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]
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Rubi [C] time = 6.4707, 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 2528
Rule 2524
Rule 12
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 6688
Rule 6742
Rule 2411
Rule 2344
Rule 2317
Rule 2507
Rule 2488
Rule 2506
Rule 6610
Rule 2525
Rule 44
Rule 2500
Rule 2433
Rule 2375
Rule 2374
Rule 6589
Rule 2440
Rule 2434
Rule 2499
Rule 2396
Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(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*}
Mathematica [A] time = 0.709266, 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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Maple [B] time = 0.059, size = 1633, normalized size = 7.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.57156, size = 1355, normalized size = 6.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.521285, size = 510, normalized size = 2.38 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.92577, size = 539, normalized size = 2.52 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}{\left (d i x + c i\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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