Optimal. Leaf size=732 \[ \frac{2 b^2 d^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^3}{B g^3 i^3 n (b c-a d)^5}-\frac{b^4 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^3 i^3 (a+b x)^2 (b c-a d)^5}-\frac{b^4 B n (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac{4 b^3 d (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^3 i^3 (a+b x) (b c-a d)^5}+\frac{8 b^3 B d n (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (a+b x) (b c-a d)^5}-\frac{4 b d^3 (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^3 i^3 (c+d x) (b c-a d)^5}+\frac{d^4 (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^3 i^3 (c+d x)^2 (b c-a d)^5}-\frac{B d^4 n (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (c+d x)^2 (b c-a d)^5}+\frac{8 A b B d^3 n (a+b x)}{g^3 i^3 (c+d x) (b c-a d)^5}-\frac{b^4 B^2 n^2 (c+d x)^2}{4 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac{8 b^3 B^2 d n^2 (c+d x)}{g^3 i^3 (a+b x) (b c-a d)^5}+\frac{8 b B^2 d^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{g^3 i^3 (c+d x) (b c-a d)^5}-\frac{8 b B^2 d^3 n^2 (a+b x)}{g^3 i^3 (c+d x) (b c-a d)^5}+\frac{B^2 d^4 n^2 (a+b x)^2}{4 g^3 i^3 (c+d x)^2 (b c-a d)^5} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 9.22738, antiderivative size = 2041, normalized size of antiderivative = 2.79, number of steps used = 163, number of rules used = 31, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.689, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610, 2500, 2433, 2375, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 6688
Rule 6742
Rule 2411
Rule 2344
Rule 2317
Rule 2507
Rule 2488
Rule 2506
Rule 6610
Rule 2500
Rule 2433
Rule 2375
Rule 2374
Rule 6589
Rule 2440
Rule 2434
Rule 2499
Rule 2396
Rule 2302
Rule 30
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}{(208 c+208 d x)^3 (a g+b g x)^3} \, dx &=\int \left (\frac{b^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^3 g^3 (a+b x)^3}-\frac{3 b^3 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (a+b x)^2}+\frac{3 b^3 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^3 g^3 (c+d x)^3}-\frac{3 b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)^2}-\frac{3 b^2 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3 (c+d x)}\right ) \, dx\\ &=\frac{\left (3 b^3 d^2\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}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b^2 d^3\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}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b^3 d\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2} \, dx}{8998912 (b c-a d)^4 g^3}-\frac{\left (3 b d^3\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2} \, dx}{8998912 (b c-a d)^4 g^3}+\frac{b^3 \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^3} \, dx}{8998912 (b c-a d)^3 g^3}-\frac{d^3 \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3} \, dx}{8998912 (b c-a d)^3 g^3}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d^2 n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^2 B d^2 n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{4499456 (b c-a d)^4 g^3}-\frac{\left (3 b B d^2 n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^2} \, dx}{4499456 (b c-a d)^4 g^3}+\frac{\left (b^2 B n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{8998912 (b c-a d)^3 g^3}-\frac{\left (B d^2 n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^3} \, dx}{8998912 (b c-a d)^3 g^3}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d^2 n\right ) \int \frac{(b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^2 B d^2 n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2 (c+d x)} \, dx}{4499456 (b c-a d)^3 g^3}-\frac{\left (3 b B d^2 n\right ) \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}{4499456 (b c-a d)^3 g^3}+\frac{\left (b^2 B n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3 (c+d x)} \, dx}{8998912 (b c-a d)^2 g^3}-\frac{\left (B d^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^3} \, dx}{8998912 (b c-a d)^2 g^3}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d^2 n\right ) \int \frac{\log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{2249728 (b c-a d)^4 g^3}+\frac{\left (3 b^2 B d^2 n\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{2249728 (b c-a d)^4 g^3}-\frac{\left (3 b^2 B d n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(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}{4499456 (b c-a d)^3 g^3}-\frac{\left (3 b B d^2 n\right ) \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}{4499456 (b c-a d)^3 g^3}+\frac{\left (b^2 B n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^3}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{8998912 (b c-a d)^2 g^3}-\frac{\left (B d^2 n\right ) \int \left (\frac{b^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^3}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{8998912 (b c-a d)^2 g^3}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{\left (b^3 B d n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{8998912 (b c-a d)^4 g^3}-\frac{\left (3 b^3 B d n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{4499456 (b c-a d)^4 g^3}-\frac{\left (3 b^2 B d^2 n\right ) \int \left (\frac{A \log (a+b x)}{(a+b x) (c+d x)}+\frac{B \log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{2249728 (b c-a d)^4 g^3}+\frac{\left (3 b^2 B d^2 n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{2249728 (b c-a d)^4 g^3}+\frac{\left (b B d^3 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}{8998912 (b c-a d)^4 g^3}+\frac{\left (3 b B d^3 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}{4499456 (b c-a d)^4 g^3}+\frac{\left (b^3 B n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{8998912 (b c-a d)^3 g^3}+\frac{\left (B d^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{8998912 (b c-a d)^3 g^3}\\ &=-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{\left (3 b^3 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)}{a+b x} \, dx}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d^3 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}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 A b^2 B d^2 n\right ) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{2249728 (b c-a d)^4 g^3}-\frac{\left (3 b^2 B^2 d^2 n\right ) \int \frac{\log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{2249728 (b c-a d)^4 g^3}-\frac{\left (b^2 B^2 d n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{8998912 (b c-a d)^4 g^3}-\frac{\left (3 b^2 B^2 d n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{4499456 (b c-a d)^4 g^3}+\frac{\left (b B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{8998912 (b c-a d)^4 g^3}+\frac{\left (3 b B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{4499456 (b c-a d)^4 g^3}+\frac{\left (b^2 B^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{17997824 (b c-a d)^3 g^3}+\frac{\left (B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{17997824 (b c-a d)^3 g^3}\\ &=-\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{\left (3 b^3 B^2 d^2\right ) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{4499456 (b c-a d)^5 g^3}+\frac{\left (3 b^3 B d^2 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}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d^3 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}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 A b B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{2249728 (b c-a d)^4 g^3}-\frac{\left (b^2 B^2 d n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{8998912 (b c-a d)^3 g^3}-\frac{\left (3 b^2 B^2 d n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{4499456 (b c-a d)^3 g^3}+\frac{\left (b B^2 d^2 n^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{8998912 (b c-a d)^3 g^3}+\frac{\left (3 b B^2 d^2 n^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{4499456 (b c-a d)^3 g^3}+\frac{\left (b^2 B^2 n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{17997824 (b c-a d)^2 g^3}+\frac{\left (B^2 d^2 n^2\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{17997824 (b c-a d)^2 g^3}\\ &=-\frac{3 b^2 B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 A b^2 B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 A b^3 B d^2 n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^3 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)}{a+b x} \, dx}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 A b B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 A b^2 B d^3 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^3 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}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^2 B^2 d^2 n\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{2249728 (b c-a d)^4 g^3}-\frac{\left (b^2 B^2 d 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}{8998912 (b c-a d)^3 g^3}-\frac{\left (3 b^2 B^2 d 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}{4499456 (b c-a d)^3 g^3}+\frac{\left (b B^2 d^2 n^2\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{8998912 (b c-a d)^3 g^3}+\frac{\left (3 b B^2 d^2 n^2\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{4499456 (b c-a d)^3 g^3}+\frac{\left (b^2 B^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}{17997824 (b c-a