Optimal. Leaf size=241 \[ \frac{2 B d i \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2}+\frac{2 B^2 d i \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b^2 g^2}-\frac{d i \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^2}-\frac{2 B i (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b g^2 (a+b x)}-\frac{i (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b g^2 (a+b x)}-\frac{2 B^2 i (c+d x)}{b g^2 (a+b x)} \]
[Out]
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Rubi [B] time = 3.02775, antiderivative size = 705, normalized size of antiderivative = 2.93, number of steps used = 43, number of rules used = 20, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ \frac{2 A B d i \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^2}+\frac{2 B^2 d i \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{2 B^2 d i \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^2}-\frac{2 B^2 d i \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac{2 B^2 d i \text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right )}{b^2 g^2}-\frac{2 B d i \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2}-\frac{2 B i (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2 (a+b x)}+\frac{2 B d i \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2}+\frac{d i \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^2}-\frac{i (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^2 (a+b x)}+\frac{2 A B d i \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac{A B d i \log ^2(a+b x)}{b^2 g^2}-\frac{B^2 d i \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{B^2 d i \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{2 B^2 i (b c-a d)}{b^2 g^2 (a+b x)}-\frac{2 B^2 d i \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^2}-\frac{2 B^2 d i \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac{B^2 d i \log ^2(a+b x)}{b^2 g^2}-\frac{2 B^2 d i \log (a+b x)}{b^2 g^2}+\frac{B^2 d i \log ^2(c+d x)}{b^2 g^2}+\frac{2 B^2 d i \log (c+d x)}{b^2 g^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 6688
Rule 6742
Rule 2411
Rule 2344
Rule 2317
Rule 2507
Rule 2488
Rule 2506
Rule 6610
Rubi steps
\begin{align*} \int \frac{(60 c+60 d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx &=\int \left (\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g^2 (a+b x)^2}+\frac{60 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g^2 (a+b x)}\right ) \, dx\\ &=\frac{(60 d) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{b g^2}+\frac{(60 (b c-a d)) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b g^2}\\ &=-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{(120 B d) \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}{b^2 g^2}+\frac{(120 B (b c-a d)) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac{\left (120 B (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}-\frac{(120 B d) \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}{b^2 e g^2}\\ &=-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac{\left (120 B (b c-a d)^2\right ) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^2}-\frac{(120 B d) \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}{b^2 e g^2}\\ &=-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}-\frac{(120 B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b g^2}+\frac{\left (120 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 g^2}+\frac{(120 B (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^2}-\frac{(120 B d (b c-a d)) \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}{b^2 g^2}\\ &=-\frac{120 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}-\frac{120 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac{120 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{\left (120 B^2 d\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}{b^2 g^2}-\frac{\left (120 B^2 d\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}{b^2 g^2}+\frac{\left (120 B^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}-\frac{(120 B d (b c-a d)) \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}{b^2 g^2}\\ &=-\frac{120 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}-\frac{120 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac{120 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}-\frac{(120 A B d (b c-a d)) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac{\left (120 B^2 d (b c-a d)\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}{b^2 g^2}+\frac{\left (120 B^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}+\frac{\left (120 B^2 d\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}{b^2 e g^2}-\frac{\left (120 B^2 d\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}{b^2 e g^2}\\ &=-\frac{60 B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{120 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}-\frac{120 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac{120 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{\left (60 B^2 d\right ) \int \frac{\log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b g^2}-\frac{(120 A B d (b c-a d)) \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 )}{b^3 g^2}+\frac{\left (120 B^2 (b c-a d)^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^2}+\frac{\left (120 B^2 d\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}{b^2 e g^2}-\frac{\left (120 B^2 d\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}{b^2 e g^2}\\ &=-\frac{120 B^2 (b c-a d)}{b^2 g^2 (a+b x)}-\frac{120 B^2 d \log (a+b x)}{b^2 g^2}-\frac{60 B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{60 B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{120 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}-\frac{120 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac{120 B^2 d \log (c+d x)}{b^2 g^2}+\frac{120 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}-\frac{(120 A B d) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g^2}+\frac{\left (120 B^2 d\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b g^2}-\frac{\left (120 B^2 d\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b g^2}+\frac{\left (120 A B d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{b^3 g^2}-\frac{\left (120 B^2 d^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 g^2}+\frac{\left (120 B^2 