Optimal. Leaf size=231 \[ \frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g i^2 n (b c-a d)^2}-\frac {d (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g i^2 (c+d x) (b c-a d)^2}+\frac {2 A B d n (a+b x)}{g i^2 (c+d x) (b c-a d)^2}+\frac {2 B^2 d n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g i^2 (c+d x) (b c-a d)^2}-\frac {2 B^2 d n^2 (a+b x)}{g i^2 (c+d x) (b c-a d)^2} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 6.11, antiderivative size = 1803, normalized size of antiderivative = 7.81, number of steps used = 83, number of rules used = 31, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.689, Rules used = {2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 12, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610, 2525, 44, 2500, 2433, 2375, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 44
Rule 2301
Rule 2302
Rule 2317
Rule 2344
Rule 2374
Rule 2375
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2411
Rule 2418
Rule 2433
Rule 2434
Rule 2440
Rule 2488
Rule 2499
Rule 2500
Rule 2506
Rule 2507
Rule 2524
Rule 2525
Rule 2528
Rule 6589
Rule 6610
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(198 c+198 d x)^2 (a g+b g x)} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g (c+d x)}\right ) \, dx\\ &=\frac {b^2 \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x} \, dx}{39204 (b c-a d)^2 g}-\frac {(b d) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x} \, dx}{39204 (b c-a d)^2 g}-\frac {d \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2} \, dx}{39204 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {(b B n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac {(b B n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac {(B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^2} \, dx}{19602 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {(B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^2} \, dx}{19602 g}-\frac {(b B n) \int \frac {(b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d)^2 g}+\frac {(b B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d)^2 g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {(B n) \int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{19602 g}-\frac {(b B n) \int \frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}+\frac {(b B n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {\left (b^2 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac {(b B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{19602 (b c-a d)^2 g}-\frac {(b B n) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{19602 (b c-a d) g}+\frac {(b B n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{19602 (b c-a d) g}+\frac {(B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{19602 (b c-a d) g}\\ &=-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}+\frac {\left (b^2 B n\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac {(b B d n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}-\frac {(A b B n) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}-\frac {\left (b B^2 n\right ) \int \frac {\log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}+\frac {\left (b B^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac {\left (B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{19602 (b c-a d) g}\\ &=-\frac {b B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}+\frac {\left (b^2 B^2\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{39204 (b c-a d)^2 g}+\frac {\left (b^2 B n\right ) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x}\right ) \, dx}{19602 (b c-a d)^2 g}-\frac {(b B d n) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x}\right ) \, dx}{19602 (b c-a d)^2 g}-\frac {(A B n) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{19602 (b c-a d) g}+\frac {\left (B^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{19602 g}+\frac {\left (b B^2 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{19602 (b c-a d)^2 g}\\ &=-\frac {b B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {(A b B n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac {\left (A b^2 B n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac {\left (b^2 B^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac {(A B d n) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}-\frac {(A b B d n) \int \frac {\log (c+d x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 d n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}+\frac {\left (b B^2 n\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}+\frac {\left (B^2 n^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{19602 g}+\frac {\left (b^2 B^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac {\left (b^2 B^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 d n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}+\frac {\left (b B^2 d n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}\\ &=\frac {B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac {b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac {A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {A b B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {A b B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac {(A b B n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}-\frac {(A b B n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac {\left (b^2 B^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac {\left (b^2 B^2 n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac {(A b B d n) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19602 (b c-a d)^2 g}+\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}+\frac {\left (b^2 B^2 n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{39204 (b c-a d)^2 g}+\frac {\left (b^2 B^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 d n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{39204 (b c-a d)^2 g}+\frac {\left (b B^2 d n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 n^2\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}{19602 (b c-a d) g}+\frac {\left (b^2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}\\ &=\frac {B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac {b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac {A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {A b B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {A b B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac {(A b B n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}+\frac {\left (b B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac {\left (b B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{39204 (b c-a d)^2 g}+\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 d n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19602 (b c-a d)^2 g}\\ &=\frac {B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac {b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac {A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {A b B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {A b B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac {\left (B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{39204 (b c-a d)^2 g}-\frac {\left (B^2 d n\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{39204 (b c-a d)^2 g}-\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}+\frac {\left (B^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}\\ &=\frac {B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac {b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac {A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {A b B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log ^3(c+d x)}{117612 (b c-a d)^2 g}+\frac {A b B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39204 (b c-a d)^2 g}+\frac {b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}+\frac {\left (b B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}\\ &=\frac {B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac {b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac {A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {A b B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log (a+b x) \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log ^3(c+d x)}{117612 (b c-a d)^2 g}+\frac {A b B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39204 (b c-a d)^2 g}+\frac {b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{39204 d (b c-a d)^2 g}+\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}-\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{39204 d (b c-a d)^2 g}\\ &=\frac {B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac {b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac {A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {A b B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log (a+b x) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log ^3(c+d x)}{117612 (b c-a d)^2 g}+\frac {A b B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39204 (b c-a d)^2 g}+\frac {b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac {\left (b B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}+\frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}\\ &=\frac {B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac {b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac {A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {A b B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log (a+b x) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log ^3(c+d x)}{117612 (b c-a d)^2 g}+\frac {A b B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39204 (b c-a d)^2 g}+\frac {b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+2 \frac {\left (b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}\\ &=\frac {B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac {b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac {A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac {b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {A b B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac {A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac {b B^2 n^2 \log (a+b x) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac {b B^2 n^2 \log ^3(c+d x)}{117612 (b c-a d)^2 g}+\frac {A b B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39204 (b c-a d)^2 g}+\frac {b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac {b B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {A b B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac {b B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac {b B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}\\ \end {align*}
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Mathematica [B] time = 1.00, size = 789, normalized size = 3.42 \[ \frac {b \log (a+b x) \left (2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2-2 B^2 n \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+A^2-2 A B n+2 B^2 n^2\right )}{g i^2 (b c-a d)^2}+\frac {2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2-2 B^2 n \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+A^2-2 A B n+2 B^2 n^2}{g i^2 (c+d x) (b c-a d)}-\frac {b \log (c+d x) \left (2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2-2 B^2 n \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+A^2-2 A B n+2 B^2 n^2\right )}{g i^2 (b c-a d)^2}+\frac {\log ^2\left (\frac {a+b x}{c+d x}\right ) \left (b B^2 c n \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+b B^2 d n x \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )-a B^2 d n^2+A b B c n+A b B d n x-b B^2 d n^2 x\right )}{g i^2 (c+d x) (b c-a d)^2}-\frac {2 B n \log \left (\frac {a+b x}{c+d x}\right ) \left (-B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )-A+B n\right )}{g i^2 (c+d x) (b c-a d)}+\frac {b B^2 n^2 \log ^3\left (\frac {a+b x}{c+d x}\right )}{3 g i^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 432, normalized size = 1.87 \[ \frac {3 \, A^{2} b c - 3 \, A^{2} a d + {\left (B^{2} b d n^{2} x + B^{2} b c n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 6 \, {\left (B^{2} b c - B^{2} a d\right )} n^{2} + 3 \, {\left (B^{2} b c - B^{2} a d + {\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e)^{2} - 3 \, {\left (B^{2} a d n^{2} - A B b c n + {\left (B^{2} b d n^{2} - A B b d n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 6 \, {\left (A B b c - A B a d\right )} n + 3 \, {\left (2 \, A B b c - 2 \, A B a d + {\left (B^{2} b d n x + B^{2} b c n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 2 \, {\left (B^{2} b c - B^{2} a d\right )} n - 2 \, {\left (B^{2} a d n - A B b c + {\left (B^{2} b d n - A B b d\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 3 \, {\left (2 \, B^{2} a d n^{2} - 2 \, A B a d n + A^{2} b c + {\left (2 \, B^{2} b d n^{2} - 2 \, A B b d n + A^{2} b d\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{3 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g i^{2} x + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} g i^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 8.06, size = 314, normalized size = 1.36 \[ -\frac {1}{3} \, {\left (\frac {B^{2} b n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{3}}{b c g - a d g} - 3 \, {\left (\frac {{\left (b x + a\right )} B^{2} d n^{2}}{{\left (b c g - a d g\right )} {\left (d x + c\right )}} - \frac {A B b n + B^{2} b n}{b c g - a d g}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + \frac {3 \, {\left (A^{2} b + 2 \, A B b + B^{2} b\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b c g - a d g} + \frac {6 \, {\left (B^{2} d n^{2} - A B d n - B^{2} d n\right )} {\left (b x + a\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b c g - a d g\right )} {\left (d x + c\right )}} - \frac {3 \, {\left (2 \, B^{2} d n^{2} - 2 \, A B d n - 2 \, B^{2} d n + A^{2} d + 2 \, A B d + B^{2} d\right )} {\left (b x + a\right )}}{{\left (b c g - a d g\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right ) \left (d i x +c i \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.29, size = 1014, normalized size = 4.39 \[ B^{2} {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )^{2} + 2 \, A B {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{3} \, {\left (\frac {{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{3} - {\left (b d x + b c\right )} \log \left (d x + c\right )^{3} + 3 \, {\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + 3 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} + 6 \, b c - 6 \, a d + 6 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, b d x + {\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + 2 \, b c + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n^{2}}{b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x} - \frac {3 \, {\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x}\right )} B^{2} - \frac {{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} A B n}{b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x} + A^{2} {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.80, size = 365, normalized size = 1.58 \[ \frac {B^2\,b\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{3\,g\,i^2\,n\,{\left (a\,d-b\,c\right )}^2}-\frac {A^2-2\,A\,B\,n+2\,B^2\,n^2}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}-\frac {2\,B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (A-B\,n\right )}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}-{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {B^2}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}-\frac {B\,b\,\left (A-B\,n\right )}{g\,i^2\,n\,{\left (a\,d-b\,c\right )}^2}\right )-\frac {b\,\mathrm {atan}\left (\frac {b\,\left (2\,b\,d\,x+\frac {a^2\,d^2\,g\,i^2-b^2\,c^2\,g\,i^2}{g\,i^2\,\left (a\,d-b\,c\right )}\right )\,\left (A^2-2\,A\,B\,n+2\,B^2\,n^2\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (b\,A^2-2\,b\,A\,B\,n+2\,b\,B^2\,n^2\right )}\right )\,\left (A^2-2\,A\,B\,n+2\,B^2\,n^2\right )\,2{}\mathrm {i}}{g\,i^2\,{\left (a\,d-b\,c\right )}^2} \]
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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