Optimal. Leaf size=241 \[ \frac {2 B d i \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2}-\frac {d i \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^2}-\frac {2 B i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g^2 (a+b x)}-\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b g^2 (a+b x)}+\frac {2 B^2 d i \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g^2}-\frac {2 B^2 i (c+d x)}{b g^2 (a+b x)} \]
[Out]
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Rubi [B] time = 3.03, antiderivative size = 705, normalized size of antiderivative = 2.93, number of steps used = 43, number of rules used = 20, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 12
Rule 44
Rule 2301
Rule 2317
Rule 2344
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2411
Rule 2418
Rule 2488
Rule 2506
Rule 2507
Rule 2524
Rule 2525
Rule 2528
Rule 6610
Rule 6688
Rule 6742
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*}
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Mathematica [B] time = 2.05, size = 1407, normalized size = 5.84 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} d i x + A^{2} c i + {\left (B^{2} d i x + B^{2} c i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left (A B d i x + A B c i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, x\right ) \]
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 [F] time = 1.70, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right ) \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}}{\left (b g x +a g \right )^{2}}\, dx \]
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 \[ 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 \relax (e)^{2} + {\left (B^{2} b^{2} d^{2} i \log \relax (e)^{2} + 2 \, A B b^{2} d^{2} i \log \relax (e)\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 \relax (e)^{2} + A B b^{2} c d i \log \relax (e)\right )} x + 2 \, {\left (B^{2} b^{2} c^{2} i \log \relax (e) + {\left (B^{2} b^{2} d^{2} i \log \relax (e) + A B b^{2} d^{2} i\right )} x^{2} + {\left (2 \, B^{2} b^{2} c d i \log \relax (e) + 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 \relax (e) - a b c d i + a^{2} d^{2} i\right )} B^{2} + {\left (B^{2} b^{2} d^{2} i \log \relax (e) + A B b^{2} d^{2} i\right )} x^{2} + {\left (A B b^{2} c d i + {\left ({\left (2 \, i \log \relax (e) - 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} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (c\,i+d\,i\,x\right )\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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