Optimal. Leaf size=132 \[ -\frac{4 B \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{b g}+\frac{8 B^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b g}-\frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{b g} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 4.0796, antiderivative size = 740, normalized size of antiderivative = 5.61, number of steps used = 46, number of rules used = 23, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.676, Rules used = {2524, 12, 2528, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2499, 2302, 30, 2396, 2433, 2374, 6589, 2500, 2440, 2434, 2375, 2317} \[ -\frac{4 A B \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) \left (-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )\right )}{b g}-\frac{8 B^2 \text{PolyLog}\left (3,-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{8 B^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{4 B^2 \log \left (\frac{1}{(a+b x)^2}\right ) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \log \left ((c+d x)^2\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{\log (a g+b g x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{b g}-\frac{4 A B \log (a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 A B \log ^2(g (a+b x))}{b g}+\frac{2 B^2 \log ^2(a g+b g x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{b g}+\frac{4 B^2 \log (a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )\right )}{b g}-\frac{4 B^2 \log (c+d x) \log ^2(g (a+b x))}{b g}+\frac{4 B^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{B^2 \log ^2\left ((c+d x)^2\right ) \log (g (a+b x))}{b g}+\frac{B^2 \log ^2\left ((c+d x)^2\right ) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{4 B^2 \log ^2(a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{B^2 \log ^2\left (\frac{1}{(a+b x)^2}\right ) \log (c+d x)}{b g}+\frac{B^2 \log ^2\left (\frac{1}{(a+b x)^2}\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{4 B^2 \log \left (\frac{1}{(a+b x)^2}\right ) \log (c+d x) \log (g (a+b x))}{b g}+\frac{4 B^2 \log ^3(g (a+b x))}{3 b g} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2524
Rule 12
Rule 2528
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 6688
Rule 6742
Rule 2499
Rule 2302
Rule 30
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 2500
Rule 2440
Rule 2434
Rule 2375
Rule 2317
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{a g+b g x} \, dx &=\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B) \int \frac{(a+b x)^2 \left (\frac{2 d e (c+d x)}{(a+b x)^2}-\frac{2 b e (c+d x)^2}{(a+b x)^3}\right ) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{e (c+d x)^2} \, dx}{b g}\\ &=\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B) \int \frac{(a+b x)^2 \left (\frac{2 d e (c+d x)}{(a+b x)^2}-\frac{2 b e (c+d x)^2}{(a+b x)^3}\right ) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{(c+d x)^2} \, dx}{b e g}\\ &=\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B) \int \frac{2 (b c-a d) e \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b e g}\\ &=\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(4 B (b c-a d)) \int \frac{\left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b g}\\ &=\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(4 B (b c-a d)) \int \left (\frac{b \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{(b c-a d) (a+b x)}+\frac{d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{(b c-a d) (c+d x)}\right ) \, dx}{b g}\\ &=\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(4 B) \int \frac{\left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{a+b x} \, dx}{g}-\frac{(4 B d) \int \frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(4 B) \int \left (\frac{A \log (a g+b g x)}{-a-b x}+\frac{B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log (a g+b g x)}{-a-b x}\right ) \, dx}{g}-\frac{(4 B d) \int \left (\frac{A \log (a g+b g x)}{c+d x}+\frac{B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log (a g+b g x)}{c+d x}\right ) \, dx}{b g}\\ &=\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(4 A B) \int \frac{\log (a g+b g x)}{-a-b x} \, dx}{g}-\frac{\left (4 B^2\right ) \int \frac{\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log (a g+b g x)}{-a-b x} \, dx}{g}-\frac{(4 A B d) \int \frac{\log (a g+b g x)}{c+d x} \, dx}{b g}-\frac{\left (4 B^2 d\right ) \int \frac{\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=-\frac{4 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}+(4 A B) \int \frac{\log \left (\frac{b g (c+d x)}{b c g-a d g}\right )}{a g+b g x} \, dx+\frac{(4 A B) \operatorname{Subst}\left (\int \frac{g \log (x)}{x} \, dx,x,a g+b g x\right )}{b g^2}+\frac{\left (4 B^2\right ) \int \frac{\log ^2(a g+b g x)}{a+b x} \, dx}{g}-\frac{\left (4 B^2 d\right ) \int \frac{\log \left (\frac{1}{(a+b x)^2}\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}-\frac{\left (4 B^2 d\right ) \int \frac{\log \left ((c+d x)^2\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}-\frac{\left (4 B^2 d\right ) \int \frac{\log ^2(a g+b g x)}{c+d x} \, dx}{b g}-\frac{\left (4 B^2 d \left (-\log \left (\frac{1}{(a+b x)^2}\right )-\log \left ((c+d x)^2\right )+\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )\right ) \int \frac{\log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=-\frac{4 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}+\left (8 B^2\right ) \int \frac{\log \left (\frac{b g (c+d x)}{b c g-a d g}\right ) \log (a g+b g x)}{a g+b g x} \, dx+\frac{\left (4 B^2\right ) \operatorname{Subst}\left (\int \frac{g \log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g^2}+\frac{(4 A B) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac{(4 A B) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}-\frac{\left (4 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{\left (-\frac{b c-a d}{d}+\frac{b x}{d}\right )^2}\right ) \log \left (-\frac{b c g-a d g}{d}+\frac{b g x}{d}\right )}{x} \, dx,x,c+d x\right )}{b g}-\frac{\left (4 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^2\right ) \log \left (\frac{-b c g+a d g}{d}+\frac{b g x}{d}\right )}{x} \, dx,x,c+d x\right )}{b g}+\left (4 B^2 \left (-\log \left (\frac{1}{(a+b x)^2}\right )-\log \left ((c+d x)^2\right )+\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )\right ) \int \frac{\log \left (\frac{b g (c+d x)}{b c g-a d g}\right )}{a g+b g x} \, dx\\ &=\frac{2 A B \log ^2(g (a+b x))}{b g}-\frac{4 B^2 \log \left (\frac{1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{4 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac{4 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{B^2 \operatorname{Subst}\left (\int \frac{\log ^2\left (x^2\right )}{\frac{-b c g+a d g}{d}+\frac{b g x}{d}} \, dx,x,c+d x\right )}{d}-\frac{\left (8 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{b c g-a d g}{d}+\frac{b g x}{d}\right )}{-\frac{b c g-a d g}{d}+\frac{b g x}{d}} \, dx,x,c+d x\right )}{d}+\frac{\left (4 B^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac{\left (8 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{b g \left (\frac{b c g-a d g}{b g}+\frac{d x}{b g}\right )}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac{\left (4 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{1}{\left (-\frac{b c-a d}{d}+\frac{b x}{d}\right )^2}\right )}{-\frac{b c-a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{d g}+\frac{\left (4 B^2 \left (-\log \left (\frac{1}{(a+b x)^2}\right )-\log \left ((c+d x)^2\right )+\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=\frac{2 A B \log ^2(g (a+b x))}{b g}-\frac{4 B^2 \log \left (\frac{1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{4 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac{4 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{8 B^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{\left (4 B^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (g (a+b x))\right )}{b g}+\frac{\left (4 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{x^2}\right ) \log \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac{\left (4 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^2\right ) \log \left (1+\frac{b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}-\frac{\left (8 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{b c g-a d g}{b g}+\frac{d x}{b g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac{\left (8 B^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=\frac{2 A B \log ^2(g (a+b x))}{b g}+\frac{4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 \log ^2\left (\frac{1}{(a+b x)^2}\right ) \log (c+d x)}{b g}-\frac{4 B^2 \log \left (\frac{1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac{4 B^2 \log ^2(g (a+b x)) \log (c+d x)}{b g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{4 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac{4 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{8 B^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \log \left ((c+d x)^2\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{8 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{\left (4 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{b c g-a d g}{b g}+\frac{d x}{b g}} \, dx,x,a g+b g x\right )}{b^2 g^2}-\frac{\left (8 B^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}+\frac{\left (B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (\frac{1}{x^2}\right )}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{b^2 g}\\ &=\frac{2 A B \log ^2(g (a+b x))}{b g}+\frac{4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 \log ^2\left (\frac{1}{(a+b x)^2}\right ) \log (c+d x)}{b g}-\frac{4 B^2 \log \left (\frac{1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac{4 B^2 \log ^2(g (a+b