Optimal. Leaf size=587 \[ -\frac{b^3 (c+d x)^4 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{4 (a+b x)^4 (b c-a d)^4}-\frac{b^3 B n (c+d x)^4 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{8 (a+b x)^4 (b c-a d)^4}+\frac{b^2 d (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x)^3 (b c-a d)^4}+\frac{2 b^2 B d n (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 (a+b x)^3 (b c-a d)^4}-\frac{3 b d^2 (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 (a+b x)^2 (b c-a d)^4}-\frac{3 b B d^2 n (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{2 (a+b x)^2 (b c-a d)^4}+\frac{d^3 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)^4}+\frac{2 B d^3 n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)^4}-\frac{b^3 B^2 n^2 (c+d x)^4}{32 (a+b x)^4 (b c-a d)^4}+\frac{2 b^2 B^2 d n^2 (c+d x)^3}{9 (a+b x)^3 (b c-a d)^4}-\frac{3 b B^2 d^2 n^2 (c+d x)^2}{4 (a+b x)^2 (b c-a d)^4}+\frac{2 B^2 d^3 n^2 (c+d x)}{(a+b x) (b c-a d)^4} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 1.40797, antiderivative size = 843, normalized size of antiderivative = 1.44, number of steps used = 29, number of rules used = 11, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6742, 2492, 44, 2514, 2490, 32, 2488, 2411, 2343, 2333, 2315} \[ \frac{13 B^2 n^2 \log (a+b x) d^4}{24 b (b c-a d)^4}+\frac{A B n \log (a+b x) d^4}{2 b (b c-a d)^4}-\frac{13 B^2 n^2 \log (c+d x) d^4}{24 b (b c-a d)^4}-\frac{A B n \log (c+d x) d^4}{2 b (b c-a d)^4}-\frac{B^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d^4}{2 b (b c-a d)^4}+\frac{B^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d^4}{2 b (b c-a d)^4}+\frac{B^2 n^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) d^4}{2 b (b c-a d)^4}+\frac{B^2 n^2 \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right ) d^4}{2 b (b c-a d)^4}+\frac{B^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d^3}{2 (b c-a d)^4 (a+b x)}+\frac{25 B^2 n^2 d^3}{24 b (b c-a d)^3 (a+b x)}+\frac{A B n d^3}{2 b (b c-a d)^3 (a+b x)}-\frac{B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d^2}{4 b (b c-a d)^2 (a+b x)^2}-\frac{13 B^2 n^2 d^2}{48 b (b c-a d)^2 (a+b x)^2}-\frac{A B n d^2}{4 b (b c-a d)^2 (a+b x)^2}+\frac{B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d}{6 b (b c-a d) (a+b x)^3}+\frac{7 B^2 n^2 d}{72 b (b c-a d) (a+b x)^3}+\frac{A B n d}{6 b (b c-a d) (a+b x)^3}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac{B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}-\frac{B^2 n^2}{32 b (a+b x)^4}-\frac{A B n}{8 b (a+b x)^4}-\frac{A^2}{4 b (a+b x)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 2492
Rule 44
Rule 2514
Rule 2490
Rule 32
Rule 2488
Rule 2411
Rule 2343
Rule 2333
Rule 2315
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^5} \, dx &=\int \left (\frac{A^2}{(a+b x)^5}+\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5}+\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5}\right ) \, dx\\ &=-\frac{A^2}{4 b (a+b x)^4}+(2 A B) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx+B^2 \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx\\ &=-\frac{A^2}{4 b (a+b x)^4}-\frac{A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{(A B (b c-a d) n) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{2 b}+\frac{\left (B^2 (b c-a d) n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b}\\ &=-\frac{A^2}{4 b (a+b x)^4}-\frac{A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{(A B (b c-a d) n) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b}+\frac{\left (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)^5}-\frac{b d \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^5 (a+b x)}-\frac{d^5 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b}\\ &=-\frac{A^2}{4 b (a+b x)^4}-\frac{A B n}{8 b (a+b x)^4}+\frac{A B d n}{6 b (b c-a d) (a+b x)^3}-\frac{A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}+\frac{A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac{A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}-\frac{A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac{A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{1}{2} \left (B^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx+\frac{\left (B^2 d^4 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{2 (b c-a d)^4}-\frac{\left (B^2 d^5 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b (b c-a d)^4}-\frac{\left (B^2 d^3 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3}+\frac{\left (B^2 d^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{2 (b c-a d)^2}-\frac{\left (B^2 d