Optimal. Leaf size=466 \[ \frac {3 B^2 n^2 (b c-a d) (-a d h-b c h+2 b d g) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b^2 d^2}-\frac {3 B^2 h n^2 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b^2 d^2}+\frac {3 B n (b c-a d) (-a d h-b c h+2 b d g) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 b^2 d^2}-\frac {(b g-a h)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{2 b^2 h}-\frac {3 B h n (a+b x) (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 b^2 d}+\frac {(g+h x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{2 h}-\frac {3 B^3 n^3 (b c-a d) (-a d h-b c h+2 b d g) \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}-\frac {3 B^3 h n^3 (b c-a d)^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 2.11, antiderivative size = 1030, normalized size of antiderivative = 2.21, number of steps used = 35, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {6742, 2492, 72, 2514, 2486, 31, 2488, 2411, 2343, 2333, 2315, 2506, 6610} \[ \frac {(g+h x)^2 A^3}{2 h}-\frac {3 B (b c-a d) h n x A^2}{2 b d}-\frac {3 B (b g-a h)^2 n \log (a+b x) A^2}{2 b^2 h}+\frac {3 B (d g-c h)^2 n \log (c+d x) A^2}{2 d^2 h}+\frac {3 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) A^2}{2 h}+\frac {3 B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) A}{2 h}+\frac {3 B^2 (b c-a d)^2 h n^2 \log (c+d x) A}{b^2 d^2}-\frac {3 B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) A}{b^2 d}+\frac {3 B^2 (b g-a h)^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 ) A}{b^2 h}-\frac {3 B^2 (d g-c h)^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 ) A}{d^2 h}-\frac {3 B^2 (d g-c h)^2 n^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) A}{d^2 h}-\frac {3 B^2 (b g-a h)^2 n^2 \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right ) A}{b^2 h}+\frac {B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\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^2 d}+\frac {3 B^3 (b g-a h)^2 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 b^2 h}-\frac {3 B^3 (d g-c h)^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 d^2 h}-\frac {3 B^3 (b c-a d)^2 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^2 d^2}-\frac {3 B^3 (b c-a d)^2 h n^3 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}-\frac {3 B^3 (b g-a h)^2 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{b^2 h}-\frac {3 B^3 (d g-c h)^2 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 )}{d^2 h}-\frac {3 B^3 (b g-a h)^2 n^3 \text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{b^2 h}+\frac {3 B^3 (d g-c h)^2 n^3 \text {PolyLog}\left (3,1-\frac {b c-a d}{b (c+d x)}\right )}{d^2 h} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 72
Rule 2315
Rule 2333
Rule 2343
Rule 2411
Rule 2486
Rule 2488
Rule 2492
Rule 2506
Rule 2514
Rule 6610
Rule 6742
Rubi steps
\begin {align*} \int (g+h x) \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)+3 A^2 B (g+h x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 A B^2 (g+h x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 (g+h x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac {A^3 (g+h x)^2}{2 h}+\left (3 A^2 B\right ) \int (g+h x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+\left (3 A B^2\right ) \int (g+h x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+B^3 \int (g+h x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac {A^3 (g+h x)^2}{2 h}+\frac {3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac {\left (3 A^2 B (b c-a d) n\right ) \int \frac {(g+h x)^2}{(a+b x) (c+d x)} \, dx}{2 h}-\frac {\left (3 A B^2 (b c-a d) n\right ) \int \frac {(g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{h}-\frac {\left (3 B^3 (b c-a d) n\right ) \int \frac {(g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{2 h}\\ &=\frac {A^3 (g+h x)^2}{2 h}+\frac {3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac {\left (3 A^2 B (b c-a d) n\right ) \int \left (\frac {h^2}{b d}+\frac {(b g-a h)^2}{b (b c-a d) (a+b x)}+\frac {(d g-c h)^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{2 h}-\frac {\left (3 A B^2 (b c-a d) n\right ) \int \left (\frac {h^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {(b g-a h)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)}+\frac {(d g-c h)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d (-b c+a d) (c+d x)}\right ) \, dx}{h}-\frac {\left (3 B^3 (b c-a d) n\right ) \int \left (\frac {h^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {(b g-a h)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)}+\frac {(d g-c h)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{d (-b c+a d) (c+d x)}\right ) \, dx}{2 h}\\ &=-\frac {3 A^2 B (b c-a d) h n x}{2 b d}+\frac {A^3 (g+h x)^2}{2 h}-\frac {3 A^2 B (b g-a h)^2 n \log (a+b x)}{2 b^2 h}+\frac {3 A^2 B (d g-c h)^2 n \log (c+d x)}{2 d^2 h}+\frac {3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac {\left (3 A B^2 (b c-a d) h n\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b d}-\frac {\left (3 B^3 (b c-a d) h n\right ) \int \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{2 b d}-\frac {\left (3 A B^2 (b g-a h)^2 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{b h}-\frac {\left (3 B^3 (b g-a h)^2 n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{2 b h}+\frac {\left (3 A B^2 (d g-c h)^2 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{d h}+\frac {\left (3 B^3 (d g-c h)^2 n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 d h}\\ &=-\frac {3 A^2 B (b c-a d) h n x}{2 b d}+\frac {A^3 (g+h x)^2}{2 h}-\frac {3 A^2 B (b g-a h)^2 n \log (a+b x)}{2 b^2 h}+\frac {3 A^2 B (d g-c h)^2 n \log (c+d x)}{2 d^2 h}-\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^2 d}+\frac {3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 A B^2 (b g-a h)^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 )}{b^2 h}-\frac {3 A B^2 (d g-c h)^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 )}{d^2 h}-\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^2 d}+\frac {3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 B^3 (b g-a h)^2 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 b^2 h}-\frac {3 B^3 (d g-c h)^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 d^2 h}+\frac {B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {\left (3 A B^2 (b c-a d)^2 h n^2\right ) \int \frac {1}{c+d x} \, dx}{b^2 d}+\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} \, dx}{b^2 d}-\frac {\left (3 A B^2 (b c-a d) (b g-a h)^2 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^2 h}-\frac {\left (3 B^3 (b c-a d) (b g-a h)^2 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^2 h}+\frac {\left (3 A B^2 (b c-a d) (d g-c h)^2 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^2 h}+\frac {\left (3 B^3 (b c-a d) (d g-c h)^2 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^2 h}\\ &=-\frac {3 A^2 B (b c-a d) h n x}{2 b d}+\frac {A^3 (g+h x)^2}{2 h}-\frac {3 A^2 B (b g-a h)^2 n \log (a+b x)}{2 b^2 h}+\frac {3 A^2 B (d g-c h)^2 n \log (c+d x)}{2 d^2 h}+\frac {3 A B^2 (b c-a d)^2 h n^2 \log (c+d x)}{b^2 d^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^2 d}+\frac {3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 A B^2 (b g-a h)^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 )}{b^2 h}-\frac {3 A B^2 (d g-c h)^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 )}{d^2 h}-\frac {3 B^3 (b c-a d)^2 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^2 d^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^2 d}+\frac {3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 B^3 (b g-a h)^2 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 b^2 h}-\frac {3 B^3 (d g-c h)^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 d^2 h}+\frac {B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac {3 B^3 (b g-a h)^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}{d (a+b x)}\right )}{b^2 h}-\frac {3 B^3 (d g-c h)^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 )}{d^2 h}-\frac {\left (3 A B^2 (b c-a d) (b g-a h)^2 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^3 h}+\frac {\left (3 A B^2 (b c-a d) (d g-c h)^2 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^3 h}+\frac {\left (3 B^3 (b c-a d)^3 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^2 d^2}+\frac {\left (3 B^3 (b c-a d) (b g-a h)^2 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^2 h}+\frac {\left (3 B^3 (b c-a d) (d g-c h)^2 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^2 h}\\ &=-\frac {3 A^2 B (b c-a d) h n x}{2 b d}+\frac {A^3 (g+h x)^2}{2 h}-\frac {3 A^2 B (b g-a h)^2 n \log (a+b x)}{2 b^2 h}+\frac {3 A^2 B (d g-c h)^2 n \log (c+d x)}{2 d^2 h}+\frac {3 A B^2 (b c-a d)^2 h n^2 \log (c+d x)}{b^2 d^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^2 d}+\frac {3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 A B^2 (b g-a h)^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 )}{b^2 h}-\frac {3 A B^2 (d g-c h)^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 )}{d^2 h}-\frac {3 B^3 (b c-a d)^2 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^2 d^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^2 d}+\frac {3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 B^3 (b g-a h)^2 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 b^2 h}-\frac {3 B^3 (d g-c h)^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 d^2 h}+\frac {B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac {3 B^3 (b g-a h)^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}{d (a+b x)}\right )}{b^2 h}-\frac {3 B^3 (d g-c h)^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 )}{d^2 h}-\frac {3 B^3 (b g-a h)^2 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 h}+\frac {3 B^3 (d g-c h)^2 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^2 h}+\frac {\left (3 A B^2 (b c-a d) (b g-a h)^2 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^3 h}-\frac {\left (3 A B^2 (b c-a d) (d g-c h)^2 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^3 h}+\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}{b x}\right )}{x \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )} \, dx,x,c+d x\right )}{b^2 d^3}\\ &=-\frac {3 A^2 B (b c-a d) h n x}{2 b d}+\frac {A^3 (g+h x)^2}{2 h}-\frac {3 A^2 B (b g-a h)^2 n \log (a+b x)}{2 b^2 h}+\frac {3 A^2 B (d g-c h)^2 n \log (c+d x)}{2 d^2 h}+\frac {3 A B^2 (b c-a d)^2 h n^2 \log (c+d x)}{b^2 d^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^2 d}+\frac {3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 A B^2 (b g-a h)^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 )}{b^2 h}-\frac {3 A B^2 (d g-c h)^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 )}{d^2 h}-\frac {3 B^3 (b c-a d)^2 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^2 d^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^2 d}+\frac {3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 B^3 (b g-a h)^2 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 b^2 h}-\frac {3 B^3 (d g-c h)^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 d^2 h}+\frac {B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac {3 B^3 (b g-a h)^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}{d (a+b x)}\right )}{b^2 h}-\frac {3 B^3 (d g-c h)^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 )}{d^2 h}-\frac {3 B^3 (b g-a h)^2 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 h}+\frac {3 B^3 (d g-c h)^2 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^2 h}+\frac {\left (3 A B^2 (b c-a d) (b g-a h)^2 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^3 h}-\frac {\left (3 A B^2 (b c-a d) (d g-c h)^2 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^3 h}-\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}\right )}{\left (\frac {-b c+a d}{d}+\frac {b}{d x}\right ) x} \, dx,x,\frac {1}{c+d x}\right )}{b^2 d^3}\\ &=-\frac {3 A^2 B (b c-a d) h n x}{2 b d}+\frac {A^3 (g+h x)^2}{2 h}-\frac {3 A^2 B (b g-a h)^2 n \log (a+b x)}{2 b^2 h}+\frac {3 A^2 B (d g-c h)^2 n \log (c+d x)}{2 d^2 h}+\frac {3 A B^2 (b c-a d)^2 h n^2 \log (c+d x)}{b^2 d^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^2 d}+\frac {3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 A B^2 (b g-a h)^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 )}{b^2 h}-\frac {3 A B^2 (d g-c h)^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 )}{d^2 h}-\frac {3 B^3 (b c-a d)^2 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^2 d^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^2 d}+\frac {3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 B^3 (b g-a h)^2 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 b^2 h}-\frac {3 B^3 (d g-c h)^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 d^2 h}+\frac {B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac {3 A B^2 (d g-c h)^2 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 h}-\frac {3 A B^2 (b g-a h)^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 h}-\frac {3 B^3 (b g-a h)^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}{d (a+b x)}\right )}{b^2 h}-\frac {3 B^3 (d g-c h)^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 )}{d^2 h}-\frac {3 B^3 (b g-a h)^2 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 h}+\frac {3 B^3 (d g-c h)^2 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^2 h}-\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}\right )}{\frac {b}{d}+\frac {(-b c+a d) x}{d}} \, dx,x,\frac {1}{c+d x}\right )}{b^2 d^3}\\ &=-\frac {3 A^2 B (b c-a d) h n x}{2 b d}+\frac {A^3 (g+h x)^2}{2 h}-\frac {3 A^2 B (b g-a h)^2 n \log (a+b x)}{2 b^2 h}+\frac {3 A^2 B (d g-c h)^2 n \log (c+d x)}{2 d^2 h}+\frac {3 A B^2 (b c-a d)^2 h n^2 \log (c+d x)}{b^2 d^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^2 d}+\frac {3 A^2 B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 A B^2 (b g-a h)^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 )}{b^2 h}-\frac {3 A B^2 (d g-c h)^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 )}{d^2 h}-\frac {3 B^3 (b c-a d)^2 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^2 d^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^2 d}+\frac {3 A B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac {3 B^3 (b g-a h)^2 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 b^2 h}-\frac {3 B^3 (d g-c h)^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 d^2 h}+\frac {B^3 (g+h x)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac {3 A B^2 (d g-c h)^2 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 h}-\frac {3 B^3 (b c-a d)^2 h n^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}-\frac {3 A B^2 (b g-a h)^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 h}-\frac {3 B^3 (b g-a h)^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}{d (a+b x)}\right )}{b^2 h}-\frac {3 B^3 (d g-c h)^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 )}{d^2 h}-\frac {3 B^3 (b g-a h)^2 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 h}+\frac {3 B^3 (d g-c h)^2 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{d^2 h}\\ \end {align*}
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Mathematica [F] time = 3.29, size = 0, normalized size = 0.00 \[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{3} h x + A^{3} g + {\left (B^{3} h x + B^{3} g\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3} + 3 \, {\left (A B^{2} h x + A B^{2} g\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 3 \, {\left (A^{2} B h x + A^{2} B g\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 8.96, size = 0, normalized size = 0.00 \[ \int \left (h x +g \right ) \left (B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A \right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (g+h\,x\right )\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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