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 = 7.37, 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 12
Rule 30
Rule 44
Rule 2301
Rule 2302
Rule 2317
Rule 2344
Rule 2374
Rule 2375
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2411
Rule 2418
Rule 2433
Rule 2434
Rule 2440
Rule 2488
Rule 2499
Rule 2500
Rule 2506
Rule 2507
Rule 2524
Rule 2525
Rule 2528
Rule 6589
Rule 6610
Rule 6688
Rule 6742
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*}
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Mathematica [A] time = 1.11, 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.
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fricas [A] time = 0.66, size = 540, normalized size = 1.57 \[ -\frac {3 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - 24 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b c d + 3 \, {\left (6 \, A^{2} + 14 \, A B + 15 \, B^{2}\right )} a^{2} d^{2} - 4 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x + B^{2} a^{2} d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} - 6 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} d^{2} x^{2} - B^{2} b^{2} c^{2} + 4 \, B^{2} a b c d + 2 \, A B a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} c d + 2 \, {\left (A B + B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 6 \, {\left ({\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} a b d^{2}\right )} x - 6 \, {\left ({\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, A^{2} a^{2} d^{2} - {\left (2 \, A B + B^{2}\right )} b^{2} c^{2} + 8 \, {\left (A B + B^{2}\right )} a b c d + 2 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} c d + 2 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{12 \, {\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x + {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 2144, normalized size = 6.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.11, size = 2115, normalized size = 6.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.30, size = 981, normalized size = 2.86 \[ {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,d^2\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )}{g^3\,i\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B^2\,x\,\left (a\,d-b\,c\right )}{g^3\,i\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B\,d^2\,\left (2\,A+3\,B\right )}{2\,g^3\,i\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )-\frac {\frac {6\,A^2\,a\,d-2\,A^2\,b\,c+15\,B^2\,a\,d-B^2\,b\,c+14\,A\,B\,a\,d-2\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (2\,b\,d\,A^2+6\,b\,d\,A\,B+7\,b\,d\,B^2\right )}{a\,d-b\,c}}{x^2\,\left (2\,b^3\,c\,g^3\,i-2\,a\,b^2\,d\,g^3\,i\right )+x\,\left (4\,a\,b^2\,c\,g^3\,i-4\,a^2\,b\,d\,g^3\,i\right )-2\,a^3\,d\,g^3\,i+2\,a^2\,b\,c\,g^3\,i}+\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,d^2\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )\,\left (2\,A+3\,B\right )}{g^3\,i\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {B^2}{b\,d\,g^3\,i\,\left (a\,d-b\,c\right )}+\frac {B\,x\,\left (2\,A+3\,B\right )\,\left (a\,d-b\,c\right )}{g^3\,i\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B^2\,d^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g^3\,i\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d^2\,\mathrm {atan}\left (\frac {d^2\,\left (A^2+3\,A\,B+\frac {7\,B^2}{2}\right )\,\left (2\,i\,a^3\,d^3\,g^3-2\,i\,a^2\,b\,c\,d^2\,g^3-2\,i\,a\,b^2\,c^2\,d\,g^3+2\,i\,b^3\,c^3\,g^3\right )\,1{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,d^2+6\,A\,B\,d^2+7\,B^2\,d^2\right )}+\frac {b\,d^3\,x\,\left (i\,a^2\,d^2\,g^3-2\,i\,a\,b\,c\,d\,g^3+i\,b^2\,c^2\,g^3\right )\,\left (A^2+3\,A\,B+\frac {7\,B^2}{2}\right )\,4{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,d^2+6\,A\,B\,d^2+7\,B^2\,d^2\right )}\right )\,\left (A^2+3\,A\,B+\frac {7\,B^2}{2}\right )\,2{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 14.18, size = 1488, normalized size = 4.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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