n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx}{2 (b c-a d)}\\ &=-\frac{A^2}{4 b (a+b x)^4}-\frac{A B n}{8 b (a+b x)^4}+\frac{A B d n}{6 b (b c-a d) (a+b x)^3}-\frac{A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}+\frac{A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac{A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}-\frac{A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac{A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac{B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b (b c-a d) (a+b x)^3}-\frac{B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac{B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{B^2 d^4 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 )}{2 b (b c-a d)^4}+\frac{B^2 d^4 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 )}{2 b (b c-a d)^4}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{\left (B^2 d n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{6 b}-\frac{\left (B^2 d^3 n^2\right ) \int \frac{1}{(a+b x)^2} \, dx}{2 (b c-a d)^3}+\frac{\left (B^2 d^4 n^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}-\frac{\left (B^2 d^4 n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}+\frac{\left (B^2 d^2 n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d)}+\frac{\left (B^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{8 b}\\ &=-\frac{A^2}{4 b (a+b x)^4}-\frac{A B n}{8 b (a+b x)^4}+\frac{A B d n}{6 b (b c-a d) (a+b x)^3}-\frac{A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}+\frac{A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac{B^2 d^3 n^2}{2 b (b c-a d)^3 (a+b x)}+\frac{A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}-\frac{A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac{A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac{B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b (b c-a d) (a+b x)^3}-\frac{B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac{B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{B^2 d^4 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 )}{2 b (b c-a d)^4}+\frac{B^2 d^4 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 )}{2 b (b c-a d)^4}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{\left (B^2 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}{6 b}-\frac{\left (B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b c+a d}{b x}\right )}{x \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )} \, dx,x,c+d x\right )}{2 b (b c-a d)^3}+\frac{\left (B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b c-a d}{d x}\right )}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{2 b^2 (b c-a d)^3}+\frac{\left (B^2 d^2 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}{4 b (b c-a d)}+\frac{\left (B^2 (b c-a d) n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{8 b}\\ &=-\frac{A^2}{4 b (a+b x)^4}-\frac{A B n}{8 b (a+b x)^4}-\frac{B^2 n^2}{32 b (a+b x)^4}+\frac{A B d n}{6 b (b c-a d) (a+b x)^3}+\frac{7 B^2 d n^2}{72 b (b c-a d) (a+b x)^3}-\frac{A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}-\frac{13 B^2 d^2 n^2}{48 b (b c-a d)^2 (a+b x)^2}+\frac{A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac{25 B^2 d^3 n^2}{24 b (b c-a d)^3 (a+b x)}+\frac{A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}+\frac{13 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4}-\frac{A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac{13 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4}-\frac{A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac{B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b (b c-a d) (a+b x)^3}-\frac{B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac{B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{B^2 d^4 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 )}{2 b (b c-a d)^4}+\frac{B^2 d^4 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 )}{2 b (b c-a d)^4}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{\left (B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\left (\frac{-b c+a d}{d}+\frac{b}{d x}\right ) x} \, dx,x,\frac{1}{c+d x}\right )}{2 b (b c-a d)^3}-\frac{\left (B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\left (\frac{b c-a d}{b}+\frac{d}{b x}\right ) x} \, dx,x,\frac{1}{a+b x}\right )}{2 b^2 (b c-a d)^3}\\ &=-\frac{A^2}{4 b (a+b x)^4}-\frac{A B n}{8 b (a+b x)^4}-\frac{B^2 n^2}{32 b (a+b x)^4}+\frac{A B d n}{6 b (b c-a d) (a+b x)^3}+\frac{7 B^2 d n^2}{72 b (b c-a d) (a+b x)^3}-\frac{A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}-\frac{13 B^2 d^2 n^2}{48 b (b c-a d)^2 (a+b