Optimal. Leaf size=299 \[ \frac{b B (c+d x)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{g^3 (a+b x)^2 (b c-a d)^2}-\frac{b (c+d x)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)^2}+\frac{d (c+d x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{g^3 (a+b x) (b c-a d)^2}-\frac{4 A B d (c+d x)}{g^3 (a+b x) (b c-a d)^2}-\frac{4 B^2 d (c+d x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{g^3 (a+b x) (b c-a d)^2}-\frac{b B^2 (c+d x)^2}{g^3 (a+b x)^2 (b c-a d)^2}+\frac{8 B^2 d (c+d x)}{g^3 (a+b x) (b c-a d)^2} \]
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Rubi [C] time = 1.08082, antiderivative size = 578, normalized size of antiderivative = 1.93, number of steps used = 30, number of rules used = 11, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.324, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{4 B^2 d^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}+\frac{4 B^2 d^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}-\frac{2 B d^2 \log (a+b x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{b g^3 (b c-a d)^2}+\frac{2 B d^2 \log (c+d x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{b g^3 (b c-a d)^2}-\frac{2 B d \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{b g^3 (a+b x) (b c-a d)}+\frac{B \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{b g^3 (a+b x)^2}-\frac{\left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{2 b g^3 (a+b x)^2}-\frac{2 B^2 d^2 \log ^2(a+b x)}{b g^3 (b c-a d)^2}-\frac{2 B^2 d^2 \log ^2(c+d x)}{b g^3 (b c-a d)^2}+\frac{6 B^2 d^2 \log (a+b x)}{b g^3 (b c-a d)^2}+\frac{4 B^2 d^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}-\frac{6 B^2 d^2 \log (c+d x)}{b g^3 (b c-a d)^2}+\frac{4 B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}+\frac{6 B^2 d}{b g^3 (a+b x) (b c-a d)}-\frac{B^2}{b g^3 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 2528
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
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)^3} \, dx &=-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac{B \int \frac{2 (b c-a d) \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{g^2 (a+b x)^3 (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}{2 b g^3 (a+b x)^2}+\frac{(2 B (b c-a d)) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{(a+b x)^3 (c+d x)} \, dx}{b g^3}\\ &=-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac{(2 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 )}{(b c-a d) (a+b x)^3}-\frac{b d \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac{d^3 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b g^3}\\ &=-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac{(2 B) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{(a+b x)^3} \, dx}{g^3}+\frac{\left (2 B d^2\right ) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{a+b x} \, dx}{(b c-a d)^2 g^3}-\frac{\left (2 B d^3\right ) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{c+d x} \, dx}{b (b c-a d)^2 g^3}-\frac{(2 B d) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{(a+b x)^2} \, dx}{(b c-a d) g^3}\\ &=\frac{B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac{2 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac{2 B d^2 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac{B^2 \int \frac{-2 b c+2 a d}{(a+b x)^3 (c+d x)} \, dx}{b g^3}+\frac{\left (2 B^2 d^2\right ) \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 ) \log (a+b x)}{e (c+d x)^2} \, dx}{b (b c-a d)^2 g^3}-\frac{\left (2 B^2 d^2\right ) \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 ) \log (c+d x)}{e (c+d x)^2} \, dx}{b (b c-a d)^2 g^3}+\frac{\left (2 B^2 d\right ) \int \frac{2 (-b c+a d)}{(a+b x)^2 (c+d x)} \, dx}{b (b c-a d) g^3}\\ &=\frac{B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac{2 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac{2 B d^2 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac{\left (4 B^2 d\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b g^3}+\frac{\left (2 B^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b g^3}+\frac{\left (2 B^2 d^2\right ) \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 ) \log (a+b x)}{(c+d x)^2} \, dx}{b (b c-a d)^2 e g^3}-\frac{\left (2 B^2 d^2\right ) \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 ) \log (c+d x)}{(c+d x)^2} \, dx}{b (b c-a d)^2 e g^3}\\ &=\frac{B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac{2 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac{2 B d^2 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac{\left (4 B^2 d\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 g^3}+\frac{\left (2 B^2 (b c-a d)\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b g^3}+\frac{\left (2 B^2 d^2\right ) \int \left (-\frac{2 b e \log (a+b x)}{a+b x}+\frac{2 d e \log (a+b x)}{c+d x}\right ) \, dx}{b (b c-a d)^2 e g^3}-\frac{\left (2 B^2 d^2\right ) \int \left (-\frac{2 b e \log (c+d x)}{a+b x}+\frac{2 d e \log (c+d x)}{c+d x}\right ) \, dx}{b (b c-a d)^2 e g^3}\\ &=-\frac{B^2}{b g^3 (a+b x)^2}+\frac{6 B^2 d}{b (b c-a d) g^3 (a+b x)}+\frac{6 B^2 d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac{6 B^2 d^2 \log (c+d x)}{b (b c-a d)^2 g^3}+\frac{B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac{2 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac{2 B d^2 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac{\left (4 B^2 d^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{(b c-a d)^2 g^3}+\frac{\left (4 B^2 d^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{(b c-a d)^2 g^3}+\frac{\left (4 B^2 d^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b (b c-a d)^2 g^3}-\frac{\left (4 B^2 d^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b (b c-a d)^2 g^3}\\ &=-\frac{B^2}{b g^3 (a+b x)^2}+\frac{6 B^2 d}{b (b c-a d) g^3 (a+b x)}+\frac{6 B^2 d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac{6 B^2 d^2 \log (c+d x)}{b (b c-a