Optimal. Leaf size=343 \[ -\frac{b^2 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^3 i (a+b x)^2 (b c-a d)^3}-\frac{b^2 B (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i (a+b x)^2 (b c-a d)^3}+\frac{d^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g^3 i (b c-a d)^3}+\frac{2 b d (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{g^3 i (a+b x) (b c-a d)^3}+\frac{4 b B d (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (a+b x) (b c-a d)^3}-\frac{b^2 B^2 (c+d x)^2}{4 g^3 i (a+b x)^2 (b c-a d)^3}+\frac{4 b B^2 d (c+d x)}{g^3 i (a+b x) (b c-a d)^3} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 7.36569, antiderivative size = 1899, normalized size of antiderivative = 5.54, number of steps used = 117, number of rules used = 31, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.738, 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 (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(90 c+90 d x) (a g+b g x)^3} \, dx &=\int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d) g^3 (a+b x)^3}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)^2}+\frac{b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3 (c+d x)}\right ) \, dx\\ &=\frac{\left (b d^2\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{90 (b c-a d)^3 g^3}-\frac{d^3 \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{90 (b c-a d)^3 g^3}-\frac{(b d) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{90 (b c-a d)^2 g^3}+\frac{b \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{90 (b c-a d) g^3}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{\left (B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{e (a+b x)} \, dx}{45 (b c-a d)^3 g^3}+\frac{\left (B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{e (a+b x)} \, dx}{45 (b c-a d)^3 g^3}-\frac{(B d) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{45 (b c-a d)^2 g^3}+\frac{B \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{90 (b c-a d) g^3}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{B \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{90 g^3}-\frac{(B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{45 (b c-a d) g^3}-\frac{\left (B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{45 (b c-a d)^3 e g^3}+\frac{\left (B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{45 (b c-a d)^3 e g^3}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{B \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^3}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{90 g^3}-\frac{(B d) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{45 (b c-a d) g^3}-\frac{\left (B d^2\right ) \int \frac{(b c-a d) e \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{45 (b c-a d)^3 e g^3}+\frac{\left (B d^2\right ) \int \frac{(b c-a d) e \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{45 (b c-a d)^3 e g^3}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{\left (b B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{90 (b c-a d)^3 g^3}+\frac{\left (b B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{90 (b c-a d)^3 g^3}-\frac{\left (B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{45 (b c-a d)^3 g^3}-\frac{(b B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{90 (b c-a d)^2 g^3}-\frac{(b B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{45 (b c-a d)^2 g^3}-\frac{\left (B d^2\right ) \int \frac{\log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{45 (b c-a d)^2 g^3}+\frac{\left (B d^2\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{45 (b c-a d)^2 g^3}+\frac{(b B) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{90 (b c-a d) g^3}\\ &=-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{90 (b c-a d)^3 g^3}+\frac{\left (B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{90 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{45 (b c-a d)^3 g^3}+\frac{\left (B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{90 (b c-a d)^2 g^3}-\frac{\left (B^2 d\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{45 (b c-a d)^2 g^3}-\frac{\left (B d^2\right ) \int \left (\frac{A \log (a+b x)}{(a+b x) (c+d x)}+\frac{B \log (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)}\right ) \, dx}{45 (b c-a d)^2 g^3}+\frac{\left (B d^2\right ) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{45 (b c-a d)^2 g^3}+\frac{B^2 \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{180 (b c-a d) g^3}\\ &=-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{B^2 \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{180 g^3}+\frac{\left (b B d^2\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (B d^3\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{c+d x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (A B d^2\right ) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{45 (b c-a d)^2 g^3}-\frac{\left (B^2 d^2\right ) \int \frac{\log (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{45 (b c-a d)^2 g^3}-\frac{\left (B^2 d\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{90 (b c-a d) g^3}-\frac{\left (B^2 d\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{45 (b c-a d) g^3}-\frac{\left (B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{90 (b c-a d)^3 e g^3}+\frac{\left (B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{90 (b c-a d)^3 e g^3}-\frac{\left (B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{45 (b c-a d)^3 e g^3}+\frac{\left (B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{45 (b c-a d)^3 e g^3}\\ &=-\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{B^2 \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}{180 g^3}+\frac{\left (b B d^2\right ) \int \left (\frac{A \log (c+d x)}{a+b x}+\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x}\right ) \, dx}{45 (b c-a d)^3 g^3}+\frac{\left (b B^2 d^2\right ) \int \frac{\log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{90 (b c-a d)^3 g^3}-\frac{\left (B d^3\right ) \int \left (\frac{A \log (c+d x)}{c+d x}+\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x}\right ) \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (A B d^2\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 )}{45 b (b c-a d)^2 g^3}-\frac{\left (B^2 d\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}{90 (b c-a d) g^3}-\frac{\left (B^2 d\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}{45 (b c-a d) g^3}-\frac{\left (B^2 d^2\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{90 (b c-a d)^3 e g^3}+\frac{\left (B^2 d^2\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{90 (b c-a d)^3 e g^3}-\frac{\left (B^2 d^2\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{45 (b c-a d)^3 e g^3}+\frac{\left (B^2 d^2\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{45 (b c-a d)^3 e g^3}\\ &=-\frac{B^2}{360 (b c-a d) g^3 (a+b x)^2}+\frac{7 B^2 d}{180 (b c-a d)^2 g^3 (a+b x)}+\frac{7 B^2 d^2 \log (a+b x)}{180 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{7 B^2 d^2 \log (c+d x)}{180 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{\left (A B d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{45 (b c-a d)^3 g^3}+\frac{\left (A b B d^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (b B^2 d^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{90 (b c-a d)^3 g^3}+\frac{\left (b B^2 d^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{90 (b c-a d)^3 g^3}-\frac{\left (b B^2 d^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{45 (b c-a d)^3 g^3}+\frac{\left (b B^2 d^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{45 (b c-a d)^3 g^3}+\frac{\left (b B^2 d^2\right ) \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (A B d^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{45 (b c-a d)^3 g^3}+\frac{\left (A B d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{45 b (b c-a d)^3 g^3}+\frac{\left (B^2 d^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{90 (b c-a d)^3 g^3}-\frac{\left (B^2 d^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{90 (b c-a d)^3 g^3}+\frac{\left (B^2 d^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^3\right ) \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{45 (b c-a d)^3 g^3}+\frac{\left (B^2 d^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{45 (b c-a d)^2 g^3}\\ &=-\frac{B^2}{360 (b c-a d) g^3 (a+b x)^2}+\frac{7 B^2 d}{180 (b c-a d)^2 g^3 (a+b x)}+\frac{7 B^2 d^2 \log (a+b x)}{180 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(a+b x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{7 B^2 d^2 \log (c+d x)}{180 (b c-a d)^3 g^3}+\frac{A B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}-\frac{\left (A B d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{45 (b c-a d)^3 g^3}-\frac{\left (A B d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{90 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{90 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{45 (b c-a d)^3 g^3}+\frac{\left (b B^2 d^2\right ) \int \frac{\log ^2(c+d x)}{a+b x} \, dx}{90 (b c-a d)^3 g^3}-\frac{\left (b B^2 d^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{90 (b c-a d)^3 g^3}+\frac{\left (b B^2 d^2\right ) \int \frac{\log (a+b x) \log (c+d x)}{a+b x} \, dx}{45 (b c-a d)^3 g^3}+\frac{\left (b B^2 d^2\right ) \int \frac{\log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (b B^2 d^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (A B d^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{90 (b c-a d)^3 g^3}-\frac{\left (B^2 d^3\right ) \int \frac{\log ^2(c+d x)}{c+d x} \, dx}{90 (b c-a d)^3 g^3}-\frac{\left (B^2 d^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^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}{45 (b c-a d)^2 g^3}+\frac{\left (b B^2 d^2 \left (-\log (a+b x)-\log \left (\frac{1}{c+d x}\right )+\log \left (\frac{e (a+b x)}{c+d x}\right )\right )\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{45 (b c-a