d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 g^2}+\frac{\left (120 B^2 d (b c-a d)\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}{b^2 g^2}\\ &=-\frac{120 B^2 (b c-a d)}{b^2 g^2 (a+b x)}-\frac{120 B^2 d \log (a+b x)}{b^2 g^2}-\frac{60 A B d \log ^2(a+b x)}{b^2 g^2}-\frac{60 B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{60 B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{120 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}-\frac{120 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac{120 B^2 d \log (c+d x)}{b^2 g^2}-\frac{120 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac{120 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{120 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac{120 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac{120 B^2 d \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 )}{b^2 g^2}-\frac{(120 A B d) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g^2}+\frac{\left (120 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g^2}+\frac{\left (120 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 g^2}+\frac{\left (120 B^2 d\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g^2}+\frac{\left (120 B^2 d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 g^2}-\frac{\left (120 B^2 d (b c-a d)\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}{b^2 g^2}\\ &=-\frac{120 B^2 (b c-a d)}{b^2 g^2 (a+b x)}-\frac{120 B^2 d \log (a+b x)}{b^2 g^2}-\frac{60 A B d \log ^2(a+b x)}{b^2 g^2}+\frac{60 B^2 d \log ^2(a+b x)}{b^2 g^2}-\frac{60 B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{60 B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{120 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}-\frac{120 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac{120 B^2 d \log (c+d x)}{b^2 g^2}-\frac{120 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac{120 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{60 B^2 d \log ^2(c+d x)}{b^2 g^2}+\frac{120 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac{120 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac{120 A B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^2}+\frac{120 B^2 d \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 )}{b^2 g^2}+\frac{120 B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^2 g^2}+\frac{\left (120 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g^2}+\frac{\left (120 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 g^2}\\ &=-\frac{120 B^2 (b c-a d)}{b^2 g^2 (a+b x)}-\frac{120 B^2 d \log (a+b x)}{b^2 g^2}-\frac{60 A B d \log ^2(a+b x)}{b^2 g^2}+\frac{60 B^2 d \log ^2(a+b x)}{b^2 g^2}-\frac{60 B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{60 B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g^2}-\frac{120 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}-\frac{120 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}-\frac{60 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2 (a+b x)}+\frac{60 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac{120 B^2 d \log (c+d x)}{b^2 g^2}-\frac{120 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac{120 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}+\frac{60 B^2 d \log ^2(c+d x)}{b^2 g^2}+\frac{120 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac{120 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac{120 A B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^2}-\frac{120 B^2 d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^2}-\frac{120 B^2 d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac{120 B^2 d \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 )}{b^2 g^2}+\frac{120 B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^2 g^2}\\ \end{align*}
Mathematica [B] time = 2.46368, size = 1407, normalized size = 5.84 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.008, size = 0, normalized size = 0. \begin{align*} \int{\frac{dix+ci}{ \left ( bgx+ag \right ) ^{2}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} A^{2} d i{\left (\frac{a}{b^{3} g^{2} x + a b^{2} g^{2}} + \frac{\log \left (b x + a\right )}{b^{2} g^{2}}\right )} - 2 \, A B c i{\left (\frac{\log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac{1}{b^{2} g^{2} x + a b g^{2}} + \frac{d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac{d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac{A^{2} c i}{b^{2} g^{2} x + a b g^{2}} - \frac{{\left ({\left (b c i - a d i\right )} B^{2} -{\left (B^{2} b d i x + B^{2} a d i\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2}}{b^{3} g^{2} x + a b^{2} g^{2}} - \int -\frac{B^{2} b^{2} c^{2} i \log \left (e\right )^{2} +{\left (B^{2} b^{2} d^{2} i \log \left (e\right )^{2} + 2 \, A B b^{2} d^{2} i \log \left (e\right )\right )} x^{2} +{\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left (b x + a\right )^{2} + 2 \,{\left (B^{2} b^{2} c d i \log \left (e\right )^{2} + A B b^{2} c d i \log \left (e\right )\right )} x + 2 \,{\left (B^{2} b^{2} c^{2} i \log \left (e\right ) +{\left (B^{2} b^{2} d^{2} i \log \left (e\right ) + A B b^{2} d^{2} i\right )} x^{2} +{\left (2 \, B^{2} b^{2} c d i \log \left (e\right ) + A B b^{2} c d i\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (b^{2} c^{2} i \log \left (e\right ) - a b c d i + a^{2} d^{2} i\right )} B^{2} +{\left (B^{2} b^{2} d^{2} i \log \left (e\right ) + A B b^{2} d^{2} i\right )} x^{2} +{\left (A B b^{2} c d i +{\left ({\left (2 \, i \log \left (e\right ) - i\right )} b^{2} c d + a b d^{2} i\right )} B^{2}\right )} x +{\left (2 \, B^{2} b^{2} d^{2} i x^{2} + 2 \,{\left (b^{2} c d i + a b d^{2} i\right )} B^{2} x +{\left (b^{2} c^{2} i + a^{2} d^{2} i\right )} B^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b^{4} d g^{2} x^{3} + a^{2} b^{2} c g^{2} +{\left (b^{4} c g^{2} + 2 \, a b^{3} d g^{2}\right )} x^{2} +{\left (2 \, a b^{3} c g^{2} + a^{2} b^{2} d g^{2}\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A^{2} d i x + A^{2} c i +{\left (B^{2} d i x + B^{2} c i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left (A B d i x + A B c i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}{\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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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