Optimal. Leaf size=128 \[ -\frac{2 B \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{b g}+\frac{2 B^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b g}-\frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )^2}{b g} \]
[Out]
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Rubi [B] time = 4.42328, antiderivative size = 719, normalized size of antiderivative = 5.62, number of steps used = 47, number of rules used = 24, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {2524, 12, 2528, 2418, 2390, 2301, 2394, 2393, 2391, 6691, 6741, 6742, 2499, 2302, 30, 2396, 2433, 2374, 6589, 2500, 2440, 2434, 2375, 2317} \[ -\frac{2 A B \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) \left (-\log \left (\frac{e (c+d x)}{a+b x}\right )+\log \left (\frac{1}{a+b x}\right )+\log (c+d x)\right )}{b g}-\frac{2 B^2 \text{PolyLog}\left (3,-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \log \left (\frac{1}{a+b x}\right ) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \log (c+d x) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{\log (a g+b g x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )^2}{b g}-\frac{2 A B \log (a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{A B \log ^2(g (a+b x))}{b g}+\frac{B^2 \log ^2(a g+b g x) \log \left (\frac{e (c+d x)}{a+b x}\right )}{b g}+\frac{2 B^2 \log (a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (-\log \left (\frac{e (c+d x)}{a+b x}\right )+\log \left (\frac{1}{a+b x}\right )+\log (c+d x)\right )}{b g}-\frac{B^2 \log (c+d x) \log ^2(g (a+b x))}{b g}+\frac{B^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{B^2 \log ^2(c+d x) \log (g (a+b x))}{b g}+\frac{B^2 \log ^2(c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{B^2 \log ^2(a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{B^2 \log ^2\left (\frac{1}{a+b x}\right ) \log (c+d x)}{b g}+\frac{B^2 \log ^2\left (\frac{1}{a+b x}\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \log \left (\frac{1}{a+b x}\right ) \log (c+d x) \log (g (a+b x))}{b g}+\frac{B^2 \log ^3(g (a+b x))}{3 b g} \]
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 6691
Rule 6741
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)}{a+b x}\right )\right )^2}{a g+b g x} \, dx &=\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B) \int \frac{(a+b x) \left (\frac{d e}{a+b x}-\frac{b e (c+d x)}{(a+b x)^2}\right ) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{e (c+d x)} \, dx}{b g}\\ &=\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B) \int \frac{(a+b x) \left (\frac{d e}{a+b x}-\frac{b e (c+d x)}{(a+b x)^2}\right ) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{c+d x} \, dx}{b e g}\\ &=\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B) \int \frac{\left (d e-\frac{b e (c+d x)}{a+b x}\right ) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{c+d x} \, dx}{b e g}\\ &=\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B) \int \frac{(b c-a d) e \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\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)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B (b c-a d)) \int \frac{\left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\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)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B (b c-a d)) \int \left (\frac{b \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\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)}{a+b x}\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)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B) \int \frac{\left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{a+b x} \, dx}{g}-\frac{(2 B d) \int \frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\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)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 B) \int \left (\frac{A \log (a g+b g x)}{-a-b x}+\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right ) \log (a g+b g x)}{-a-b x}\right ) \, dx}{g}-\frac{(2 B d) \int \left (\frac{A \log (a g+b g x)}{c+d x}+\frac{B \log \left (\frac{e (c+d x)}{a+b x}\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)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{(2 A B) \int \frac{\log (a g+b g x)}{-a-b x} \, dx}{g}-\frac{\left (2 B^2\right ) \int \frac{\log \left (\frac{e (c+d x)}{a+b x}\right ) \log (a g+b g x)}{-a-b x} \, dx}{g}-\frac{(2 A B d) \int \frac{\log (a g+b g x)}{c+d x} \, dx}{b g}-\frac{\left (2 B^2 d\right ) \int \frac{\log \left (\frac{e (c+d x)}{a+b x}\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=-\frac{2 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)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}+\frac{B^2 \log \left (\frac{e (c+d x)}{a+b x}\right ) \log ^2(a g+b g x)}{b g}+(2 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{(2 A B) \operatorname{Subst}\left (\int \frac{g \log (x)}{x} \, dx,x,a g+b g x\right )}{b g^2}+\frac{B^2 \int \frac{\log ^2(a g+b g x)}{a+b x} \, dx}{g}-\frac{\left (B^2 d\right ) \int \frac{\log ^2(a g+b g x)}{c+d x} \, dx}{b g}-\frac{\left (2 B^2 d\right ) \int \frac{\log \left (\frac{1}{a+b x}\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}-\frac{\left (2 B^2 d\right ) \int \frac{\log (c+d x) \log (a g+b g x)}{c+d x} \, dx}{b g}-\frac{\left (2 B^2 d \left (-\log \left (\frac{1}{a+b x}\right )-\log (c+d x)+\log \left (\frac{e (c+d x)}{a+b x}\right )\right )\right ) \int \frac{\log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=-\frac{2 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{B^2 \log \left (\frac{e (c+d x)}{a+b x}\right ) \log ^2(a g+b g x)}{b g}+\left (2 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{B^2 \operatorname{Subst}\left (\int \frac{g \log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g^2}+\frac{(2 A B) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac{(2 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 (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{-\frac{b c-a d}{d}+\frac{b x}{d}}\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 (2 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 )}{x} \, dx,x,c+d x\right )}{b g}+\left (2 B^2 \left (-\log \left (\frac{1}{a+b x}\right )-\log (c+d x)+\log \left (\frac{e (c+d x)}{a+b x}\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{A B \log ^2(g (a+b x))}{b g}-\frac{2 B^2 \log \left (\frac{1}{a+b x}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2(c+d x)}{b g}-\frac{2 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{B^2 \log \left (\frac{e (c+d x)}{a+b x}\right ) \log ^2(a g+b g x)}{b g}-\frac{2 