d)^2 g^3}+\frac{4 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}+\frac{4 B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac{B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac{2 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac{2 B d^2 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac{\left (4 B^2 d^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{(b c-a d)^2 g^3}-\frac{\left (4 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b (b c-a d)^2 g^3}-\frac{\left (4 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b (b c-a d)^2 g^3}-\frac{\left (4 B^2 d^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b (b c-a d)^2 g^3}\\ &=-\frac{B^2}{b g^3 (a+b x)^2}+\frac{6 B^2 d}{b (b c-a d) g^3 (a+b x)}+\frac{6 B^2 d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac{2 B^2 d^2 \log ^2(a+b x)}{b (b c-a d)^2 g^3}-\frac{6 B^2 d^2 \log (c+d x)}{b (b c-a d)^2 g^3}+\frac{4 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac{2 B^2 d^2 \log ^2(c+d x)}{b (b c-a d)^2 g^3}+\frac{4 B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac{B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac{2 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac{2 B d^2 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac{\left (4 B^2 d^2\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 (b c-a d)^2 g^3}-\frac{\left (4 B^2 d^2\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 (b c-a d)^2 g^3}\\ &=-\frac{B^2}{b g^3 (a+b x)^2}+\frac{6 B^2 d}{b (b c-a d) g^3 (a+b x)}+\frac{6 B^2 d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac{2 B^2 d^2 \log ^2(a+b x)}{b (b c-a d)^2 g^3}-\frac{6 B^2 d^2 \log (c+d x)}{b (b c-a d)^2 g^3}+\frac{4 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac{2 B^2 d^2 \log ^2(c+d x)}{b (b c-a d)^2 g^3}+\frac{4 B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac{B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac{2 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac{2 B d^2 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac{4 B^2 d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac{4 B^2 d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}\\ \end{align*}
Mathematica [C] time = 0.484111, size = 452, normalized size = 1.51 \[ -\frac{\left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2-\frac{2 B \left (-2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+2 B d^2 (a+b x)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )+2 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )+(b c-a d)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )+2 d (a+b x) (a d-b c) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )-B \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+4 B d (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )}{(b c-a d)^2}}{2 b g^3 (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.07, size = 664, normalized size = 2.2 \begin{align*} -{\frac{{A}^{2}}{2\,b \left ( bx+a \right ) ^{2}{g}^{3}}}-{\frac{{B}^{2}}{b \left ( bx+a \right ) ^{2}{g}^{3}}}+{\frac{{B}^{2}}{b \left ( bx+a \right ) ^{2}{g}^{3}}\ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) }-{\frac{{B}^{2}}{2\,b \left ( bx+a \right ) ^{2}{g}^{3}} \left ( \ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) \right ) ^{2}}-6\,{\frac{{B}^{2}d}{b{g}^{3} \left ( ad-bc \right ) \left ( bx+a \right ) }}-3\,{\frac{{B}^{2}{d}^{2}}{b{g}^{3} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }\ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) }+{\frac{{B}^{2}{d}^{2}}{2\,b{g}^{3} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) } \left ( \ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) \right ) ^{2}}+2\,{\frac{{B}^{2}d}{b{g}^{3} \left ( ad-bc \right ) \left ( bx+a \right ) }\ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) }-{\frac{AB}{b \left ( bx+a \right ) ^{2}{g}^{3}}\ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) }+{\frac{AB{a}^{2}{d}^{2}}{b{g}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}-2\,{\frac{ABadc}{{g}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}+{\frac{bAB{c}^{2}}{{g}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}+2\,{\frac{AB{d}^{2}a}{b{g}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}-2\,{\frac{ABdc}{{g}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}+2\,{\frac{AB{d}^{3}a}{b{g}^{3} \left ( ad-bc \right ) ^{3}}\ln \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) }-2\,{\frac{AB{d}^{2}c}{{g}^{3} \left ( ad-bc \right ) ^{3}}\ln \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55689, size = 1351, normalized size = 4.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.09039, size = 846, normalized size = 2.83 \begin{align*} -\frac{{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b^{2} c^{2} - 2 \,{\left (A^{2} - 4 \, A B + 8 \, B^{2}\right )} a b c d +{\left (A^{2} - 6 \, A B + 14 \, B^{2}\right )} a^{2} d^{2} -{\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x - B^{2} b^{2} c^{2} + 2 \, B^{2} a b c d\right )} \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} + 4 \,{\left ({\left (A B - 3 \, B^{2}\right )} b^{2} c d -{\left (A B - 3 \, B^{2}\right )} a b d^{2}\right )} x - 2 \,{\left ({\left (A B - 3 \, B^{2}\right )} b^{2} d^{2} x^{2} -{\left (A B - B^{2}\right )} b^{2} c^{2} + 2 \,{\left (A B - 2 \, B^{2}\right )} a b c d - 2 \,{\left (B^{2} b^{2} c d -{\left (A B - 2 \, B^{2}\right )} a b d^{2}\right )} x\right )} \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 \,{\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \,{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x +{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.77515, size = 877, normalized size = 2.93 \begin{align*} \text{result too large to display} \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}}{{\left (b g x + a g\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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