Optimal. Leaf size=393 \[ \frac {b^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 h (b g-a h)^2}+\frac {B h n (a+b x) (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(g+h x) (b g-a h)^2 (d g-c h)}+\frac {B n (b c-a d) (-a d h-b c h+2 b d g) \log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(b g-a h)^2 (d g-c h)^2}-\frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 h (g+h x)^2}+\frac {B^2 n^2 (b c-a d) (-a d h-b c h+2 b d g) \text {Li}_2\left (\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {B^2 h n^2 (b c-a d)^2 \log \left (\frac {g+h x}{c+d x}\right )}{(b g-a h)^2 (d g-c h)^2} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 1.63, antiderivative size = 968, normalized size of antiderivative = 2.46, number of steps used = 29, number of rules used = 16, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {6742, 2492, 72, 2514, 2488, 2411, 2343, 2333, 2315, 2490, 36, 31, 2494, 2394, 2393, 2391} \[ -\frac {A^2}{2 h (g+h x)^2}+\frac {b^2 B n \log (a+b x) A}{h (b g-a h)^2}-\frac {B d^2 n \log (c+d x) A}{h (d g-c h)^2}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) A}{h (g+h x)^2}+\frac {B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x) A}{(b g-a h)^2 (d g-c h)^2}-\frac {B (b c-a d) n A}{(b g-a h) (d g-c h) (g+h x)}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}-\frac {B^2 (b c-a d)^2 h n^2 \log (c+d x)}{(b g-a h)^2 (d g-c h)^2}-\frac {b^2 B^2 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (b g-a h)^2}+\frac {B^2 d^2 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (d g-c h)^2}+\frac {B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h)^2 (d g-c h) (g+h x)}+\frac {B^2 (b c-a d)^2 h n^2 \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}-\frac {B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac {B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac {B^2 (b c-a d) (2 b d g-b c h-a d h) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac {B^2 d^2 n^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{h (d g-c h)^2}-\frac {B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \text {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \text {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {b^2 B^2 n^2 \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{h (b g-a h)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 36
Rule 72
Rule 2315
Rule 2333
Rule 2343
Rule 2391
Rule 2393
Rule 2394
Rule 2411
Rule 2488
Rule 2490
Rule 2492
Rule 2494
Rule 2514
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx &=\int \left (\frac {A^2}{(g+h x)^3}+\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3}+\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3}\right ) \, dx\\ &=-\frac {A^2}{2 h (g+h x)^2}+(2 A B) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx+B^2 \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^3} \, dx\\ &=-\frac {A^2}{2 h (g+h x)^2}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (g+h x)^2}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac {(A B (b c-a d) n) \int \frac {1}{(a+b x) (c+d x) (g+h x)^2} \, dx}{h}+\frac {\left (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 {A^2}{2 h (g+h x)^2}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (g+h x)^2}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac {(A B (b c-a d) n) \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}{h}+\frac {\left (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 {A^2}{2 h (g+h x)^2}-\frac {A B (b c-a d) n}{(b g-a h) (d g-c h) (g+h x)}+\frac {A b^2 B n \log (a+b x)}{h (b g-a h)^2}-\frac {A B d^2 n \log (c+d x)}{h (d g-c h)^2}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (g+h x)^2}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac {A B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac {\left (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 (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 (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 (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 {A^2}{2 h (g+h x)^2}-\frac {A B (b c-a d) n}{(b g-a h) (d g-c h) (g+h x)}+\frac {A b^2 B n \log (a+b x)}{h (b g-a h)^2}-\frac {A B d^2 n \log (c+d x)}{h (d g-c h)^2}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (g+h x)^2}+\frac {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 {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 {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 {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac {A B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac {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 (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 (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 (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 (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 (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^2}{2 h (g+h x)^2}-\frac {A B (b c-a d) n}{(b g-a h) (d g-c h) (g+h x)}+\frac {A b^2 B n \log (a+b x)}{h (b g-a h)^2}-\frac {A B d^2 n \log (c+d x)}{h (d g-c h)^2}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (g+h x)^2}+\frac {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 {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 {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 {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac {A B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}-\frac {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 {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 {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 (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 (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 (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 (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 (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 (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 {A^2}{2 h (g+h x)^2}-\frac {A B (b c-a d) n}{(b g-a h) (d g-c h) (g+h x)}+\frac {A b^2 B n \log (a+b x)}{h (b g-a h)^2}-\frac {A B d^2 n \log (c+d x)}{h (d g-c h)^2}-\frac {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 {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (g+h x)^2}+\frac {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 {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 {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 {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac {A B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac {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 {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 {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 {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 (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 (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 (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 (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 {A^2}{2 h (g+h x)^2}-\frac {A B (b c-a d) n}{(b g-a h) (d g-c h) (g+h x)}+\frac {A b^2 B n \log (a+b x)}{h (b g-a h)^2}-\frac {A B d^2 n \log (c+d x)}{h (d g-c h)^2}-\frac {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 {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (g+h x)^2}+\frac {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 {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 {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 {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac {A B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac {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 {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 {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 {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 {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 {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 {\left (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 (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^2}{2 h (g+h x)^2}-\frac {A B (b c-a d) n}{(b g-a h) (d g-c h) (g+h x)}+\frac {A b^2 B n \log (a+b x)}{h (b g-a h)^2}-\frac {A B d^2 n \log (c+d x)}{h (d g-c h)^2}-\frac {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 {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (g+h x)^2}+\frac {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 {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 {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 {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}+\frac {A B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac {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 {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 {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 {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 {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 {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 {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 {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}\\ \end {align*}
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Mathematica [B] time = 6.46, size = 15406, normalized size = 39.20 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \, A B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{2}}{h^{3} x^{3} + 3 \, g h^{2} x^{2} + 3 \, g^{2} h x + g^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (h x + g\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.96, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A \right )^{2}}{\left (h x +g \right )^{3}}\, dx \]
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 \[ -\frac {1}{2} \, B^{2} {\left (\frac {\log \left ({\left (d x + c\right )}^{n}\right )^{2}}{h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h} + 2 \, \int -\frac {d h x \log \relax (e)^{2} + c h \log \relax (e)^{2} + {\left (d h x + c h\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + 2 \, {\left (d h x \log \relax (e) + c h \log \relax (e)\right )} \log \left ({\left (b x + a\right )}^{n}\right ) + {\left (d g n + {\left (h n - 2 \, h \log \relax (e)\right )} d x - 2 \, c h \log \relax (e) - 2 \, {\left (d h x + c h\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{d h^{4} x^{4} + c g^{3} h + {\left (3 \, d g h^{3} + c h^{4}\right )} x^{3} + 3 \, {\left (d g^{2} h^{2} + c g h^{3}\right )} x^{2} + {\left (d g^{3} h + 3 \, c g^{2} h^{2}\right )} x}\,{d x}\right )} + \frac {{\left (\frac {b^{2} e n \log \left (b x + a\right )}{b^{2} g^{2} h - 2 \, a b g h^{2} + a^{2} h^{3}} - \frac {d^{2} e n \log \left (d x + c\right )}{d^{2} g^{2} h - 2 \, c d g h^{2} + c^{2} h^{3}} - \frac {{\left (2 \, a b d^{2} e g n - a^{2} d^{2} e h n - {\left (2 \, c d e g n - c^{2} e h n\right )} b^{2}\right )} \log \left (h x + g\right )}{{\left (d^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} a^{2} - 2 \, {\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} a b + {\left (d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2}\right )} b^{2}} + \frac {b c e n - a d e n}{{\left (d g^{2} h - c g h^{2}\right )} a - {\left (d g^{3} - c g^{2} h\right )} b + {\left ({\left (d g h^{2} - c h^{3}\right )} a - {\left (d g^{2} h - c g h^{2}\right )} b\right )} x}\right )} A B}{e} - \frac {A B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h} - \frac {A^{2}}{2 \, {\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} \]
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 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^2}{{\left (g+h\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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