^{2} x^{2} + 3 \, A B^{2} a^{2} b x + A B^{2} a^{3}\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 3 \,{\left (A^{2} B b^{3} x^{3} + 3 \, A^{2} B a b^{2} x^{2} + 3 \, A^{2} B a^{2} b x + A^{2} B a^{3}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{3}{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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