d)^3 g^3}\\ &=-\frac{B^2}{360 (b c-a d) g^3 (a+b x)^2}+\frac{7 B^2 d}{180 (b c-a d)^2 g^3 (a+b x)}+\frac{7 B^2 d^2 \log (a+b x)}{180 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(a+b x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x)}{60 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{7 B^2 d^2 \log (c+d x)}{180 (b c-a d)^3 g^3}+\frac{A B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(c+d x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}-\frac{\left (A B d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,c+d x\right )}{90 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{90 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{90 (b c-a d)^3 g^3}+\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )}{x} \, dx,x,a+b x\right )}{45 (b c-a d)^3 g^3}+\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{-\frac{-b c+a d}{b}+\frac{d x}{b}}\right ) \log \left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )}{x} \, dx,x,a+b x\right )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^3 \left (-\log (a+b x)-\log \left (\frac{1}{c+d x}\right )+\log \left (\frac{e (a+b x)}{c+d x}\right )\right )\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{45 (b c-a d)^3 g^3}\\ &=-\frac{B^2}{360 (b c-a d) g^3 (a+b x)^2}+\frac{7 B^2 d}{180 (b c-a d)^2 g^3 (a+b x)}+\frac{7 B^2 d^2 \log (a+b x)}{180 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(a+b x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x)}{60 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{7 B^2 d^2 \log (c+d x)}{180 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log ^2(a+b x) \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(c+d x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{90 (b c-a d)^3 g^3}-\frac{\left (B^2 d^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 )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{90 b (b c-a d)^3 g^3}-\frac{\left (B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{1}{-\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 )}{45 b (b c-a d)^3 g^3}+\frac{\left (B^2 d^3\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 )}{45 b (b c-a d)^3 g^3}-\frac{\left (B^2 d^2 \left (-\log (a+b x)-\log \left (\frac{1}{c+d x}\right )+\log \left (\frac{e (a+b x)}{c+d x}\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 )}{45 (b c-a d)^3 g^3}\\ &=-\frac{B^2}{360 (b c-a d) g^3 (a+b x)^2}+\frac{7 B^2 d}{180 (b c-a d)^2 g^3 (a+b x)}+\frac{7 B^2 d^2 \log (a+b x)}{180 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(a+b x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x)}{60 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{7 B^2 d^2 \log (c+d x)}{180 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log ^2(a+b x) \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(c+d x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^3(c+d x)}{270 (b c-a d)^3 g^3}+\frac{A B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{x}\right ) \log \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{x} \, dx,x,c+d x\right )}{45 (b c-a d)^3 g^3}+\frac{\left (B^2 d^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 )}{45 (b c-a d)^3 g^3}+\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^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 )}{45 (b c-a d)^3 g^3}\\ &=-\frac{B^2}{360 (b c-a d) g^3 (a+b x)^2}+\frac{7 B^2 d}{180 (b c-a d)^2 g^3 (a+b x)}+\frac{7 B^2 d^2 \log (a+b x)}{180 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(a+b x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{7 B^2 d^2 \log (c+d x)}{180 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log ^2(a+b x) \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(c+d x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^3(c+d x)}{270 (b c-a d)^3 g^3}+\frac{A B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}-\frac{\left (b B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (\frac{1}{x}\right )}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{90 (b c-a d)^3 g^3}-\frac{\left (b B^2 d\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 )}{90 (b c-a d)^3 g^3}+\frac{\left (B^2 d^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 )}{45 (b c-a d)^3 g^3}\\ &=-\frac{B^2}{360 (b c-a d) g^3 (a+b x)^2}+\frac{7 B^2 d}{180 (b c-a d)^2 g^3 (a+b x)}+\frac{7 B^2 d^2 \log (a+b x)}{180 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(a+b x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{7 B^2 d^2 \log (c+d x)}{180 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log ^2(a+b x) \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(c+d x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^3(c+d x)}{270 (b c-a d)^3 g^3}+\frac{A B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}-\frac{\left (B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{x}\right ) \log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{45 (b c-a d)^3 g^3}+\frac{\left (B^2 