Optimal. Leaf size=562 \[ -\frac {B^2 d^3 n^2 (a+b x)^2}{4 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {6 A b B d^2 n (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}+\frac {6 b B^2 d^2 n^2 (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {2 b^3 B^2 n^2 (c+d x)}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {6 b B^2 d^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}+\frac {B d^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {2 b^3 B n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {d^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{B (b c-a d)^4 g^2 i^3 n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.32, antiderivative size = 562, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {2561, 2395,
2333, 2332, 2342, 2341, 2339, 30} \begin {gather*} -\frac {b^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac {2 b^3 B n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac {b^2 d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{B g^2 i^3 n (b c-a d)^4}-\frac {d^3 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}+\frac {B d^3 n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}+\frac {3 b d^2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^2 i^3 (c+d x) (b c-a d)^4}-\frac {6 A b B d^2 n (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4}-\frac {2 b^3 B^2 n^2 (c+d x)}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac {B^2 d^3 n^2 (a+b x)^2}{4 g^2 i^3 (c+d x)^2 (b c-a d)^4}-\frac {6 b B^2 d^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g^2 i^3 (c+d x) (b c-a d)^4}+\frac {6 b B^2 d^2 n^2 (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2332
Rule 2333
Rule 2339
Rule 2341
Rule 2342
Rule 2395
Rule 2561
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}{(207 c+207 d x)^3 (a g+b g x)^2} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)^2}-\frac {b^3 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^2 g^2 (c+d x)^3}+\frac {2 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)^2}+\frac {b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b^3 d\right ) \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}{2956581 (b c-a d)^4 g^2}+\frac {\left (b^2 d^2\right ) \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}{2956581 (b c-a d)^4 g^2}+\frac {b^3 \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2} \, dx}{8869743 (b c-a d)^3 g^2}+\frac {\left (2 b d^2\right ) \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}{8869743 (b c-a d)^3 g^2}+\frac {d^2 \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3} \, dx}{8869743 (b c-a d)^2 g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B d n\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) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B n\right ) \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)^2 (c+d x)} \, dx}{8869743 (b c-a d)^3 g^2}+\frac {(4 b B d 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}{8869743 (b c-a d)^3 g^2}+\frac {(B d 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)^3} \, dx}{8869743 (b c-a d)^2 g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 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)^2 (c+d x)} \, dx}{8869743 (b c-a d)^2 g^2}+\frac {(4 b B d 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}{8869743 (b c-a d)^2 g^2}+\frac {(B d 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)^3} \, dx}{8869743 (b c-a d) g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^3 g^2}-\frac {\left (2 b^2 B d 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) (c+d x)} \, dx}{2956581 (b c-a d)^3 g^2}+\frac {\left (2 b^2 B n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b 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 (a+b x)}+\frac {d^2 \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}{8869743 (b c-a d)^2 g^2}+\frac {(4 b B d 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}{8869743 (b c-a d)^2 g^2}+\frac {(B d n) \int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (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)^3}-\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)^2}-\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{8869743 (b c-a d) g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {\left (b^3 B d n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (2 b^3 B d n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (4 b^3 B d n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^2 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^3 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{8869743 (b c-a d)^3 g^2}+\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^3 g^2}-\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^3 g^2}-\frac {\left (b B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{8869743 (b c-a d)^3 g^2}-\frac {\left (4 b B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{8869743 (b c-a d)^3 g^2}-\frac {\left (B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{8869743 (b c-a d)^2 g^2}\\ &=-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^3 B d 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}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B d^2 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)}{c+d x} \, dx}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d n\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{2956581 (b c-a d)^3 g^2}+\frac {\left (2 b^2 B^2 d 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}{2956581 (b c-a d)^3 g^2}-\frac {\left (b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (4 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{8869743 (b c-a d)^3 g^2}-\frac {\left (b B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8869743 (b c-a d)^3 g^2}-\frac {\left (4 b B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8869743 (b c-a d)^3 g^2}-\frac {\left (B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{17739486 (b c-a d)^2 g^2}\\ &=\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {\left (b^3 B^2 d\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^3 B d 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}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B d^2 n\right ) \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}{2956581 (b c-a d)^4 g^2}+\frac {(2 A b B d n) \text {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 )}{2956581 (b c-a d)^3 g^2}-\frac {\left (b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (4 