x)^2}+\frac{A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac{25 B^2 d^3 n^2}{24 b (b c-a d)^3 (a+b x)}+\frac{A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}+\frac{13 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4}-\frac{A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac{13 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4}-\frac{A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac{B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b (b c-a d) (a+b x)^3}-\frac{B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac{B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{B^2 d^4 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 )}{2 b (b c-a d)^4}+\frac{B^2 d^4 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 )}{2 b (b c-a d)^4}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{\left (B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\frac{b}{d}+\frac{(-b c+a d) x}{d}} \, dx,x,\frac{1}{c+d x}\right )}{2 b (b c-a d)^3}-\frac{\left (B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\frac{d}{b}+\frac{(b c-a d) x}{b}} \, dx,x,\frac{1}{a+b x}\right )}{2 b^2 (b c-a d)^3}\\ &=-\frac{A^2}{4 b (a+b x)^4}-\frac{A B n}{8 b (a+b x)^4}-\frac{B^2 n^2}{32 b (a+b x)^4}+\frac{A B d n}{6 b (b c-a d) (a+b x)^3}+\frac{7 B^2 d n^2}{72 b (b c-a d) (a+b x)^3}-\frac{A B d^2 n}{4 b (b c-a d)^2 (a+b x)^2}-\frac{13 B^2 d^2 n^2}{48 b (b c-a d)^2 (a+b x)^2}+\frac{A B d^3 n}{2 b (b c-a d)^3 (a+b x)}+\frac{25 B^2 d^3 n^2}{24 b (b c-a d)^3 (a+b x)}+\frac{A B d^4 n \log (a+b x)}{2 b (b c-a d)^4}+\frac{13 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4}-\frac{A B d^4 n \log (c+d x)}{2 b (b c-a d)^4}-\frac{13 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4}-\frac{A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^4}-\frac{B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac{B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b (b c-a d) (a+b x)^3}-\frac{B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac{B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{B^2 d^4 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 )}{2 b (b c-a d)^4}+\frac{B^2 d^4 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 )}{2 b (b c-a d)^4}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{B^2 d^4 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{B^2 d^4 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{2 b (b c-a d)^4}\\ \end{align*}
Mathematica [A] time = 1.07058, size = 1011, normalized size = 1.72 \[ -\frac{9 \left (8 A^2+4 B n A+16 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) A+B^2 n^2+8 B^2 \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2+4 B^2 n \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) (b c-a d)^4-4 B d n (a+b x) \left (12 A+7 B n+12 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) (b c-a d)^3+6 B d^2 n (a+b x)^2 \left (12 A+13 B n+12 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) (b c-a d)^2-12 B d^3 n (a+b x)^3 \left (12 A+25 B n+12 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) (b c-a d)-12 B n \log (a+b x) \left (-3 \left (4 A+B n+4 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) (b c-a d)^3+4 B d n (a+b x) (b c-a d)^2+12 B d^3 n (a+b x)^3+6 B d^2 (a d-b c) n (a+b x)^2\right ) (b c-a d)+72 b B^2 n^2 \left (\left (c^4-d^4 x^4\right ) b^3-4 a d \left (c^3+d^3 x^3\right ) b^2+6 a^2 d^2 \left (c^2-d^2 x^2\right ) b-4 a^3 d^3 (c+d x)\right ) \log ^2(a+b x)+72 b B^2 n^2 \left (\left (c^4-d^4 x^4\right ) b^3-4 a d \left (c^3+d^3 x^3\right ) b^2+6 a^2 d^2 \left (c^2-d^2 x^2\right ) b-4 a^3 d^3 (c+d x)\right ) \log ^2(c+d x)-12 B d^4 n (a+b x)^4 \log (a+b x) \left (12 A+25 B n+12 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+12 B d^4 n (a+b x)^4 \log (c+d x) \left (12 A+25 B n+12 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+12 B n \log (c+d x) \left (-12 B n \log (a+b x) (b c-a d)^4-3 \left (4 A+B n+4 B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) (b c-a d)^4+4 B d n (a+b x) (b c-a d)^3-6 B d^2 n (a+b x)^2 (b c-a d)^2+12 B d^3 n (a+b x)^3 (b c-a d)+12 B d^4 n (a+b x)^4 \log (a+b x)\right )}{288 b (b c-a d)^4 (a+b x)^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 3.333, size = 33370, normalized size = 56.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.05568, size = 3021, normalized size = 5.15 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.69859, size = 5029, normalized size = 8.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (b x + a\right )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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