Optimal. Leaf size=457 \[ -\frac {b^4 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 i^2 (a+b x)^3 (b c-a d)^5}+\frac {2 b^3 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x)^2 (b c-a d)^5}-\frac {6 b^2 d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x) (b c-a d)^5}+\frac {d^4 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (c+d x) (b c-a d)^5}-\frac {4 b d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (b c-a d)^5}-\frac {b^4 B (c+d x)^3}{9 g^4 i^2 (a+b x)^3 (b c-a d)^5}+\frac {b^3 B d (c+d x)^2}{g^4 i^2 (a+b x)^2 (b c-a d)^5}-\frac {6 b^2 B d^2 (c+d x)}{g^4 i^2 (a+b x) (b c-a d)^5}-\frac {B d^4 (a+b x)}{g^4 i^2 (c+d x) (b c-a d)^5}+\frac {2 b B d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )}{g^4 i^2 (b c-a d)^5} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 1.36, antiderivative size = 705, normalized size of antiderivative = 1.54, number of steps used = 36, 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} \[ -\frac {4 b B d^3 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g^4 i^2 (b c-a d)^5}-\frac {4 b B d^3 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g^4 i^2 (b c-a d)^5}-\frac {4 b d^3 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (b c-a d)^5}-\frac {d^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (c+d x) (b c-a d)^4}+\frac {4 b d^3 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (b c-a d)^5}-\frac {3 b d^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x) (b c-a d)^4}+\frac {b d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x)^2 (b c-a d)^3}-\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 i^2 (a+b x)^3 (b c-a d)^2}+\frac {B d^3}{g^4 i^2 (c+d x) (b c-a d)^4}-\frac {13 b B d^2}{3 g^4 i^2 (a+b x) (b c-a d)^4}+\frac {2 b B d^3 \log ^2(a+b x)}{g^4 i^2 (b c-a d)^5}+\frac {2 b B d^3 \log ^2(c+d x)}{g^4 i^2 (b c-a d)^5}-\frac {10 b B d^3 \log (a+b x)}{3 g^4 i^2 (b c-a d)^5}-\frac {4 b B d^3 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g^4 i^2 (b c-a d)^5}+\frac {10 b B d^3 \log (c+d x)}{3 g^4 i^2 (b c-a d)^5}-\frac {4 b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g^4 i^2 (b c-a d)^5}+\frac {2 b B d}{3 g^4 i^2 (a+b x)^2 (b c-a d)^3}-\frac {b B}{9 g^4 i^2 (a+b x)^3 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
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 )}{(46 c+46 d x)^2 (a g+b g x)^4} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^2 g^4 (a+b x)^4}-\frac {b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1058 (b c-a d)^3 g^4 (a+b x)^3}+\frac {3 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)^2}-\frac {b^2 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)^2}+\frac {b d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b^2 d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{529 (b c-a d)^5 g^4}+\frac {\left (b d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{529 (b c-a d)^5 g^4}+\frac {\left (3 b^2 d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{2116 (b c-a d)^4 g^4}+\frac {d^4 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{2116 (b c-a d)^4 g^4}-\frac {\left (b^2 d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{1058 (b c-a d)^3 g^4}+\frac {b^2 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{2116 (b c-a d)^2 g^4}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^3\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}{529 (b c-a d)^5 g^4}-\frac {\left (b B d^3\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}{529 (b c-a d)^5 g^4}+\frac {\left (3 b B d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2116 (b c-a d)^4 g^4}+\frac {\left (B d^3\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{2116 (b c-a d)^4 g^4}-\frac {(b B d) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2116 (b c-a d)^3 g^4}+\frac {(b B) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{6348 (b c-a d)^2 g^4}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {\left (3 b B d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{2116 (b c-a d)^3 g^4}+\frac {\left (B d^3\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{2116 (b c-a d)^3 g^4}-\frac {(b B d) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2116 (b c-a d)^2 g^4}+\frac {(b B) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{6348 (b c-a d) g^4}+\frac {\left (b B d^3\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}{529 (b c-a d)^5 e g^4}-\frac {\left (b B d^3\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}{529 (b c-a d)^5 e g^4}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {\left (3 b B 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}{2116 (b c-a d)^3 g^4}+\frac {\left (B d^3\right ) \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}{2116 (b c-a d)^3 g^4}-\frac {(b B d) \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}{2116 (b c-a d)^2 g^4}+\frac {(b B) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{6348 (b c-a d) g^4}+\frac {\left (b B d^3\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}{529 (b c-a d)^5 e g^4}-\frac {\left (b B d^3\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}{529 (b c-a d)^5 e g^4}\\ &=-\frac {b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac {13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac {B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {\left (b^2 B d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{529 (b c-a d)^5 g^4}-\frac {\left (b^2 B d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{529 (b c-a d)^5 g^4}-\frac {\left (b B d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{529 (b c-a d)^5 g^4}\\ &=-\frac {b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac {13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac {B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}-\frac {b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}-\frac {b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{529 (b c-a d)^5 g^4}+\frac {\left (b^2 B d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{529 (b c-a d)^5 g^4}\\ &=-\frac {b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac {13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac {B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}+\frac {b B d^3 \log ^2(a+b x)}{1058 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}-\frac {b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {b B d^3 \log ^2(c+d x)}{1058 (b c-a d)^5 g^4}-\frac {b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{529 (b c-a d)^5 g^4}\\ &=-\frac {b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac {13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac {B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}+\frac {b B d^3 \log ^2(a+b x)}{1058 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}-\frac {b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {b B d^3 \log ^2(c+d x)}{1058 (b c-a d)^5 g^4}-\frac {b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}-\frac {b B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}-\frac {b B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}\\ \end {align*}
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Mathematica [C] time = 1.34, size = 520, normalized size = 1.14 \[ -\frac {36 b d^3 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {9 d^3 (a d-b c) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-36 b d^3 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {27 b d^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {9 b d (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}+\frac {3 b (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^3}+\frac {27 b^2 B c d^2}{a+b x}-18 b B d^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+18 b B d^3 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+\frac {12 b B d^2 (b c-a d)}{a+b x}-\frac {6 b B d (b c-a d)^2}{(a+b x)^2}+\frac {b B (b c-a d)^3}{(a+b x)^3}-\frac {27 a b B d^3}{a+b x}+30 b B d^3 \log (a+b x)+\frac {9 a B d^4}{c+d x}-\frac {9 b B c d^3}{c+d x}-30 b B d^3 \log (c+d x)}{9 g^4 i^2 (b c-a d)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 1019, normalized size = 2.23 \[ -\frac {{\left (3 \, A + B\right )} b^{4} c^{4} - 9 \, {\left (2 \, A + B\right )} a b^{3} c^{3} d + 54 \, {\left (A + B\right )} a^{2} b^{2} c^{2} d^{2} - 5 \, {\left (6 \, A + 11 \, B\right )} a^{3} b c d^{3} - 9 \, {\left (A - B\right )} a^{4} d^{4} + 6 \, {\left ({\left (6 \, A + 5 \, B\right )} b^{4} c d^{3} - {\left (6 \, A + 5 \, B\right )} a b^{3} d^{4}\right )} x^{3} + 3 \, {\left ({\left (6 \, A + 11 \, B\right )} b^{4} c^{2} d^{2} + 8 \, {\left (3 \, A + B\right )} a b^{3} c d^{3} - {\left (30 \, A + 19 \, B\right )} a^{2} b^{2} d^{4}\right )} x^{2} + 18 \, {\left (B b^{4} d^{4} x^{4} + B a^{3} b c d^{3} + {\left (B b^{4} c d^{3} + 3 \, B a b^{3} d^{4}\right )} x^{3} + 3 \, {\left (B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} x^{2} + {\left (3 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - {\left ({\left (6 \, A + 5 \, B\right )} b^{4} c^{3} d - 27 \, {\left (2 \, A + 3 \, B\right )} a b^{3} c^{2} d^{2} - 3 \, {\left (6 \, A - 19 \, B\right )} a^{2} b^{2} c d^{3} + {\left (66 \, A + 19 \, B\right )} a^{3} b d^{4}\right )} x + 3 \, {\left (2 \, {\left (6 \, A + 5 \, B\right )} b^{4} d^{4} x^{4} + B b^{4} c^{4} - 6 \, B a b^{3} c^{3} d + 18 \, B a^{2} b^{2} c^{2} d^{2} + 12 \, A a^{3} b c d^{3} - 3 \, B a^{4} d^{4} + 2 \, {\left ({\left (6 \, A + 11 \, B\right )} b^{4} c d^{3} + 9 \, {\left (2 \, A + B\right )} a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} + 3 \, {\left (2 \, A + 3 \, B\right )} a b^{3} c d^{3} + 6 \, A a^{2} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (B b^{4} c^{3} d - 9 \, B a b^{3} c^{2} d^{2} - 18 \, {\left (A + B\right )} a^{2} b^{2} c d^{3} - 6 \, {\left (A - B\right )} a^{3} b d^{4}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{9 \, {\left ({\left (b^{8} c^{5} d - 5 \, a b^{7} c^{4} d^{2} + 10 \, a^{2} b^{6} c^{3} d^{3} - 10 \, a^{3} b^{5} c^{2} d^{4} + 5 \, a^{4} b^{4} c d^{5} - a^{5} b^{3} d^{6}\right )} g^{4} i^{2} x^{4} + {\left (b^{8} c^{6} - 2 \, a b^{7} c^{5} d - 5 \, a^{2} b^{6} c^{4} d^{2} + 20 \, a^{3} b^{5} c^{3} d^{3} - 25 \, a^{4} b^{4} c^{2} d^{4} + 14 \, a^{5} b^{3} c d^{5} - 3 \, a^{6} b^{2} d^{6}\right )} g^{4} i^{2} x^{3} + 3 \, {\left (a b^{7} c^{6} - 4 \, a^{2} b^{6} c^{5} d + 5 \, a^{3} b^{5} c^{4} d^{2} - 5 \, a^{5} b^{3} c^{2} d^{4} + 4 \, a^{6} b^{2} c d^{5} - a^{7} b d^{6}\right )} g^{4} i^{2} x^{2} + {\left (3 \, a^{2} b^{6} c^{6} - 14 \, a^{3} b^{5} c^{5} d + 25 \, a^{4} b^{4} c^{4} d^{2} - 20 \, a^{5} b^{3} c^{3} d^{3} + 5 \, a^{6} b^{2} c^{2} d^{4} + 2 \, a^{7} b c d^{5} - a^{8} d^{6}\right )} g^{4} i^{2} x + {\left (a^{3} b^{5} c^{6} - 5 \, a^{4} b^{4} c^{5} d + 10 \, a^{5} b^{3} c^{4} d^{2} - 10 \, a^{6} b^{2} c^{3} d^{3} + 5 \, a^{7} b c^{2} d^{4} - a^{8} c d^{5}\right )} g^{4} i^{2}\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 = 0.05, size = 2068, normalized size = 4.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.45, size = 2560, normalized size = 5.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.44, size = 1679, normalized size = 3.67 \[ \text {result too large to display} \]
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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