Optimal. Leaf size=392 \[ -\frac {2 A B d^2 n (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}+\frac {2 B^2 d^2 n^2 (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {2 b^2 B^2 n^2 (c+d x)}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 B^2 d^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {2 b^2 B n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}+\frac {d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^3 g^2 i^2 n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.26, antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps
used = 10, 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^2 (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^2 (a+b x) (b c-a d)^3}-\frac {2 b^2 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^2 (a+b x) (b c-a d)^3}+\frac {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^2 (c+d x) (b c-a d)^3}-\frac {2 A B d^2 n (a+b x)}{g^2 i^2 (c+d x) (b c-a d)^3}-\frac {2 b d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g^2 i^2 n (b c-a d)^3}-\frac {2 b^2 B^2 n^2 (c+d x)}{g^2 i^2 (a+b x) (b c-a d)^3}-\frac {2 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^2 (c+d x) (b c-a d)^3}+\frac {2 B^2 d^2 n^2 (a+b x)}{g^2 i^2 (c+d x) (b c-a d)^3} \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}{(199 c+199 d x)^2 (a g+b g x)^2} \, 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}{39601 (b c-a d)^2 g^2 (a+b x)^2}-\frac {2 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 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}{39601 (b c-a d)^2 g^2 (c+d x)^2}+\frac {2 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2 (c+d x)}\right ) \, dx\\ &=-\frac {\left (2 b^2 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}{39601 (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} \, dx}{39601 (b c-a d)^3 g^2}+\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)^2} \, dx}{39601 (b c-a d)^2 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)^2} \, dx}{39601 (b c-a d)^2 g^2}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {(4 b B d 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}{39601 (b c-a d)^3 g^2}-\frac {(4 b B d 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}{39601 (b c-a d)^3 g^2}+\frac {(2 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 )}{(a+b x)^2 (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}+\frac {(2 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}{39601 (b c-a d)^2 g^2}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {(4 b B d 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}{39601 (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 ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{39601 (b c-a d)^3 g^2}+\frac {(2 b B n) \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}{39601 (b c-a d) g^2}+\frac {(2 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}{39601 (b c-a d) g^2}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {(4 b B d 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}{39601 (b c-a d)^2 g^2}-\frac {(4 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)}{(a+b x) (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}+\frac {(2 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 )}{(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}{39601 (b c-a d) g^2}+\frac {(2 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}{39601 (b c-a d) g^2}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 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} \, dx}{39601 (b c-a d)^2 g^2}+\frac {(4 b B d 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}{39601 (b c-a d)^2 g^2}-\frac {(4 b B d 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}{39601 (b c-a d)^2 g^2}-\frac {\left (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)^2} \, dx}{39601 (b c-a d)^2 g^2}\\ &=-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {\left (4 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} \, dx}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b 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}{39601 (b c-a d)^3 g^2}+\frac {(4 A b B d n) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}+\frac {\left (4 b 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}{39601 (b c-a d)^2 g^2}+\frac {\left (2 b B^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{39601 (b c-a d)^2 g^2}\\ &=\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {\left (2 b^2 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}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b^2 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}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b 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}{39601 (b c-a d)^3 g^2}+\frac {(4 A 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 )}{39601 (b c-a d)^2 g^2}+\frac {\left (2 b B^2 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{39601 (b c-a d) g^2}-\frac {\left (2 B^2 d n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{39601 (b c-a d) g^2}\\ &=\frac {2 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {(4 A b B d n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 A b^2 B d n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b^2 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}{39601 (b c-a d)^3 g^2}-\frac {\left (4 A 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 A b B d^2 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b 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}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b 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}{39601 (b c-a d)^2 g^2}+\frac {\left (2 b 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}{39601 (b c-a d) g^2}-\frac {\left (2 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}{39601 (b c-a d) g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {(4 A b B d n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{39601 (b c-a d)^3 g^2}+\frac {(4 A b B d n) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b^2 B^2 d n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b^2 B^2 d n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}+\frac {\left (4 A b B d^2 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{39601 (b c-a d)^3 g^2}-\frac {\left (2 b^2 B^2 d n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b B^2 d^2 n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b 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}{39601 (b c-a d)^2 g^2}-\frac {\left (4 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 ) \int \frac {\log (c+d x)}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {(4 A b B d n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b 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}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b 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}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (2 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b B^2 d n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 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 ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n^2 \log ^3(c+d x)}{118803 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n^2 \log ^3(c+d x)}{118803 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b^2 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b^2 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n^2 \log ^3(c+d x)}{118803 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n^2 \log ^3(c+d x)}{118803 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {4 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}-2 \frac {\left (4 b 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 )}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 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}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n^2 \log ^3(c+d x)}{118803 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}+\frac {4 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(870\) vs. \(2(392)=784\).
