Optimal. Leaf size=611 \[ -\frac {6 B^3 d^2 n^3 (c+d x)}{(b c-a d)^3 (a+b x)}+\frac {3 b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac {2 b^2 B^3 n^3 (c+d x)^3}{27 (b c-a d)^3 (a+b x)^3}-\frac {6 B^2 d^2 n^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^3 (a+b x)}+\frac {3 b B^2 d n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{2 (b c-a d)^3 (a+b x)^2}-\frac {2 b^2 B^2 n^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{9 (b c-a d)^3 (a+b x)^3}-\frac {3 B d^2 n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^3 (a+b x)}+\frac {3 b B d n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 (b c-a d)^3 (a+b x)^2}-\frac {b^2 B n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 (b c-a d)^3 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 (b c-a d)^3 (a+b x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.32, antiderivative size = 611, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2573, 2549,
2395, 2342, 2341} \begin {gather*} -\frac {2 b^2 B^2 n^2 (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{9 (a+b x)^3 (b c-a d)^3}-\frac {b^2 B n (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{3 (a+b x)^3 (b c-a d)^3}-\frac {b^2 (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{3 (a+b x)^3 (b c-a d)^3}-\frac {6 B^2 d^2 n^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)^3}+\frac {3 b B^2 d n^2 (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{2 (a+b x)^2 (b c-a d)^3}-\frac {3 B d^2 n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)^3}-\frac {d^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x) (b c-a d)^3}+\frac {3 b B d n (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {b d (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^2 (b c-a d)^3}-\frac {2 b^2 B^3 n^3 (c+d x)^3}{27 (a+b x)^3 (b c-a d)^3}-\frac {6 B^3 d^2 n^3 (c+d x)}{(a+b x) (b c-a d)^3}+\frac {3 b B^3 d n^3 (c+d x)^2}{4 (a+b x)^2 (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2341
Rule 2342
Rule 2395
Rule 2549
Rule 2573
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx &=\int \left (\frac {A^3}{(a+b x)^4}+\frac {3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}+\frac {3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}+\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}\right ) \, dx\\ &=-\frac {A^3}{3 b (a+b x)^3}+\left (3 A^2 B\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx+\left (3 A B^2\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx+B^3 \int \frac {\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {\left (A^2 B (b c-a d) n\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b}+\frac {\left (2 A B^2 (b c-a d) n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4 (c+d x)} \, dx}{b}+\frac {\left (B^3 (b c-a d) n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4 (c+d x)} \, dx}{b}\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {\left (A^2 B (b c-a d) n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b}+\frac {\left (2 A B^2 (b c-a d) n\right ) \int \left (\frac {b \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)^4}-\frac {b d \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b}+\frac {\left (B^3 (b c-a d) n\right ) \int \left (\frac {b \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)^4}-\frac {b d \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b}\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B n}{3 b (a+b x)^3}+\frac {A^2 B d n}{2 b (b c-a d) (a+b x)^2}-\frac {A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac {A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}+\frac {A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\left (2 A B^2 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx+\left (B^3 n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx-\frac {\left (2 A B^2 d^3 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{(b c-a d)^3}-\frac {\left (B^3 d^3 n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{(b c-a d)^3}+\frac {\left (2 A B^2 d^4 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b (b c-a d)^3}+\frac {\left (B^3 d^4 n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b (b c-a d)^3}+\frac {\left (2 A B^2 d^2 n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{(b c-a d)^2}+\frac {\left (B^3 d^2 n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{(b c-a d)^2}-\frac {\left (2 A B^2 d n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{b c-a d}-\frac {\left (B^3 d n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{b c-a d}\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B n}{3 b (a+b x)^3}+\frac {A^2 B d n}{2 b (b c-a d) (a+b x)^2}-\frac {A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac {A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}+\frac {A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}-\frac {2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {2 A B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {2 A B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {B^3 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {\left (b B^3 d n\right ) \int \frac {(c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{(b c-a d)^2}+\frac {\left (B^3 d^2 n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{(b c-a d)^2}-\frac {\left (A B^2 d n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b}+\frac {\left (2 A B^2 d^2 n^2\right ) \int \frac {1}{(a+b