Optimal. Leaf size=128 \[ -\frac {2 B \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{b g}+\frac {2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g} \]
[Out]
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Rubi [B] time = 4.42, antiderivative size = 719, normalized size of antiderivative = 5.62, number of steps used = 47, number of rules used = 24, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, 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 12
Rule 30
Rule 2301
Rule 2302
Rule 2317
Rule 2374
Rule 2375
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2418
Rule 2433
Rule 2434
Rule 2440
Rule 2499
Rule 2500
Rule 2524
Rule 2528
Rule 6589
Rule 6691
Rule 6741
Rule 6742
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*}
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Mathematica [A] time = 0.29, size = 251, normalized size = 1.96 \[ \frac {A^2 \log (a+b x)+2 A B \log (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )+2 A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )-2 A B \log (a+b x) \log \left (\frac {c}{d}+x\right )+2 A B \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{a d-b c}\right )-A B \log ^2\left (\frac {a}{b}+x\right )+2 A B \log \left (\frac {a}{b}+x\right ) \log (a+b x)-2 B^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \log \left (\frac {e (c+d x)}{a+b x}\right )-B^2 \log \left (\frac {a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac {e (c+d x)}{a+b x}\right )+2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (\frac {d e x + c e}{b x + a}\right )^{2} + 2 \, A B \log \left (\frac {d e x + c e}{b x + a}\right ) + A^{2}}{b g x + a g}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 906, normalized size = 7.08 \[ -\frac {B^{2} a d \ln \left (-\frac {\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b}{d e}+1\right ) \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )^{2}}{\left (a d -b c \right ) b g}+\frac {B^{2} c \ln \left (-\frac {\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b}{d e}+1\right ) \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )^{2}}{\left (a d -b c \right ) g}-\frac {2 A B a d \ln \left (-\frac {-d e +\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b}{d e}\right ) \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{\left (a d -b c \right ) b g}+\frac {2 A B c \ln \left (-\frac {-d e +\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b}{d e}\right ) \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{\left (a d -b c \right ) g}-\frac {2 B^{2} a d \polylog \left (2, \frac {\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b}{d e}\right ) \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{\left (a d -b c \right ) b g}+\frac {2 B^{2} c \polylog \left (2, \frac {\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b}{d e}\right ) \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{\left (a d -b c \right ) g}-\frac {A^{2} a d \ln \left (-d e +\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b \right )}{\left (a d -b c \right ) b g}+\frac {A^{2} c \ln \left (-d e +\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b \right )}{\left (a d -b c \right ) g}-\frac {2 A B a d \dilog \left (-\frac {-d e +\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b}{d e}\right )}{\left (a d -b c \right ) b g}+\frac {2 A B c \dilog \left (-\frac {-d e +\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b}{d e}\right )}{\left (a d -b c \right ) g}+\frac {2 B^{2} a d \polylog \left (3, \frac {\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b}{d e}\right )}{\left (a d -b c \right ) b g}-\frac {2 B^{2} c \polylog \left (3, \frac {\left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right ) b}{d e}\right )}{\left (a d -b c \right ) g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {B^{2} \log \left (b x + a\right ) \log \left (d x + c\right )^{2}}{b g} + \frac {A^{2} \log \left (b g x + a g\right )}{b g} - \int -\frac {B^{2} b c \log \relax (e)^{2} + 2 \, A B b c \log \relax (e) + {\left (B^{2} b d x + B^{2} b c\right )} \log \left (b x + a\right )^{2} + {\left (B^{2} b d \log \relax (e)^{2} + 2 \, A B b d \log \relax (e)\right )} x - 2 \, {\left (B^{2} b c \log \relax (e) + A B b c + {\left (B^{2} b d \log \relax (e) + A B b d\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (B^{2} b c \log \relax (e) + A B b c + {\left (B^{2} b d \log \relax (e) + A B b d\right )} x - {\left (2 \, B^{2} b d x + {\left (b c + a d\right )} B^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b^{2} d g x^{2} + a b c g + {\left (b^{2} c g + a b d g\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\right )}^2}{a\,g+b\,g\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A^{2}}{a + b x}\, dx + \int \frac {B^{2} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}^{2}}{a + b x}\, dx + \int \frac {2 A B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}{a + b x}\, dx}{g} \]
Verification of antiderivative is not currently implemented for this CAS.
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