d^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 )}{45 (b c-a d)^3 g^3}\\ &=-\frac{B^2}{360 (b c-a d) g^3 (a+b x)^2}+\frac{7 B^2 d}{180 (b c-a d)^2 g^3 (a+b x)}+\frac{7 B^2 d^2 \log (a+b x)}{180 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(a+b x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{7 B^2 d^2 \log (c+d x)}{180 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log ^2(a+b x) \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(c+d x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^3(c+d x)}{270 (b c-a d)^3 g^3}+\frac{A B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (\frac{1}{c+d x}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}+2 \frac{\left (B^2 d^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 )}{45 (b c-a d)^3 g^3}\\ &=-\frac{B^2}{360 (b c-a d) g^3 (a+b x)^2}+\frac{7 B^2 d}{180 (b c-a d)^2 g^3 (a+b x)}+\frac{7 B^2 d^2 \log (a+b x)}{180 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(a+b x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{90 (b c-a d)^3 g^3}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{180 (b c-a d) g^3 (a+b x)^2}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^2 g^3 (a+b x)}+\frac{B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{30 (b c-a d)^3 g^3}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{180 (b c-a d) g^3 (a+b x)^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^2 g^3 (a+b x)}+\frac{d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{90 (b c-a d)^3 g^3}-\frac{7 B^2 d^2 \log (c+d x)}{180 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log ^2(a+b x) \log (c+d x)}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{45 (b c-a d)^3 g^3}-\frac{B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{30 (b c-a d)^3 g^3}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{90 (b c-a d)^3 g^3}-\frac{A B d^2 \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(c+d x)}{60 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{90 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^3(c+d x)}{270 (b c-a d)^3 g^3}+\frac{A B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{90 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}-\frac{B^2 d^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{A B d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{30 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (\frac{1}{c+d x}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}-\frac{B^2 d^2 \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{45 (b c-a d)^3 g^3}+\frac{B^2 d^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{45 (b c-a d)^3 g^3}\\ \end{align*}
Mathematica [A] time = 1.13258, size = 318, normalized size = 0.93 \[ \frac{-6 B \left (-2 a^2 A d^2-4 a b d (A d x+B (c+d x))+b^2 \left (B \left (c^2-2 c d x-3 d^2 x^2\right )-2 A d^2 x^2\right )\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )-6 d^2 \left (2 A^2+6 A B+7 B^2\right ) (a+b x)^2 \log (c+d x)+6 d \left (2 A^2+6 A B+7 B^2\right ) (a+b x) (b c-a d)-3 \left (2 A^2+2 A B+B^2\right ) (b c-a d)^2+6 d^2 \left (2 A^2+6 A B+7 B^2\right ) (a+b x)^2 \log (a+b x)-6 B (b c-a d) (-6 a A d-7 a B d+2 A b (c-2 d x)+b B (c-6 d x)) \log \left (\frac{e (a+b x)}{c+d x}\right )+4 B^2 d^2 (a+b x)^2 \log ^3\left (\frac{e (a+b x)}{c+d x}\right )}{12 g^3 i (a+b x)^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.059, size = 2144, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 2.12, size = 2855, normalized size = 8.32 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.527888, size = 1130, normalized size = 3.29 \begin{align*} -\frac{3 \,{\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - 24 \,{\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b c d + 3 \,{\left (6 \, A^{2} + 14 \, A B + 15 \, B^{2}\right )} a^{2} d^{2} - 4 \,{\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x + B^{2} a^{2} d^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{3} - 6 \,{\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} d^{2} x^{2} - B^{2} b^{2} c^{2} + 4 \, B^{2} a b c d + 2 \, A B a^{2} d^{2} + 2 \,{\left (B^{2} b^{2} c d + 2 \,{\left (A B + B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} - 6 \,{\left ({\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} b^{2} c d -{\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} a b d^{2}\right )} x - 6 \,{\left ({\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, A^{2} a^{2} d^{2} -{\left (2 \, A B + B^{2}\right )} b^{2} c^{2} + 8 \,{\left (A B + B^{2}\right )} a b c d + 2 \,{\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} c d + 2 \,{\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{12 \,{\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x +{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 12.3273, size = 1488, normalized size = 4.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{3}{\left (d i x + c i\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]