x)^2} \, dx}{(b c-a d)^2}+\frac {\left (2 B^3 d^2 n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{(b c-a d)^2}-\frac {\left (2 A B^2 d^3 n^2\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b (b c-a d)^2}+\frac {\left (2 A B^2 d^3 n^2\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{b (b c-a d)^2}-\frac {\left (2 B^3 d^3 n^2\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{b (b c-a d)^2}+\frac {\left (2 B^3 d^3 n^2\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{b (b c-a d)^2}+\frac {\left (2 A B^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b}+\frac {\left (2 B^3 (b c-a d) n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b}\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B n}{3 b (a+b x)^3}+\frac {A^2 B d n}{2 b (b c-a d) (a+b x)^2}-\frac {A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac {2 A B^2 d^2 n^2}{b (b c-a d)^2 (a+b x)}-\frac {A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}+\frac {A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}-\frac {2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac {2 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {2 A B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {2 A B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {B^3 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {\left (A B^2 d n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b}-\frac {\left (b B^3 d n^2\right ) \int \frac {(c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{(b c-a d)^2}+\frac {\left (2 A B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b x}\right )}{x \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )} \, dx,x,c+d x\right )}{b (b c-a d)^2}+\frac {\left (2 B^3 d^2 n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{(b c-a d)^2}-\frac {\left (2 A B^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b c-a d}{d x}\right )}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^2 (b c-a d)^2}+\frac {\left (2 A B^2 (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b}+\frac {\left (2 B^3 (b c-a d) n^2\right ) \int \left (\frac {b \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)^4}-\frac {b d \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b}+\frac {\left (2 B^3 d^2 n^3\right ) \int \frac {1}{(a+b x)^2} \, dx}{(b c-a d)^2}+\frac {\left (2 B^3 d^3 n^3\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b (b c-a d)^2}+\frac {\left (2 B^3 d^3 n^3\right ) \int \frac {\text {Li}_2\left (1+\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{b (b c-a d)^2}\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B n}{3 b (a+b x)^3}-\frac {2 A B^2 n^2}{9 b (a+b x)^3}+\frac {A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac {5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}-\frac {A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac {11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac {2 B^3 d^2 n^3}{b (b c-a d)^2 (a+b x)}-\frac {A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac {5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}+\frac {A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac {5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}-\frac {2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac {4 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {2 A B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {2 A B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {B^3 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}+\frac {1}{3} \left (2 B^3 n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx-\frac {\left (2 B^3 d^3 n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{3 (b c-a d)^3}+\frac {\left (2 B^3 d^4 n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{3 b (b c-a d)^3}-\frac {\left (2 A B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\left (\frac {-b c+a d}{d}+\frac {b}{d x}\right ) x} \, dx,x,\frac {1}{c+d x}\right )}{b (b c-a d)^2}+\frac {\left (2 B^3 d^2 n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{3 (b c-a d)^2}+\frac {\left (2 A B^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\left (\frac {b c-a d}{b}+\frac {d}{b x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{b^2 (b c-a d)^2}-\frac {\left (2 B^3 d n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{3 (b c-a d)}-\frac {\left (b B^3 d n^3\right ) \int \frac {c+d x}{(a+b x)^3} \, dx}{2 (b c-a d)^2}+\frac {\left (2 B^3 d^2 n^3\right ) \int \frac {1}{(a+b x)^2} \, dx}{(b c-a d)^2}\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B n}{3 b (a+b x)^3}-\frac {2 A B^2 n^2}{9 b (a+b x)^3}+\frac {A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac {5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}-\frac {A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac {11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac {4 B^3 d^2 n^3}{b (b c-a d)^2 (a+b x)}+\frac {b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac {A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac {5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}+\frac {A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac {5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}+\frac {B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac {2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac {14 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac {b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {2 