Optimal. Leaf size=365 \[ -\frac {b^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac {3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac {d^3 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}+\frac {3 b d^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (c+d x) (b c-a d)^4}-\frac {b^3 B (c+d x)}{g^2 i^3 (a+b x) (b c-a d)^4}+\frac {3 b^2 B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^2 i^3 (b c-a d)^4}+\frac {B d^3 (a+b x)^2}{4 g^2 i^3 (c+d x)^2 (b c-a d)^4}-\frac {3 b B d^2 (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4} \]
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Rubi [C] time = 1.11, antiderivative size = 631, normalized size of antiderivative = 1.73, number of steps used = 32, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac {3 b^2 B d \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac {3 b^2 B d \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac {3 b^2 d \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^3}+\frac {3 b^2 d \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac {2 b d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (c+d x) (b c-a d)^3}-\frac {d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^2}-\frac {b^2 B}{g^2 i^3 (a+b x) (b c-a d)^3}+\frac {3 b^2 B d \log ^2(a+b x)}{2 g^2 i^3 (b c-a d)^4}+\frac {3 b^2 B d \log ^2(c+d x)}{2 g^2 i^3 (b c-a d)^4}+\frac {3 b^2 B d \log (a+b x)}{2 g^2 i^3 (b c-a d)^4}-\frac {3 b^2 B d \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac {3 b^2 B d \log (c+d x)}{2 g^2 i^3 (b c-a d)^4}-\frac {3 b^2 B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}+\frac {5 b B d}{2 g^2 i^3 (c+d x) (b c-a d)^3}+\frac {B d}{4 g^2 i^3 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(52 c+52 d x)^3 (a g+b g x)^2} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)^2}-\frac {3 b^3 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^2 g^2 (c+d x)^3}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)^2}+\frac {3 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2 (c+d x)}\right ) \, dx\\ &=-\frac {\left (3 b^3 d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{140608 (b c-a d)^4 g^2}+\frac {\left (3 b^2 d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{140608 (b c-a d)^4 g^2}+\frac {b^3 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{140608 (b c-a d)^3 g^2}+\frac {\left (b d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{70304 (b c-a d)^3 g^2}+\frac {d^2 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{140608 (b c-a d)^2 g^2}\\ &=-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac {3 b^2 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac {\left (3 b^2 B 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}{140608 (b c-a d)^4 g^2}-\frac {\left (3 b^2 B 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}{140608 (b c-a d)^4 g^2}+\frac {\left (b^2 B\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{140608 (b c-a d)^3 g^2}+\frac {(b B d) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{70304 (b c-a d)^3 g^2}+\frac {(B d) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{281216 (b c-a d)^2 g^2}\\ &=-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac {3 b^2 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac {\left (b^2 B\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{140608 (b c-a d)^2 g^2}+\frac {(b B d) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{70304 (b c-a d)^2 g^2}+\frac {(B d) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{281216 (b c-a d) g^2}+\frac {\left (3 b^2 B 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}{140608 (b c-a d)^4 e g^2}-\frac {\left (3 b^2 B 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}{140608 (b c-a d)^4 e g^2}\\ &=-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac {3 b^2 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac {\left (b^2 B\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}{140608 (b c-a d)^2 g^2}+\frac {(b B d) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{70304 (b c-a d)^2 g^2}+\frac {(B d) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{281216 (b c-a d) g^2}+\frac {\left (3 b^2 B 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}{140608 (b c-a d)^4 e g^2}-\frac {\left (3 b^2 B 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}{140608 (b c-a d)^4 e g^2}\\ &=-\frac {b^2 B}{140608 (b c-a d)^3 g^2 (a+b x)}+\frac {B d}{562432 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d}{281216 (b c-a d)^3 g^2 (c+d x)}+\frac {3 b^2 B d \log (a+b x)}{281216 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac {3 b^2 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}-\frac {3 b^2 B d \log (c+d x)}{281216 (b c-a d)^4 g^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac {\left (3 b^3 B d\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{140608 (b c-a d)^4 g^2}-\frac {\left (3 b^3 B d\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{140608 (b c-a d)^4 g^2}-\frac {\left (3 b^2 B d^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{140608 (b c-a d)^4 g^2}+\frac {\left (3 b^2 B d^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{140608 (b c-a d)^4 g^2}\\ &=-\frac {b^2 B}{140608 (b c-a d)^3 g^2 (a+b x)}+\frac {B d}{562432 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d}{281216 (b c-a d)^3 g^2 (c+d x)}+\frac {3 b^2 B d \log (a+b x)}{281216 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac {3 b^2 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}-\frac {3 b^2 B d \log (c+d x)}{281216 (b c-a d)^4 g^2}-\frac {3 b^2 B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}-\frac {3 b^2 B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{140608 (b c-a d)^4 g^2}+\frac {\left (3 b^2 B d\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{140608 (b c-a d)^4 g^2}+\frac {\left (3 b^2 B d\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{140608 (b c-a d)^4 g^2}+\frac {\left (3 b^3 B d\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{140608 (b c-a d)^4 g^2}+\frac {\left (3 b^2 B d^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{140608 (b c-a d)^4 g^2}\\ &=-\frac {b^2 B}{140608 (b c-a d)^3 g^2 (a+b x)}+\frac {B d}{562432 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d}{281216 (b c-a d)^3 g^2 (c+d x)}+\frac {3 b^2 B d \log (a+b x)}{281216 (b c-a d)^4 g^2}+\frac {3 b^2 B d \log ^2(a+b x)}{281216 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac {3 b^2 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}-\frac {3 b^2 B d \log (c+d x)}{281216 (b c-a d)^4 g^2}-\frac {3 b^2 B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac {3 b^2 B d \log ^2(c+d x)}{281216 (b c-a d)^4 g^2}-\frac {3 b^2 B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{140608 (b c-a d)^4 g^2}+\frac {\left (3 b^2 B 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 )}{140608 (b c-a d)^4 g^2}+\frac {\left (3 b^2 B 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 )}{140608 (b c-a d)^4 g^2}\\ &=-\frac {b^2 B}{140608 (b c-a d)^3 g^2 (a+b x)}+\frac {B d}{562432 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d}{281216 (b c-a d)^3 g^2 (c+d x)}+\frac {3 b^2 B d \log (a+b x)}{281216 (b c-a d)^4 g^2}+\frac {3 b^2 B d \log ^2(a+b x)}{281216 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac {3 b^2 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}-\frac {3 b^2 B d \log (c+d x)}{281216 (b c-a d)^4 g^2}-\frac {3 b^2 B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac {3 b^2 B d \log ^2(c+d x)}{281216 (b c-a d)^4 g^2}-\frac {3 b^2 B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{140608 (b c-a d)^4 g^2}-\frac {3 b^2 B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{140608 (b c-a d)^4 g^2}-\frac {3 b^2 B d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{140608 (b c-a d)^4 g^2}\\ \end {align*}
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Mathematica [C] time = 0.75, size = 452, normalized size = 1.24 \[ \frac {-12 b^2 d \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {4 b^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+12 b^2 d \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {8 b d (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-\frac {2 d (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(c+d x)^2}-\frac {4 b^3 B c}{a+b x}+6 b^2 B d \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-6 b^2 B d \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+\frac {4 a b^2 B d}{a+b x}+6 b^2 B d \log (a+b x)-\frac {8 a b B d^2}{c+d x}+\frac {2 b B d (b c-a d)}{c+d x}+\frac {B d (b c-a d)^2}{(c+d x)^2}+\frac {8 b^2 B c d}{c+d x}-6 b^2 B d \log (c+d x)}{4 g^2 i^3 (b c-a d)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 672, normalized size = 1.84 \[ -\frac {4 \, {\left (A + B\right )} b^{3} c^{3} + 3 \, {\left (2 \, A - 5 \, B\right )} a b^{2} c^{2} d - 12 \, {\left (A - B\right )} a^{2} b c d^{2} + {\left (2 \, A - B\right )} a^{3} d^{3} + 6 \, {\left ({\left (2 \, A - B\right )} b^{3} c d^{2} - {\left (2 \, A - B\right )} a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (B b^{3} d^{3} x^{3} + B a b^{2} c^{2} d + {\left (2 \, B b^{3} c d^{2} + B a b^{2} d^{3}\right )} x^{2} + {\left (B b^{3} c^{2} d + 2 \, B a b^{2} c d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, {\left ({\left (6 \, A - B\right )} b^{3} c^{2} d - 2 \, {\left (2 \, A + B\right )} a b^{2} c d^{2} - {\left (2 \, A - 3 \, B\right )} a^{2} b d^{3}\right )} x + 2 \, {\left (3 \, {\left (2 \, A - B\right )} b^{3} d^{3} x^{3} + 2 \, B b^{3} c^{3} + 6 \, A a b^{2} c^{2} d - 6 \, B a^{2} b c d^{2} + B a^{3} d^{3} + 3 \, {\left (4 \, A b^{3} c d^{2} + {\left (2 \, A - 3 \, B\right )} a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (2 \, {\left (A + B\right )} b^{3} c^{2} d + 4 \, {\left (A - B\right )} a b^{2} c d^{2} - B a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} g^{2} i^{3} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} g^{2} i^{3} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} g^{2} i^{3} x + {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4}\right )} g^{2} i^{3}\right )}} \]
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 = 1729, normalized size = 4.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.59, size = 1721, normalized size = 4.72 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.29, size = 983, normalized size = 2.69 \[ \frac {A\,b^2\,c^2}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {A\,a^2\,d^2}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^4}+\frac {B\,a^2\,d^2}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {B\,b^2\,c^2}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {B\,a\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {B\,b\,c\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,A\,b^2\,d^2\,x^2}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,d^2\,x^2}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {5\,A\,a\,b\,c\,d}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {11\,B\,a\,b\,c\,d}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,b^2\,d^2\,x^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,A\,a\,b\,d^2\,x}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {9\,B\,a\,b\,d^2\,x}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {9\,A\,b^2\,c\,d\,x}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,c\,d\,x}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,a\,b\,c\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,a\,b\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,b^2\,c\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {A\,b^2\,d\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,6{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4}-\frac {B\,b^2\,d\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,3{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 50.56, size = 1562, normalized size = 4.28 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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