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(x)}{\frac{-b c g+a d g}{d}+\frac{b g x}{d}} \, dx,x,c+d x\right )}{d}-\frac{\left (2 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{B^2 \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac{\left (2 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 (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{1}{-\frac{b c-a d}{d}+\frac{b x}{d}}\right )}{-\frac{b c-a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{d g}+\frac{\left (2 B^2 \left (-\log \left (\frac{1}{a+b x}\right )-\log (c+d x)+\log \left (\frac{e (c+d x)}{a+b x}\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{A B \log ^2(g (a+b x))}{b g}-\frac{2 B^2 \log \left (\frac{1}{a+b x}\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(c+d x)}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2(c+d x)}{b g}-\frac{2 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{B^2 \log \left (\frac{e (c+d x)}{a+b x}\right ) \log ^2(a g+b g x)}{b g}-\frac{2 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{B^2 \operatorname{Subst}\left (\int x^2 \, dx,x,\log (g (a+b x))\right )}{b g}+\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{x}\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 (2 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 (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \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 (2 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{A B \log ^2(g (a+b x))}{b g}+\frac{B^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 \log ^2\left (\frac{1}{a+b x}\right ) \log (c+d x)}{b g}-\frac{2 B^2 \log \left (\frac{1}{a+b x}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac{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(c+d x)}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2(c+d x)}{b g}-\frac{2 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{B^2 \log \left (\frac{e (c+d x)}{a+b x}\right ) \log ^2(a g+b g x)}{b g}-\frac{2 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{\left (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 (2 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}\right )}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{b^2 g}\\ &=\frac{A B \log ^2(g (a+b x))}{b g}+\frac{B^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 \log ^2\left (\frac{1}{a+b x}\right ) \log (c+d x)}{b g}-\frac{2 B^2 \log \left (\frac{1}{a+b x}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac{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(c+d x)}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2(c+d x)}{b g}+\frac{B^2 \log ^2\left (\frac{1}{a+b x}\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{2 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{B^2 \log \left (\frac{e (c+d x)}{a+b x}\right ) \log ^2(a g+b g x)}{b g}-\frac{2 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{\left (2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{x}\right ) \log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac{\left (2 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{A B \log ^2(g (a+b x))}{b g}+\frac{B^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 \log ^2\left (\frac{1}{a+b x}\right ) \log (c+d x)}{b g}-\frac{2 B^2 \log \left (\frac{1}{a+b x}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac{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(c+d x)}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2(c+d x)}{b g}+\frac{B^2 \log ^2\left (\frac{1}{a+b x}\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{2 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{B^2 \log \left (\frac{e (c+d x)}{a+b x}\right ) \log ^2(a g+b g x)}{b g}-\frac{2 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \log \left (\frac{1}{a+b x}\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{\left (2 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 (2 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{A B \log ^2(g (a+b x))}{b g}+\frac{B^2 \log ^3(g (a+b x))}{3 b g}-\frac{B^2 \log ^2\left (\frac{1}{a+b x}\right ) \log (c+d x)}{b g}-\frac{2 B^2 \log \left (\frac{1}{a+b x}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac{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(c+d x)}{b g}-\frac{B^2 \log (g (a+b x)) \log ^2(c+d x)}{b g}+\frac{B^2 \log ^2\left (\frac{1}{a+b x}\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{B^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{2 A B \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac{2 B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \log (a g+b g x)}{b g}+\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \log (a g+b g x)}{b g}-\frac{B^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac{B^2 \log \left (\frac{e (c+d x)}{a+b x}\right ) \log ^2(a g+b g x)}{b g}-\frac{2 A B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \log \left (\frac{1}{a+b x}\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \left (\log \left (\frac{1}{a+b x}\right )+\log (c+d x)-\log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}+\frac{2 B^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{2 B^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}\\ \end{align*}
Mathematica [A] time = 0.276452, size = 251, normalized size = 1.96 \[ \frac{2 A B \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )-2 B^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \log \left (\frac{e (c+d x)}{a+b x}\right )+2 B^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )+A^2 \log (a+b x)+2 A B \log (a+b x) \log \left (\frac{e (c+d x)}{a+b x}\right )-2 A B \log (a+b x) \log \left (\frac{c}{d}+x\right )+2 A B \log \left (\frac{c}{d}+x\right ) \log \left (\frac{d (a+b x)}{a d-b c}\right )-A B \log ^2\left (\frac{a}{b}+x\right )+2 A B \log \left (\frac{a}{b}+x\right ) \log (a+b x)-B^2 \log \left (\frac{a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac{e (c+d x)}{a+b x}\right )}{b g} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.067, size = 906, normalized size = 7.1 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{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 ) +{\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 - 2 \,{\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 ) + 2 \,{\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 -{\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 e x + c e}{b x + a}\right )^{2} + 2 \, A B \log \left (\frac{d e x + c e}{b x + a}\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 )} e}{b x + a}\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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