x)) \log (c+d x)}{b g}+\frac{B^2 \log ^2\left (\frac{1}{(a+b x)^2}\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{4 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac{4 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{8 B^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \log \left ((c+d x)^2\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{8 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{8 B^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{\left (4 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{x^2}\right ) \log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac{\left (8 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1+\frac{d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=\frac{2 A B \log ^2(g (a+b x))}{b g}+\frac{4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 \log ^2\left (\frac{1}{(a+b x)^2}\right ) \log (c+d x)}{b g}-\frac{4 B^2 \log \left (\frac{1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac{4 B^2 \log ^2(g (a+b x)) \log (c+d x)}{b g}+\frac{B^2 \log ^2\left (\frac{1}{(a+b x)^2}\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{4 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac{4 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{4 B^2 \log \left (\frac{1}{(a+b x)^2}\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \log \left ((c+d x)^2\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{8 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{8 B^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{\left (8 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 )}{b g}-\frac{\left (8 B^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=\frac{2 A B \log ^2(g (a+b x))}{b g}+\frac{4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 \log ^2\left (\frac{1}{(a+b x)^2}\right ) \log (c+d x)}{b g}-\frac{4 B^2 \log \left (\frac{1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac{4 B^2 \log ^2(g (a+b x)) \log (c+d x)}{b g}+\frac{B^2 \log ^2\left (\frac{1}{(a+b x)^2}\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac{4 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{4 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac{4 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{4 B^2 \log \left (\frac{1}{(a+b x)^2}\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \left (\log \left (\frac{1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{4 B^2 \log \left ((c+d x)^2\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{8 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{8 B^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}\\ \end{align*}
Mathematica [A] time = 0.330015, size = 257, normalized size = 1.95 \[ \frac{4 A B \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )-4 B^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+8 B^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )+A^2 \log (a+b x)+2 A B \log (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )-4 A B \log (a+b x) \log \left (\frac{c}{d}+x\right )+4 A B \log \left (\frac{c}{d}+x\right ) \log \left (\frac{d (a+b x)}{a d-b c}\right )-2 A B \log ^2\left (\frac{a}{b}+x\right )+4 A B \log \left (\frac{a}{b}+x\right ) \log (a+b x)-B^2 \log \left (\frac{a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{b g} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.391, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{bgx+ag} \left ( A+B\ln \left ({\frac{e \left ( dx+c \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}} \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*} \frac{4 \, B^{2} \log \left (b x + a\right ) \log \left (d x + c\right )^{2}}{b g} + \frac{A^{2} \log \left (b g x + a g\right )}{b g} - \int -\frac{B^{2} b c \log \left (e\right )^{2} + 2 \, A B b c \log \left (e\right ) + 4 \,{\left (B^{2} b d x + B^{2} b c\right )} \log \left (b x + a\right )^{2} +{\left (B^{2} b d \log \left (e\right )^{2} + 2 \, A B b d \log \left (e\right )\right )} x - 4 \,{\left (B^{2} b c \log \left (e\right ) + A B b c +{\left (B^{2} b d \log \left (e\right ) + A B b d\right )} x\right )} \log \left (b x + a\right ) + 4 \,{\left (B^{2} b c \log \left (e\right ) + A B b c +{\left (B^{2} b d \log \left (e\right ) + A B b d\right )} x - 2 \,{\left (2 \, B^{2} b d x +{\left (b c + a d\right )} B^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b^{2} d g x^{2} + a b c g +{\left (b^{2} c g + a b d g\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{B^{2} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2} + 2 \, A B \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + A^{2}}{b g x + a g}, 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 (B \log \left (\frac{{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2}}{b g x + a g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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