Optimal. Leaf size=214 \[ \frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g i^2 (b c-a d)^2}-\frac {d (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g i^2 (c+d x) (b c-a d)^2}+\frac {2 A B d (a+b x)}{g i^2 (c+d x) (b c-a d)^2}+\frac {2 B^2 d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{g i^2 (c+d x) (b c-a d)^2}-\frac {2 B^2 d (a+b x)}{g i^2 (c+d x) (b c-a d)^2} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 6.47, antiderivative size = 1687, normalized size of antiderivative = 7.88, number of steps used = 87, number of rules used = 31, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.738, Rules used = {2528, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610, 2525, 44, 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}{(96 c+96 d x)^2 (a g+b g x)} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g (c+d x)}\right ) \, dx\\ &=\frac {b^2 \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{9216 (b c-a d)^2 g}-\frac {(b d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{9216 (b c-a d)^2 g}-\frac {d \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^2} \, dx}{9216 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {(b B) \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}{4608 (b c-a d)^2 g}+\frac {(b B) \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}{4608 (b c-a d)^2 g}-\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) (c+d x)^2} \, dx}{4608 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {B \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^2} \, dx}{4608 g}-\frac {(b B) \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}{4608 (b c-a d)^2 e g}+\frac {(b B) \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}{4608 (b c-a d)^2 e g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {B \int \left (\frac {b^2 \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 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d 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 (c+d x)}\right ) \, dx}{4608 g}-\frac {(b B) \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}{4608 (b c-a d)^2 e g}+\frac {(b B) \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}{4608 (b c-a d)^2 e g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {\left (b^2 B\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{4608 (b c-a d)^2 g}+\frac {(b B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{4608 (b c-a d)^2 g}-\frac {(b B) \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}{4608 (b c-a d) g}+\frac {(b B) \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}{4608 (b c-a d) g}+\frac {(B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{4608 (b c-a d) g}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}+\frac {\left (b B^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}{4608 (b c-a d)^2 g}-\frac {\left (b B^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}{4608 (b c-a d)^2 g}-\frac {(b B) \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}{4608 (b c-a d) g}+\frac {(b B) \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}{4608 (b c-a d) g}+\frac {B^2 \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{4608 (b c-a d) g}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}+\frac {B^2 \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{4608 g}+\frac {\left (b^2 B\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}{4608 (b c-a d)^2 g}-\frac {(b B d) \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}{4608 (b c-a d)^2 g}-\frac {(A b B) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{4608 (b c-a d) g}-\frac {\left (b B^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}{4608 (b c-a d) g}+\frac {\left (b B^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}{4608 (b c-a d)^2 e g}-\frac {\left (b B^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}{4608 (b c-a d)^2 e g}\\ &=-\frac {b B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}+\frac {B^2 \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{4608 g}+\frac {\left (b^2 B\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}{4608 (b c-a d)^2 g}+\frac {\left (b^2 B^2\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{9216 (b c-a d)^2 g}-\frac {(b B d) \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}{4608 (b c-a d)^2 g}-\frac {(A B) \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 )}{4608 (b c-a d) g}+\frac {\left (b B^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}{4608 (b c-a d)^2 e g}-\frac {\left (b B^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}{4608 (b c-a d)^2 e g}\\ &=\frac {B^2}{4608 (b c-a d) g (c+d x)}+\frac {b B^2 \log (a+b x)}{4608 (b c-a d)^2 g}-\frac {b B^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 )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {(A b B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{4608 (b c-a d)^2 g}+\frac {\left (A b^2 B\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4608 (b c-a d)^2 g}+\frac {\left (b^2 B^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{4608 (b c-a d)^2 g}-\frac {\left (b^2 B^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4608 (b c-a d)^2 g}+\frac {\left (b^2 B^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{4608 (b c-a d)^2 g}+\frac {(A B d) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{4608 (b c-a d)^2 g}-\frac {(A b B d) \int \frac {\log (c+d x)}{c+d x} \, dx}{4608 (b c-a d)^2 g}-\frac {\left (b B^2 d\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{4608 (b c-a d)^2 g}+\frac {\left (b B^2 d\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{4608 (b c-a d)^2 g}-\frac {\left (b B^2 d\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{4608 (b c-a d)^2 g}+\frac {\left (b B^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}{4608 (b c-a d) g}\\ &=\frac {B^2}{4608 (b c-a d) g (c+d x)}+\frac {b B^2 \log (a+b x)}{4608 (b c-a d)^2 g}-\frac {A b B \log ^2(a+b x)}{9216 (b c-a d)^2 g}-\frac {b B^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 )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {A b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {A b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^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 )}{4608 (b c-a d)^2 g}-\frac {(A b B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{4608 (b c-a d)^2 g}-\frac {(A b B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{4608 (b c-a d)^2 g}+\frac {\left (b B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{4608 (b c-a d)^2 g}+\frac {\left (b B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{4608 (b c-a d)^2 g}+\frac {\left (b^2 B^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{9216 (b c-a d)^2 g}+\frac {\left (b^2 B^2\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{4608 (b c-a d)^2 g}+\frac {\left (b^2 B^2\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{4608 (b c-a d)^2 g}+\frac {\left (b^2 B^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{4608 (b c-a d)^2 g}-\frac {(A b B d) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4608 (b c-a d)^2 g}-\frac {\left (b B^2 d\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{9216 (b c-a d)^2 g}+\frac {\left (b B^2 d\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4608 (b c-a d)^2 g}-\frac {\left (b B^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}{4608 (b c-a d) g}+\frac {\left (b^2 B^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}{4608 (b c-a d)^2 g}\\ &=\frac {B^2}{4608 (b c-a d) g (c+d x)}+\frac {b B^2 \log (a+b x)}{4608 (b c-a d)^2 g}-\frac {A b B \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x)}{9216 (b c-a d)^2 g}-\frac {b B^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 )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {A b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^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)}{4608 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {A b B \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {A b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4608 (b c-a d)^2 g}-\frac {(A b B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4608 (b c-a d)^2 g}-\frac {\left (b B^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{9216 (b c-a d)^2 g}+\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}+\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}+\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}+\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}-\frac {\left (b B^2 d\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{4608 (b c-a d)^2 g}-\frac {\left (b B^2 d \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}{4608 (b c-a d)^2 g}\\ &=\frac {B^2}{4608 (b c-a d) g (c+d x)}+\frac {b B^2 \log (a+b x)}{4608 (b c-a d)^2 g}-\frac {A b B \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x)}{9216 (b c-a d)^2 g}-\frac {b B^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 )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x) \log (c+d x)}{9216 (b c-a d)^2 g}+\frac {A b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^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)}{4608 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {A b B \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {A b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4608 (b c-a d)^2 g}-\frac {\left (b B^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{9216 (b c-a d)^2 g}-\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}-\frac {\left (B^2 d\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 )}{9216 (b c-a d)^2 g}-\frac {\left (B^2 d\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 )}{4608 (b c-a d)^2 g}+\frac {\left (B^2 d\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 )}{4608 (b c-a d)^2 g}-\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}\\ &=\frac {B^2}{4608 (b c-a d) g (c+d x)}+\frac {b B^2 \log (a+b x)}{4608 (b c-a d)^2 g}-\frac {A b B \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x)}{9216 (b c-a d)^2 g}-\frac {b B^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 )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x) \log (c+d x)}{9216 (b c-a d)^2 g}+\frac {A b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^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)}{4608 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {A b B \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log ^3(c+d x)}{27648 (b c-a d)^2 g}+\frac {A b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9216 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4608 (b c-a d)^2 g}-\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}+\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}+\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}-\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}\\ &=\frac {B^2}{4608 (b c-a d) g (c+d x)}+\frac {b B^2 \log (a+b x)}{4608 (b c-a d)^2 g}-\frac {A b B \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {b B^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 )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x) \log (c+d x)}{9216 (b c-a d)^2 g}+\frac {A b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^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)}{4608 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {A b B \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log ^3(c+d x)}{27648 (b c-a d)^2 g}+\frac {A b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9216 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^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 )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4608 (b c-a d)^2 g}+\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}-\frac {\left (b^2 B^2\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 )}{9216 d (b c-a d)^2 g}-\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{9216 d (b c-a d)^2 g}\\ &=\frac {B^2}{4608 (b c-a d) g (c+d x)}+\frac {b B^2 \log (a+b x)}{4608 (b c-a d)^2 g}-\frac {A b B \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {b B^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 )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x) \log (c+d x)}{9216 (b c-a d)^2 g}+\frac {A b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^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)}{4608 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {A b B \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log ^3(c+d x)}{27648 (b c-a d)^2 g}+\frac {A b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9216 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4608 (b c-a d)^2 g}-\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}+\frac {\left (b B^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 )}{4608 (b c-a d)^2 g}\\ &=\frac {B^2}{4608 (b c-a d) g (c+d x)}+\frac {b B^2 \log (a+b x)}{4608 (b c-a d)^2 g}-\frac {A b B \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {b B^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 )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x) \log (c+d x)}{9216 (b c-a d)^2 g}+\frac {A b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^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)}{4608 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {A b B \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log ^3(c+d x)}{27648 (b c-a d)^2 g}+\frac {A b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9216 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4608 (b c-a d)^2 g}+2 \frac {\left (b B^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 )}{4608 (b c-a d)^2 g}\\ &=\frac {B^2}{4608 (b c-a d) g (c+d x)}+\frac {b B^2 \log (a+b x)}{4608 (b c-a d)^2 g}-\frac {A b B \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {b B^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 )}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9216 (b c-a d)^2 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d) g (c+d x)}-\frac {b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4608 (b c-a d)^2 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d) g (c+d x)}+\frac {b \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9216 (b c-a d)^2 g}-\frac {b B^2 \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log ^2(a+b x) \log (c+d x)}{9216 (b c-a d)^2 g}+\frac {A b B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b B^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)}{4608 (b c-a d)^2 g}+\frac {b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{4608 (b c-a d)^2 g}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9216 (b c-a d)^2 g}-\frac {A b B \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log ^2(c+d x)}{9216 (b c-a d)^2 g}+\frac {b B^2 \log (a+b x) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9216 (b c-a d)^2 g}-\frac {b B^2 \log ^3(c+d x)}{27648 (b c-a d)^2 g}+\frac {A b B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9216 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {A b B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}-\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^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 )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4608 (b c-a d)^2 g}+\frac {b B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4608 (b c-a d)^2 g}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 187, normalized size = 0.87 \[ \frac {3 b \left (A^2-2 A B+2 B^2\right ) (c+d x) \log (a+b x)-3 \left (A^2-2 A B+2 B^2\right ) (a d+b (c+d x) \log (c+d x)-b c)+3 B (A b (c+d x)-B d (a+b x)) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+6 B (A-B) (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )+b B^2 (c+d x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )}{3 g i^2 (c+d x) (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 236, normalized size = 1.10 \[ \frac {{\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b c - 3 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a d + 3 \, {\left (A B b c - B^{2} a d + {\left (A B - B^{2}\right )} b d x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, {\left (A^{2} b c + {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b d x - 2 \, {\left (A B - B^{2}\right )} a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g i^{2} x + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} g i^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.44, size = 349, normalized size = 1.63 \[ -\frac {{\left (B^{2} b e \log \left (\frac {b x e + a e}{d x + c}\right )^{3} + 3 \, A B b e \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 3 \, A^{2} b e \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {3 \, {\left (b x e + a e\right )} B^{2} d \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} - \frac {6 \, {\left (b x e + a e\right )} A B d \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {6 \, {\left (b x e + a e\right )} B^{2} d \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {3 \, {\left (b x e + a e\right )} A^{2} d}{d x + c} + \frac {6 \, {\left (b x e + a e\right )} A B d}{d x + c} - \frac {6 \, {\left (b x e + a e\right )} B^{2} d}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{3 \, {\left (b c g - a d g\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1633, normalized size = 7.63 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.92, size = 1004, normalized size = 4.69 \[ B^{2} {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2} + 2 \, A B {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {1}{3} \, B^{2} {\left (\frac {3 \, {\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x} - \frac {{\left (b d x + b c\right )} \log \left (b x + a\right )^{3} - {\left (b d x + b c\right )} \log \left (d x + c\right )^{3} + 3 \, {\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + 3 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} + 6 \, b c - 6 \, a d + 6 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, b d x + {\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + 2 \, b c + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x}\right )} + A^{2} {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} - \frac {{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} A B}{b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.28, size = 423, normalized size = 1.98 \[ {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B\,b\,\left (A-B\right )}{g\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,\left (a\,d-b\,c\right )}{b\,d\,g\,i^2\,\left (\frac {x}{b}+\frac {c}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {A^2-2\,A\,B+2\,B^2}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}+\frac {B^2\,b\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {2\,B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (A-B\right )\,\left (a\,d-b\,c\right )}{b\,d\,g\,i^2\,\left (\frac {x}{b}+\frac {c}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {b\,\mathrm {atan}\left (\frac {b\,\left (2\,b\,d\,x+\frac {a^2\,d^2\,g\,i^2-b^2\,c^2\,g\,i^2}{g\,i^2\,\left (a\,d-b\,c\right )}\right )\,\left (A^2-2\,A\,B+2\,B^2\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (b\,A^2-2\,b\,A\,B+2\,b\,B^2\right )}\right )\,\left (A^2-2\,A\,B+2\,B^2\right )\,2{}\mathrm {i}}{g\,i^2\,{\left (a\,d-b\,c\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.19, size = 539, normalized size = 2.52 \[ \frac {B^{2} b \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a^{2} d^{2} g i^{2} - 6 a b c d g i^{2} + 3 b^{2} c^{2} g i^{2}} + \frac {\left (- 2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a c d g i^{2} + a d^{2} g i^{2} x - b c^{2} g i^{2} - b c d g i^{2} x} + \left (A^{2} - 2 A B + 2 B^{2}\right ) \left (- \frac {b \log {\left (x + \frac {- \frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} + \frac {b \log {\left (x + \frac {\frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} - \frac {1}{a c d g i^{2} - b c^{2} g i^{2} + x \left (a d^{2} g i^{2} - b c d g i^{2}\right )}\right ) + \frac {\left (A B b c + A B b d x - B^{2} a d - B^{2} b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{2} c d^{2} g i^{2} + a^{2} d^{3} g i^{2} x - 2 a b c^{2} d g i^{2} - 2 a b c d^{2} g i^{2} x + b^{2} c^{3} g i^{2} + b^{2} c^{2} d g i^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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