time = 0.79, size = 870, normalized size = 2.22 \begin {gather*} -\frac {2 b B^2 d n^2 (a+b x) (c+d x) \log ^3\left (\frac {a+b x}{c+d x}\right )+3 B n \log ^2\left (\frac {a+b x}{c+d x}\right ) \left (2 a A b c d+b^2 B c^2 n-a^2 B d^2 n+2 A b^2 c d x+2 a A b d^2 x+2 b^2 B c d n x-2 a b B d^2 n x+2 A b^2 d^2 x^2+2 b B d (a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 b B d n (a+b x) (c+d x) \log \left (\frac {a+b x}{c+d x}\right )\right )+6 B (b c-a d) n \log \left (\frac {a+b x}{c+d x}\right ) \left (A b c+a A d+b B c n-a B d n+2 A b d x+B (a d+b (c+2 d x)) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n (b c+a d+2 b d x) \log \left (\frac {a+b x}{c+d x}\right )\right )+6 b d (a+b x) (c+d x) \log (a+b x) \left (A^2+2 B^2 n^2+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\right )+3 b (b c-a d) (c+d x) \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 )+3 d (b c-a d) (a+b x) \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 )-6 b d (a+b x) (c+d x) \left (A^2+2 B^2 n^2+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\right ) \log (c+d x)}{3 (b c-a d)^3 g^2 i^2 (a+b x) (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (b g x +a g \right )^{2} \left (d i x +c i \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1854 vs.
\(2 (371) = 742\).
time = 0.43, size = 1854, normalized size = 4.73 \begin {gather*} B^{2} {\left (\frac {2 \, b d x + b c + a d}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} g^{2} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} g^{2} x + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} g^{2}} + \frac {2 \, b d \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}} - \frac {2 \, b d \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )^{2} + 2 \, A B {\left (\frac {2 \, b d x + b c + a d}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} g^{2} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} g^{2} x + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} g^{2}} + \frac {2 \, b d \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}} - \frac {2 \, b d \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {2}{3} \, {\left (\frac {{\left (3 \, b^{2} c^{2} - 3 \, a^{2} d^{2} + {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right )^{3} + 3 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )^{3} + 6 \, {\left (b^{2} c d - a b d^{2}\right )} x + 6 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, b^{2} d^{2} x^{2} + 2 \, a b c d + {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )\right )} n^{2}}{a b^{3} c^{4} g^{2} - 3 \, a^{2} b^{2} c^{3} d g^{2} + 3 \, a^{3} b c^{2} d^{2} g^{2} - a^{4} c d^{3} g^{2} + {\left (b^{4} c^{3} d g^{2} - 3 \, a b^{3} c^{2} d^{2} g^{2} + 3 \, a^{2} b^{2} c d^{3} g^{2} - a^{3} b d^{4} g^{2}\right )} x^{2} + {\left (b^{4} c^{4} g^{2} - 2 \, a b^{3} c^{3} d g^{2} + 2 \, a^{3} b c d^{3} g^{2} - a^{4} d^{4} g^{2}\right )} x} + \frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )^{2}\right )} n \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )}{a b^{3} c^{4} g^{2} - 3 \, a^{2} b^{2} c^{3} d g^{2} + 3 \, a^{3} b c^{2} d^{2} g^{2} - a^{4} c d^{3} g^{2} + {\left (b^{4} c^{3} d g^{2} - 3 \, a b^{3} c^{2} d^{2} g^{2} + 3 \, a^{2} b^{2} c d^{3} g^{2} - a^{3} b d^{4} g^{2}\right )} x^{2} + {\left (b^{4} c^{4} g^{2} - 2 \, a b^{3} c^{3} d g^{2} + 2 \, a^{3} b c d^{3} g^{2} - a^{4} d^{4} g^{2}\right )} x}\right )} B^{2} + \frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )^{2}\right )} A B n}{a b^{3} c^{4} g^{2} - 3 \, a^{2} b^{2} c^{3} d g^{2} + 3 \, a^{3} b c^{2} d^{2} g^{2} - a^{4} c d^{3} g^{2} + {\left (b^{4} c^{3} d g^{2} - 3 \, a b^{3} c^{2} d^{2} g^{2} + 3 \, a^{2} b^{2} c d^{3} g^{2} - a^{3} b d^{4} g^{2}\right )} x^{2} + {\left (b^{4} c^{4} g^{2} - 2 \, a b^{3} c^{3} d g^{2} + 2 \, a^{3} b c d^{3} g^{2} - a^{4} d^{4} g^{2}\right )} x} + A^{2} {\left (\frac {2 \, b d x + b c + a d}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} g^{2} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} g^{2} x + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} g^{2}} + \frac {2 \, b d \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}} - \frac {2 \, b d \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 740, normalized size = 1.89 \begin {gather*} \frac {3 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - 3 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + B^{2} a b c d n^{2} + {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n^{2} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 6 \, {\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} n^{2} + 3 \, {\left (2 \, {\left (A B + B^{2}\right )} b^{2} d^{2} n x^{2} + 2 \, {\left (A B + B^{2}\right )} a b c d n + {\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} n^{2} + 2 \, {\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2} + {\left ({\left (A B + B^{2}\right )} b^{2} c d + {\left (A B + B^{2}\right )} a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 6 \, {\left ({\left (A B + B^{2}\right )} b^{2} c^{2} - 2 \, {\left (A B + B^{2}\right )} a b c d + {\left (A B + B^{2}\right )} a^{2} d^{2}\right )} n + 6 \, {\left ({\left (A^{2} + 2 \, A B + B^{2}\right )} b^{2} c d - {\left (A^{2} + 2 \, A B + B^{2}\right )} a b d^{2} + 2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2}\right )} x + 6 \, {\left ({\left (A^{2} + 2 \, A B + B^{2}\right )} a b c d + {\left (B^{2} b^{2} c^{2} + B^{2} a^{2} d^{2}\right )} n^{2} + {\left (2 \, B^{2} b^{2} d^{2} n^{2} + {\left (A^{2} + 2 \, A B + B^{2}\right )} b^{2} d^{2}\right )} x^{2} + {\left ({\left (A B + B^{2}\right )} b^{2} c^{2} - {\left (A B + B^{2}\right )} a^{2} d^{2}\right )} n + {\left ({\left (A^{2} + 2 \, A B + B^{2}\right )} b^{2} c d + {\left (A^{2} + 2 \, A B + B^{2}\right )} a b d^{2} + 2 \, {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n^{2} + 2 \, {\left ({\left (A B + B^{2}\right )} b^{2} c d - {\left (A B + B^{2}\right )} a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{3 \, {\left ({\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} g^{2} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} g^{2} x + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\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 [A]
time = 257.14, size = 164, normalized size = 0.42 \begin {gather*} {\left (\frac {{\left (d x + c\right )} B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{{\left (b x + a\right )} g^{2}} + \frac {2 \, {\left (B^{2} n^{2} + A B n + B^{2} n\right )} {\left (d x + c\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )} g^{2}} + \frac {{\left (2 \, B^{2} n^{2} + 2 \, A B n + 2 \, B^{2} n + A^{2} + 2 \, A B + B^{2}\right )} {\left (d x + c\right )}}{{\left (b x + a\right )} g^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (a\,g+b\,g\,x\right )}^2\,{\left (c\,i+d\,i\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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