A B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 A B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {B^3 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {\left (2 A B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\frac {b}{d}+\frac {(-b c+a d) x}{d}} \, dx,x,\frac {1}{c+d x}\right )}{b (b c-a d)^2}+\frac {\left (2 A B^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\frac {d}{b}+\frac {(b c-a d) x}{b}} \, dx,x,\frac {1}{a+b x}\right )}{b^2 (b c-a d)^2}-\frac {\left (B^3 d n^3\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b}+\frac {\left (2 B^3 d^2 n^3\right ) \int \frac {1}{(a+b x)^2} \, dx}{3 (b c-a d)^2}-\frac {\left (2 B^3 d^3 n^3\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{3 b (b c-a d)^2}+\frac {\left (2 B^3 d^3 n^3\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{3 b (b c-a d)^2}+\frac {\left (2 B^3 (b c-a d) n^3\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b}\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B n}{3 b (a+b x)^3}-\frac {2 A B^2 n^2}{9 b (a+b x)^3}+\frac {A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac {5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}-\frac {A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac {11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac {14 B^3 d^2 n^3}{3 b (b c-a d)^2 (a+b x)}+\frac {b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac {A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac {5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}+\frac {A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac {5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}+\frac {B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac {2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac {14 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac {b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {2 A B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 A B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {B^3 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 A B^2 d^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {2 A B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {\left (B^3 d n^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{3 b}+\frac {\left (2 B^3 d^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b x}\right )}{x \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )} \, dx,x,c+d x\right )}{3 b (b c-a d)^2}-\frac {\left (2 B^3 d^3 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b c-a d}{d x}\right )}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{3 b^2 (b c-a d)^2}+\frac {\left (2 B^3 (b c-a d) n^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{9 b}\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B n}{3 b (a+b x)^3}-\frac {2 A B^2 n^2}{9 b (a+b x)^3}-\frac {2 B^3 n^3}{27 b (a+b x)^3}+\frac {A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac {5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}+\frac {5 B^3 d n^3}{18 b (b c-a d) (a+b x)^2}-\frac {A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac {11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac {47 B^3 d^2 n^3}{9 b (b c-a d)^2 (a+b x)}+\frac {b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac {A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac {5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}-\frac {5 B^3 d^3 n^3 \log (a+b x)}{9 b (b c-a d)^3}+\frac {A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac {5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}+\frac {5 B^3 d^3 n^3 \log (c+d x)}{9 b (b c-a d)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}+\frac {B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac {2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac {14 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac {b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {2 A B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 A B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {B^3 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 A B^2 d^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {2 A B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {\left (2 B^3 d^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\left (\frac {-b c+a d}{d}+\frac {b}{d x}\right ) x} \, dx,x,\frac {1}{c+d x}\right )}{3 b (b c-a d)^2}+\frac {\left (2 B^3 d^3 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\left (\frac {b c-a d}{b}+\frac {d}{b x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{3 b^2 (b c-a d)^2}\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B n}{3 b (a+b x)^3}-\frac {2 A B^2 n^2}{9 b (a+b x)^3}-\frac {2 B^3 n^3}{27 b (a+b x)^3}+\frac {A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac {5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}+\frac {5 B^3 d n^3}{18 b (b c-a d) (a+b x)^2}-\frac {A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac {11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac {47 B^3 d^2 n^3}{9 b (b c-a d)^2 (a+b x)}+\frac {b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac {A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac {5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}-\frac {5 B^3 d^3 n^3 \log (a+b x)}{9 b (b c-a d)^3}+\frac {A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac {5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}+\frac {5 B^3 d^3 n^3 \log (c+d x)}{9 b (b c-a d)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}+\frac {B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac {2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac {14 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac {b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {2 A B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 A B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {B^3 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 A B^2 d^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {2 A B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {\left (2 B^3 d^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b}\right )}{\frac {b}{d}+\frac {(-b c+a d) x}{d}} \, dx,x,\frac {1}{c+d x}\right )}{3 b (b c-a d)^2}+\frac {\left (2 B^3 d^3 n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\frac {d}{b}+\frac {(b c-a d) x}{b}} \, dx,x,\frac {1}{a+b x}\right )}{3 b^2 (b c-a d)^2}\\ &=-\frac {A^3}{3 b (a+b x)^3}-\frac {A^2 B n}{3 b (a+b x)^3}-\frac {2 A B^2 n^2}{9 b (a+b x)^3}-\frac {2 B^3 n^3}{27 b (a+b x)^3}+\frac {A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac {5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}+\frac {5 B^3 d n^3}{18 b (b c-a d) (a+b x)^2}-\frac {A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac {11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac {47 B^3 d^2 n^3}{9 b (b c-a d)^2 (a+b x)}+\frac {b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac {A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac {5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}-\frac {5 B^3 d^3 n^3 \log (a+b x)}{9 b (b c-a d)^3}+\frac {A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac {5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}+\frac {5 B^3 d^3 n^3 \log (c+d x)}{9 b (b c-a d)^3}-\frac {A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac {A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}+\frac {B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac {2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac {14 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac {b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {2 A B^2 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {2 A B^2 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d)^3}-\frac {A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac {B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac {b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac {B^3 d^3 n \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 d^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d)^3}-\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac {2 A B^2 d^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{3 b (b c-a d)^3}-\frac {2 A B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^3 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{3 b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac {2 B^3 d^3 n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 0.89, size = 1003, normalized size = 1.64 \begin {gather*} \frac {-36 B^3 d^3 n^3 (a+b x)^3 \log ^3(a+b x)+36 B^3 d^3 n^3 (a+b x)^3 \log ^3(c+d x)+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(c+d x) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(a+b x) \left (6 A+11 B n+6 B n \log (c+d x)+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+6 B d^3 n (a+b x)^3 \log (c+d x) \left (18 A^2+66 A B n+85 B^2 n^2+6 B (6 A+11 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-(b c-a d) \left (36 A^3 b^2 c^2-72 a A^3 b c d+36 a^2 A^3 d^2+36 A^2 b^2 B c^2 n-126 a A^2 b B c d n+198 a^2 A^2 B d^2 n+24 A b^2 B^2 c^2 n^2-138 a A b B^2 c d n^2+510 a^2 A B^2 d^2 n^2+8 b^2 B^3 c^2 n^3-73 a b B^3 c d n^3+575 a^2 B^3 d^2 n^3-54 A^2 b^2 B c d n x+270 a A^2 b B d^2 n x-90 A b^2 B^2 c d n^2 x+882 a A b B^2 d^2 n^2 x-57 b^2 B^3 c d n^3 x+1077 a b B^3 d^2 n^3 x+108 A^2 b^2 B d^2 n x^2+396 A b^2 B^2 d^2 n^2 x^2+510 b^2 B^3 d^2 n^3 x^2+6 B \left (18 A^2 (b c-a d)^2+6 A B n \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+B^2 n^2 \left (85 a^2 d^2+a b d (-23 c+147 d x)+b^2 \left (4 c^2-15 c d x+66 d^2 x^2\right )\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 B^2 \left (6 A (b c-a d)^2+B n \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+36 B^3 (b c-a d)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-6 B d^3 n (a+b x)^3 \log (a+b x) \left (18 A^2+66 A B n+85 B^2 n^2+18 B^2 n^2 \log ^2(c+d x)+6 B (6 A+11 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+6 B n \log (c+d x) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )}{108 b (b c-a d)^3 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 30.04, size = 175812, normalized size = 287.74
method | result | size |
risch | \(\text {Expression too large to display}\) | \(175812\) |
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. 3508 vs.
\(2 (606) = 1212\).
time = 0.63, size = 3508, normalized size = 5.74 \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] Leaf count of result is larger than twice the leaf count of optimal. 3044 vs.
\(2 (606) = 1212\).
time = 0.52, size = 3044, normalized size = 4.98 \begin {gather*} \text {Too large to display} \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 = 10.73, size = 2069, normalized size = 3.39 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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