d)^2 g^3}+\frac{\left (B^2 d^2 n^2\right ) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{17997824 (b c-a d)^2 g^3}\\ &=-\frac{b^2 B^2 n^2}{35995648 (b c-a d)^3 g^3 (a+b x)^2}+\frac{15 b^2 B^2 d n^2}{17997824 (b c-a d)^4 g^3 (a+b x)}+\frac{B^2 d^2 n^2}{35995648 (b c-a d)^3 g^3 (c+d x)^2}+\frac{15 b B^2 d^2 n^2}{17997824 (b c-a d)^4 g^3 (c+d x)}+\frac{15 b^2 B^2 d^2 n^2 \log (a+b x)}{8998912 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(a+b x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{15 b^2 B^2 d^2 n^2 \log (c+d x)}{8998912 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 A b^2 B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 A b^2 B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^3 B^2 d^2 n\right ) \int \frac{\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^3 B^2 d^2 n\right ) \int \frac{\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 A b^2 B d^3 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^3 B^2 d^2 n^2\right ) \int \frac{\log ^2(c+d x)}{a+b x} \, dx}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^3 n^2\right ) \int \frac{\log ^2(c+d x)}{c+d x} \, dx}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^2 n^2\right ) \int \frac{\text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{2249728 (b c-a d)^4 g^3}+\frac{\left (3 b^3 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 (c+d x)}{a+b x} \, dx}{2249728 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B^2 n^2}{35995648 (b c-a d)^3 g^3 (a+b x)^2}+\frac{15 b^2 B^2 d n^2}{17997824 (b c-a d)^4 g^3 (a+b x)}+\frac{B^2 d^2 n^2}{35995648 (b c-a d)^3 g^3 (c+d x)^2}+\frac{15 b B^2 d^2 n^2}{17997824 (b c-a d)^4 g^3 (c+d x)}+\frac{15 b^2 B^2 d^2 n^2 \log (a+b x)}{8998912 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(a+b x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{15 b^2 B^2 d^2 n^2 \log (c+d x)}{8998912 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 A b^2 B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^2 B^2 d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^n\right ) \log \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )}{x} \, dx,x,a+b x\right )}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^2 B^2 d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right ) \log \left (\left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,c+d x\right )}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^3 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}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^3 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}{2249728 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B^2 n^2}{35995648 (b c-a d)^3 g^3 (a+b x)^2}+\frac{15 b^2 B^2 d n^2}{17997824 (b c-a d)^4 g^3 (a+b x)}+\frac{B^2 d^2 n^2}{35995648 (b c-a d)^3 g^3 (c+d x)^2}+\frac{15 b B^2 d^2 n^2}{17997824 (b c-a d)^4 g^3 (c+d x)}+\frac{15 b^2 B^2 d^2 n^2 \log (a+b x)}{8998912 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(a+b x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{15 b^2 B^2 d^2 n^2 \log (c+d x)}{8998912 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (x^n\right )}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b B^2 d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )^{-n}\right )}{-\frac{-b c+a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{d \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )}{-\frac{-b c+a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2249728 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B^2 n^2}{35995648 (b c-a d)^3 g^3 (a+b x)^2}+\frac{15 b^2 B^2 d n^2}{17997824 (b c-a d)^4 g^3 (a+b x)}+\frac{B^2 d^2 n^2}{35995648 (b c-a d)^3 g^3 (c+d x)^2}+\frac{15 b B^2 d^2 n^2}{17997824 (b c-a d)^4 g^3 (c+d x)}+\frac{15 b^2 B^2 d^2 n^2 \log (a+b x)}{8998912 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(a+b x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{15 b^2 B^2 d^2 n^2 \log (c+d x)}{8998912 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B^2 d^2 n^2 \log ^3(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{x} \, dx,x,c+d x\right )}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^2 B^2 d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^n\right ) \log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^2 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{x} \, dx,x,c+d x\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2249728 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B^2 n^2}{35995648 (b c-a d)^3 g^3 (a+b x)^2}+\frac{15 b^2 B^2 d n^2}{17997824 (b c-a d)^4 g^3 (a+b x)}+\frac{B^2 d^2 n^2}{35995648 (b c-a d)^3 g^3 (c+d x)^2}+\frac{15 