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{8869743 (b c-a d)^2 g^2}-\frac {\left (b B^2 d n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8869743 (b c-a d)^2 g^2}-\frac {\left (4 b B^2 d n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8869743 (b c-a d)^2 g^2}-\frac {\left (B^2 d n^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{17739486 (b c-a d) g^2}\\ &=\frac {b^2 B^2 d \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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 A b^3 B d n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^3 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)}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 A b B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d^2 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d^2 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}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d 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}{2956581 (b c-a d)^3 g^2}-\frac {\left (b^3 B^2 d n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^3 B^2 d n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^3 B^2 d n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (2 b^3 B^2 d n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^3 B^2 d n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (4 b^3 B^2 d n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (4 b^2 B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{8869743 (b c-a d)^2 g^2}-\frac {\left (b B^2 d 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}{8869743 (b c-a d)^2 g^2}-\frac {\left (4 b B^2 d 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}{8869743 (b c-a d)^2 g^2}-\frac {\left (B^2 d n^2\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{17739486 (b c-a d) g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^3 B^2 d n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^3 B^2 d n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d^2 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^3 B^2 d n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^3 B^2 d n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^3 B^2 d n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (4 b^3 B^2 d n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d^2 n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d 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}{2956581 (b c-a d)^3 g^2}-\frac {\left (2 b^3 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 (c+d x)}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d n\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d n\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d^2 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}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d^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 \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (b B^2 d^2\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b B^2 d^2 n\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d n^2\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b B^2 d^2 n^2\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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 ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log ^3(c+d x)}{8869743 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d n\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d n\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d n^2\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log ^3(c+d x)}{8869743 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (b^3 B^2\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (b^3 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log ^3(c+d x)}{8869743 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d n\right ) \text {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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log ^3(c+d x)}{8869743 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}-2 \frac {\left (2 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log ^3(c+d x)}{8869743 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1334\) vs. \(2(562)=1124\).
time = 1.20, size = 1334, normalized size = 2.37 \begin {gather*} -\frac {4 b^2 B^2 d n^2 (a+b x) (c+d x)^2 \log ^3\left (\frac {a+b x}{c+d x}\right )+2 B n \log ^2\left (\frac {a+b x}{c+d x}\right ) \left (6 a A b^2 c^2 d+2 b^3 B c^3 n-6 a^2 b B c d^2 n+a^3 B d^3 n+6 A b^3 c^2 d x+12 a A b^2 c d^2 x+6 b^3 B c^2 d n x-12 a b^2 B c d^2 n x-3 a^2 b B d^3 n x+12 A b^3 c d^2 x^2+6 a A b^2 d^3 x^2-9 a b^2 B d^3 n x^2+6 A b^3 d^3 x^3-3 b^3 B d^3 n x^3+6 b^2 B d (a+b x) (c+d x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-6 b^2 B d n (a+b x) (c+d x)^2 \log \left (\frac {a+b x}{c+d x}\right )\right )+4 b^2 (b c-a d) (c+d x)^2 \left (A^2+2 A B n+2 B^2 n^2+B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B n (A+B n) \log \left (\frac {a+b x}{c+d x}\right )+B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (A+B n-B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+2 B (b c-a d) n \log \left (\frac {a+b x}{c+d x}\right ) \left (2 b d (a+b x) (c+d x) \left (4 A-5 B n+4 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-4 B n \log \left (\frac {a+b x}{c+d x}\right )\right )+d (b c-a d) (a+b x) \left (2 A-B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )+4 b^2 (c+d x)^2 \left (A+B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+d (b c-a d)^2 (a+b x) \left (2 A^2-2 A B n+B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-2 A+B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+6 b^2 d (a+b x) (c+d x)^2 \log (a+b x) \left (2 A^2-2 A B n+5 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-2 A+B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+2 b d (b c-a d) (a+b x) (c+d x) \left (4 A^2-10 A B n+11 B^2 n^2+4 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-4 A+5 B n) \log \left (\frac {a+b x}{c+d x}\right )+4 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-4 A+5 B n+4 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )-6 b^2 d (a+b x) (c+d x)^2 \left (2 A^2-2 A B n+5 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-2 A+B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) \log (c+d x)}{4 (b c-a d)^4 g^2 i^3 (a+b x) (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (b g x +a g \right )^{2} \left (d i x +c i \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 4053 vs. \(2 (530) = 1060\).