b B^2 d^2 n^2}{17997824 (b c-a d)^4 g^3 (c+d x)}+\frac{15 b^2 B^2 d^2 n^2 \log (a+b x)}{8998912 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(a+b x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{15 b^2 B^2 d^2 n^2 \log (c+d x)}{8998912 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log (a+b x) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B^2 d^2 n^2 \log ^3(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^3 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (x^{-n}\right )}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{4499456 (b c-a d)^5 g^3}-\frac{\left (3 b^3 B^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{4499456 (b c-a d)^5 g^3}+\frac{\left (3 b^2 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2249728 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B^2 n^2}{35995648 (b c-a d)^3 g^3 (a+b x)^2}+\frac{15 b^2 B^2 d n^2}{17997824 (b c-a d)^4 g^3 (a+b x)}+\frac{B^2 d^2 n^2}{35995648 (b c-a d)^3 g^3 (c+d x)^2}+\frac{15 b B^2 d^2 n^2}{17997824 (b c-a d)^4 g^3 (c+d x)}+\frac{15 b^2 B^2 d^2 n^2 \log (a+b x)}{8998912 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(a+b x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{15 b^2 B^2 d^2 n^2 \log (c+d x)}{8998912 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log (a+b x) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B^2 d^2 n^2 \log ^3(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B^2 d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2249728 (b c-a d)^5 g^3}+\frac{\left (3 b^2 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2249728 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B^2 n^2}{35995648 (b c-a d)^3 g^3 (a+b x)^2}+\frac{15 b^2 B^2 d n^2}{17997824 (b c-a d)^4 g^3 (a+b x)}+\frac{B^2 d^2 n^2}{35995648 (b c-a d)^3 g^3 (c+d x)^2}+\frac{15 b B^2 d^2 n^2}{17997824 (b c-a d)^4 g^3 (c+d x)}+\frac{15 b^2 B^2 d^2 n^2 \log (a+b x)}{8998912 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(a+b x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{15 b^2 B^2 d^2 n^2 \log (c+d x)}{8998912 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log (a+b x) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B^2 d^2 n^2 \log ^3(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}+2 \frac{\left (3 b^2 B^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2249728 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B^2 n^2}{35995648 (b c-a d)^3 g^3 (a+b x)^2}+\frac{15 b^2 B^2 d n^2}{17997824 (b c-a d)^4 g^3 (a+b x)}+\frac{B^2 d^2 n^2}{35995648 (b c-a d)^3 g^3 (c+d x)^2}+\frac{15 b B^2 d^2 n^2}{17997824 (b c-a d)^4 g^3 (c+d x)}+\frac{15 b^2 B^2 d^2 n^2 \log (a+b x)}{8998912 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(a+b x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{17997824 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8998912 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{17997824 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (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}{17997824 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8998912 (b c-a d)^4 g^3 (c+d x)}+\frac{3 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4499456 (b c-a d)^5 g^3}-\frac{15 b^2 B^2 d^2 n^2 \log (c+d x)}{8998912 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 A b^2 B d^2 n \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \log (a+b x) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{4499456 (b c-a d)^5 g^3}-\frac{b^2 B^2 d^2 n^2 \log ^3(c+d x)}{4499456 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{4499456 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{4499456 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 A b^2 B d^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}-\frac{3 b^2 B^2 d^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{2249728 (b c-a d)^5 g^3}+\frac{3 b^2 B^2 d^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2249728 (b c-a d)^5 g^3}\\ \end{align*}
Mathematica [B] time = 2.64962, size = 1653, normalized size = 2.26 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.737, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{3} \left ( dix+ci \right ) ^{3}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 3.28386, size = 7552, normalized size = 10.32 \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 = 0.8252, size = 6098, normalized size = 8.33 \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 (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{3}{\left (d i x + c i\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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