time = 1.01, size = 4053, normalized size = 7.21 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1656 vs. \(2 (530) = 1060\).
time = 0.45, size = 1656, normalized size = 2.95 \begin {gather*} -\frac {4 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} c^{3} + 6 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b^{2} c^{2} d + 12 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a^{2} b c d^{2} + 2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a^{3} d^{3} + 4 \, {\left (i \, B^{2} b^{3} d^{3} n^{2} x^{3} + i \, B^{2} a b^{2} c^{2} d n^{2} + {\left (2 i \, B^{2} b^{3} c d^{2} + i \, B^{2} a b^{2} d^{3}\right )} n^{2} x^{2} + {\left (i \, B^{2} b^{3} c^{2} d + 2 i \, B^{2} a b^{2} c d^{2}\right )} n^{2} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} - {\left (-8 i \, B^{2} b^{3} c^{3} - 15 i \, B^{2} a b^{2} c^{2} d + 24 i \, B^{2} a^{2} b c d^{2} - i \, B^{2} a^{3} d^{3}\right )} n^{2} + 6 \, {\left (2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} c d^{2} + 2 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a b^{2} d^{3} + 5 \, {\left (i \, B^{2} b^{3} c d^{2} - i \, B^{2} a b^{2} d^{3}\right )} n^{2} + 2 \, {\left ({\left (-i \, A B - i \, B^{2}\right )} b^{3} c d^{2} + {\left (i \, A B + i \, B^{2}\right )} a b^{2} d^{3}\right )} n\right )} x^{2} + 2 \, {\left (6 \, {\left (i \, A B + i \, B^{2}\right )} a b^{2} c^{2} d n + 3 \, {\left (-i \, B^{2} b^{3} d^{3} n^{2} + 2 \, {\left (i \, A B + i \, B^{2}\right )} b^{3} d^{3} n\right )} x^{3} + {\left (2 i \, B^{2} b^{3} c^{3} - 6 i \, B^{2} a^{2} b c d^{2} + i \, B^{2} a^{3} d^{3}\right )} n^{2} + 3 \, {\left (-3 i \, B^{2} a b^{2} d^{3} n^{2} + 2 \, {\left (2 \, {\left (i \, A B + i \, B^{2}\right )} b^{3} c d^{2} + {\left (i \, A B + i \, B^{2}\right )} a b^{2} d^{3}\right )} n\right )} x^{2} + 3 \, {\left ({\left (2 i \, B^{2} b^{3} c^{2} d - 4 i \, B^{2} a b^{2} c d^{2} - i \, B^{2} a^{2} b d^{3}\right )} n^{2} + 2 \, {\left ({\left (i \, A B + i \, B^{2}\right )} b^{3} c^{2} d + 2 \, {\left (i \, A B + i \, B^{2}\right )} a b^{2} c d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, {\left (4 \, {\left (i \, A B + i \, B^{2}\right )} b^{3} c^{3} + 15 \, {\left (-i \, A B - i \, B^{2}\right )} a b^{2} c^{2} d + 12 \, {\left (i \, A B + i \, B^{2}\right )} a^{2} b c d^{2} + {\left (-i \, A B - i \, B^{2}\right )} a^{3} d^{3}\right )} n + 3 \, {\left (6 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} c^{2} d + 4 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a b^{2} c d^{2} + 2 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a^{2} b d^{3} + {\left (13 i \, B^{2} b^{3} c^{2} d - 6 i \, B^{2} a b^{2} c d^{2} - 7 i \, B^{2} a^{2} b d^{3}\right )} n^{2} + 2 \, {\left ({\left (-i \, A B - i \, B^{2}\right )} b^{3} c^{2} d + 2 \, {\left (-i \, A B - i \, B^{2}\right )} a b^{2} c d^{2} + 3 \, {\left (i \, A B + i \, B^{2}\right )} a^{2} b d^{3}\right )} n\right )} x + 2 \, {\left (6 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b^{2} c^{2} d + 3 \, {\left (5 i \, B^{2} b^{3} d^{3} n^{2} + 2 \, {\left (-i \, A B - i \, B^{2}\right )} b^{3} d^{3} n + 2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} d^{3}\right )} x^{3} + {\left (4 i \, B^{2} b^{3} c^{3} + 12 i \, B^{2} a^{2} b c d^{2} - i \, B^{2} a^{3} d^{3}\right )} n^{2} + 3 \, {\left (6 \, {\left (-i \, A B - i \, B^{2}\right )} a b^{2} d^{3} n + 4 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} c d^{2} + 2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b^{2} d^{3} + {\left (8 i \, B^{2} b^{3} c d^{2} + 7 i \, B^{2} a b^{2} d^{3}\right )} n^{2}\right )} x^{2} + 2 \, {\left (2 \, {\left (i \, A B + i \, B^{2}\right )} b^{3} c^{3} + 6 \, {\left (-i \, A B - i \, B^{2}\right )} a^{2} b c d^{2} + {\left (i \, A B + i \, B^{2}\right )} a^{3} d^{3}\right )} n + 3 \, {\left (2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} c^{2} d + 4 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b^{2} c d^{2} + {\left (4 i \, B^{2} b^{3} c^{2} d + 8 i \, B^{2} a b^{2} c d^{2} + 3 i \, B^{2} a^{2} b d^{3}\right )} n^{2} + 2 \, {\left (2 \, {\left (i \, A B + i \, B^{2}\right )} b^{3} c^{2} d + 4 \, {\left (-i \, A B - i \, B^{2}\right )} a b^{2} c d^{2} + {\left (-i \, A B - i \, B^{2}\right )} a^{2} b d^{3}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} g^{2} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} g^{2} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} g^{2} x + {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4}\right )} g^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.14, size = 1785, normalized size = 3.18 \begin {gather*} \frac {\frac {-2\,A^2\,a^2\,d^2+10\,A^2\,a\,b\,c\,d+4\,A^2\,b^2\,c^2+2\,A\,B\,a^2\,d^2\,n-22\,A\,B\,a\,b\,c\,d\,n+8\,A\,B\,b^2\,c^2\,n-B^2\,a^2\,d^2\,n^2+23\,B^2\,a\,b\,c\,d\,n^2+8\,B^2\,b^2\,c^2\,n^2}{2\,\left (a\,d-b\,c\right )}+\frac {3\,x^2\,\left (2\,A^2\,b^2\,d^2-2\,A\,B\,b^2\,d^2\,n+5\,B^2\,b^2\,d^2\,n^2\right )}{a\,d-b\,c}+\frac {3\,x\,\left (6\,c\,A^2\,b^2\,d+2\,a\,A^2\,b\,d^2-2\,c\,A\,B\,b^2\,d\,n-6\,a\,A\,B\,b\,d^2\,n+13\,c\,B^2\,b^2\,d\,n^2+7\,a\,B^2\,b\,d^2\,n^2\right )}{2\,\left (a\,d-b\,c\right )}}{x\,\left (4\,a^3\,c\,d^3\,g^2\,i^3-6\,a^2\,b\,c^2\,d^2\,g^2\,i^3+2\,b^3\,c^4\,g^2\,i^3\right )+x^2\,\left (2\,a^3\,d^4\,g^2\,i^3-6\,a\,b^2\,c^2\,d^2\,g^2\,i^3+4\,b^3\,c^3\,d\,g^2\,i^3\right )+x^3\,\left (2\,a^2\,b\,d^4\,g^2\,i^3-4\,a\,b^2\,c\,d^3\,g^2\,i^3+2\,b^3\,c^2\,d^2\,g^2\,i^3\right )+2\,a^3\,c^2\,d^2\,g^2\,i^3+2\,a\,b^2\,c^4\,g^2\,i^3-4\,a^2\,b\,c^3\,d\,g^2\,i^3}-{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {\frac {B^2\,\left (a\,d+2\,b\,c\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {3\,B^2\,b\,d\,x}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (b\,c^2\,g^2\,i^3+2\,a\,d\,c\,g^2\,i^3\right )+x^2\,\left (a\,d^2\,g^2\,i^3+2\,b\,c\,d\,g^2\,i^3\right )+a\,c^2\,g^2\,i^3+b\,d^2\,g^2\,i^3\,x^3}+\frac {3\,B\,b^2\,d\,\left (2\,A-B\,n\right )}{2\,g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4}-\frac {3\,B^2\,b^2\,d\,\left (d\,g^2\,i^3\,n\,x^2\,\left (a\,d-b\,c\right )+\frac {a\,c\,g^2\,i^3\,n\,\left (a\,d-b\,c\right )}{b}+\frac {g^2\,i^3\,n\,x\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{b}\right )}{g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4\,\left (x\,\left (b\,c^2\,g^2\,i^3+2\,a\,d\,c\,g^2\,i^3\right )+x^2\,\left (a\,d^2\,g^2\,i^3+2\,b\,c\,d\,g^2\,i^3\right )+a\,c^2\,g^2\,i^3+b\,d^2\,g^2\,i^3\,x^3\right )}\right )-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {x\,\left (\frac {3\,b\,d\,n\,B^2}{2}+3\,A\,b\,d\,B\right )-\frac {B^2\,a\,d\,n}{2}+2\,B^2\,b\,c\,n+A\,B\,a\,d+2\,A\,B\,b\,c}{x\,\left (2\,a^3\,c\,d^3\,g^2\,i^3-3\,a^2\,b\,c^2\,d^2\,g^2\,i^3+b^3\,c^4\,g^2\,i^3\right )+x^2\,\left (a^3\,d^4\,g^2\,i^3-3\,a\,b^2\,c^2\,d^2\,g^2\,i^3+2\,b^3\,c^3\,d\,g^2\,i^3\right )+x^3\,\left (a^2\,b\,d^4\,g^2\,i^3-2\,a\,b^2\,c\,d^3\,g^2\,i^3+b^3\,c^2\,d^2\,g^2\,i^3\right )+a^3\,c^2\,d^2\,g^2\,i^3+a\,b^2\,c^4\,g^2\,i^3-2\,a^2\,b\,c^3\,d\,g^2\,i^3}-\frac {3\,B\,b^2\,d\,\left (2\,A-B\,n\right )\,\left (d\,g^2\,i^3\,n\,x^2\,{\left (a\,d-b\,c\right )}^3+\frac {g^2\,i^3\,n\,x\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^3}{b}+\frac {a\,c\,g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^3}{b}\right )}{g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4\,\left (x\,\left (2\,a^3\,c\,d^3\,g^2\,i^3-3\,a^2\,b\,c^2\,d^2\,g^2\,i^3+b^3\,c^4\,g^2\,i^3\right )+x^2\,\left (a^3\,d^4\,g^2\,i^3-3\,a\,b^2\,c^2\,d^2\,g^2\,i^3+2\,b^3\,c^3\,d\,g^2\,i^3\right )+x^3\,\left (a^2\,b\,d^4\,g^2\,i^3-2\,a\,b^2\,c\,d^3\,g^2\,i^3+b^3\,c^2\,d^2\,g^2\,i^3\right )+a^3\,c^2\,d^2\,g^2\,i^3+a\,b^2\,c^4\,g^2\,i^3-2\,a^2\,b\,c^3\,d\,g^2\,i^3\right )}\right )-\frac {B^2\,b^2\,d\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4}+\frac {b^2\,d\,\mathrm {atan}\left (\frac {b^2\,d\,\left (2\,A^2-2\,A\,B\,n+5\,B^2\,n^2\right )\,\left (2\,a^4\,d^4\,g^2\,i^3-4\,a^3\,b\,c\,d^3\,g^2\,i^3+4\,a\,b^3\,c^3\,d\,g^2\,i^3-2\,b^4\,c^4\,g^2\,i^3\right )\,3{}\mathrm {i}}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^4\,\left (6\,d\,A^2\,b^2-6\,d\,A\,B\,b^2\,n+15\,d\,B^2\,b^2\,n^2\right )}+\frac {b^3\,d^2\,x\,\left (2\,A^2-2\,A\,B\,n+5\,B^2\,n^2\right )\,\left (a^3\,d^3\,g^2\,i^3-3\,a^2\,b\,c\,d^2\,g^2\,i^3+3\,a\,b^2\,c^2\,d\,g^2\,i^3-b^3\,c^3\,g^2\,i^3\right )\,6{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4\,\left (6\,d\,A^2\,b^2-6\,d\,A\,B\,b^2\,n+15\,d\,B^2\,b^2\,n^2\right )}\right )\,\left (2\,A^2-2\,A\,B\,n+5\,B^2\,n^2\right )\,3{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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