Optimal. Leaf size=141 \[ -\frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B^2 i (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 1.94, antiderivative size = 639, normalized size of antiderivative = 4.53, number of steps used = 58, 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} \[ -\frac {B^2 d^2 i \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac {B^2 d^2 i \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac {B d^2 i \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (b c-a d)}+\frac {B d^2 i \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (b c-a d)}-\frac {B d i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (a+b x)}-\frac {B i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^3 (a+b x)^2}-\frac {d i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^3 (a+b x)}-\frac {i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b^2 g^3 (a+b x)^2}+\frac {B^2 d^2 i \log ^2(a+b x)}{2 b^2 g^3 (b c-a d)}+\frac {B^2 d^2 i \log ^2(c+d x)}{2 b^2 g^3 (b c-a d)}-\frac {B^2 d^2 i \log (a+b x)}{2 b^2 g^3 (b c-a d)}+\frac {B^2 d^2 i \log (c+d x)}{2 b^2 g^3 (b c-a d)}-\frac {B^2 d^2 i \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac {B^2 d^2 i \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac {B^2 i (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {B^2 d i}{2 b^2 g^3 (a+b x)} \]
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 {(61 c+61 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx &=\int \left (\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^3 (a+b x)^3}+\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^3 (a+b x)^2}\right ) \, dx\\ &=\frac {(61 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^3}+\frac {(61 (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)^3} \, dx}{b g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(122 B 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^3}+\frac {(61 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)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(122 B d (b c-a d)) \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^3}+\frac {\left (61 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)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(122 B d (b c-a d)) \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^3}+\frac {\left (61 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)^3}-\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)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}-\frac {(61 B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {(122 B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {\left (61 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (122 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (61 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (122 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {(61 B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (122 B^2 d\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}-\frac {\left (61 B^2 d^2\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 (b c-a d) g^3}+\frac {\left (61 B^2 d^2\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 (b c-a d) g^3}+\frac {\left (122 B^2 d^2\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 (b c-a d) g^3}-\frac {\left (122 B^2 d^2\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 (b c-a d) g^3}+\frac {\left (61 B^2 (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (122 B^2 d (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (61 B^2 (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}-\frac {\left (61 B^2 d^2\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 (b c-a d) e g^3}+\frac {\left (61 B^2 d^2\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 (b c-a d) e g^3}+\frac {\left (122 B^2 d^2\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 (b c-a d) e g^3}-\frac {\left (122 B^2 d^2\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 (b c-a d) e g^3}\\ &=-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d (b c-a 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^2 g^3}+\frac {\left (122 B^2 d (b c-a 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^2 g^3}+\frac {\left (61 B^2 (b c-a d)^2\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}{2 b^2 g^3}-\frac {\left (61 B^2 d^2\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 (b c-a d) e g^3}+\frac {\left (61 B^2 d^2\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 (b c-a d) e g^3}+\frac {\left (122 B^2 d^2\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 (b c-a d) e g^3}-\frac {\left (122 B^2 d^2\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 (b c-a d) e g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (61 B^2 d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (122 B^2 d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (61 B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac {\left (122 B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (61 B^2 d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 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^2 (b c-a d) g^3}-\frac {\left (61 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^2 (b c-a d) g^3}+\frac {\left (122 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^2 (b c-a d) g^3}+\frac {\left (122 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^2 (b c-a d) g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.91, size = 765, normalized size = 5.43 \[ -\frac {i \left (B \left (-4 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+4 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+4 d (a+b x) (a d-b c) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-2 B d^2 (a+b x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+B \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )-4 B d (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )+4 B d (a+b x) \left (2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 d (a+b x) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 d (a+b x) \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+B d (a+b x) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 B (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )+2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2-4 d (a+b x) (a d-b c) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2\right )}{4 b^2 g^3 (a+b x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.85, size = 289, normalized size = 2.05 \[ -\frac {2 \, {\left ({\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b d^{2}\right )} i x + 2 \, {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + {\left ({\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a^{2} d^{2}\right )} i + 2 \, {\left ({\left (2 \, A B + B^{2}\right )} b^{2} d^{2} i x^{2} + 2 \, {\left (2 \, A B + B^{2}\right )} b^{2} c d i x + {\left (2 \, A B + B^{2}\right )} b^{2} c^{2} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.29, size = 185, normalized size = 1.31 \[ -\frac {{\left (2 \, B^{2} i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 4 \, A B i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) + 2 \, B^{2} i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) + 2 \, A^{2} i e^{3} + 2 \, A B i e^{3} + B^{2} i e^{3}\right )} {\left (d x + c\right )}^{2} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{4 \, {\left (b x e + a e\right )}^{2} g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 865, normalized size = 6.13 \[ \frac {B^{2} a d \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}-\frac {B^{2} b c \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}+\frac {A B a d \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}-\frac {A B b c \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}+\frac {B^{2} a d \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}-\frac {B^{2} b c \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}+\frac {A^{2} a d \,e^{2} i}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}-\frac {A^{2} b c \,e^{2} i}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}+\frac {A B a d \,e^{2} i}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}-\frac {A B b c \,e^{2} i}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}+\frac {B^{2} a d \,e^{2} i}{4 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}-\frac {B^{2} b c \,e^{2} i}{4 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 2.39, size = 1987, normalized size = 14.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.18, size = 469, normalized size = 3.33 \[ -\frac {x\,\left (2\,b\,d\,i\,A^2+2\,b\,d\,i\,A\,B+b\,d\,i\,B^2\right )+A^2\,a\,d\,i+A^2\,b\,c\,i+\frac {B^2\,a\,d\,i}{2}+\frac {B^2\,b\,c\,i}{2}+A\,B\,a\,d\,i+A\,B\,b\,c\,i}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,c\,i}{2\,b^2\,g^3}+\frac {B^2\,a\,d\,i}{2\,b^3\,g^3}+\frac {B^2\,d\,i\,x}{b^2\,g^3}}{2\,a\,x+b\,x^2+\frac {a^2}{b}}-\frac {B^2\,d^2\,i}{2\,b^2\,g^3\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x\,\left (\frac {B^2\,i}{b^2\,g^3}+\frac {2\,A\,B\,i}{b^2\,g^3}\right )+\frac {A\,B\,a\,i}{b^3\,g^3}+\frac {B\,i\,\left (A\,b\,c-B\,a\,d+B\,b\,c\right )}{b^3\,d\,g^3}+\frac {B^2\,d^2\,i\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )}{b^2\,g^3\,\left (a\,d-b\,c\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B\,d^2\,i\,\mathrm {atan}\left (\frac {\left (\frac {2\,c\,b^3\,g^3+2\,a\,d\,b^2\,g^3}{2\,b^2\,g^3}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (2\,A+B\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 14.31, size = 714, normalized size = 5.06 \[ - \frac {B d^{2} i \left (2 A + B\right ) \log {\left (x + \frac {2 A B a d^{3} i + 2 A B b c d^{2} i + B^{2} a d^{3} i + B^{2} b c d^{2} i - \frac {B a^{2} d^{4} i \left (2 A + B\right )}{a d - b c} + \frac {2 B a b c d^{3} i \left (2 A + B\right )}{a d - b c} - \frac {B b^{2} c^{2} d^{2} i \left (2 A + B\right )}{a d - b c}}{4 A B b d^{3} i + 2 B^{2} b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {B d^{2} i \left (2 A + B\right ) \log {\left (x + \frac {2 A B a d^{3} i + 2 A B b c d^{2} i + B^{2} a d^{3} i + B^{2} b c d^{2} i + \frac {B a^{2} d^{4} i \left (2 A + B\right )}{a d - b c} - \frac {2 B a b c d^{3} i \left (2 A + B\right )}{a d - b c} + \frac {B b^{2} c^{2} d^{2} i \left (2 A + B\right )}{a d - b c}}{4 A B b d^{3} i + 2 B^{2} b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {\left (B^{2} c^{2} i + 2 B^{2} c d i x + B^{2} d^{2} i x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{3} d g^{3} - 2 a^{2} b c g^{3} + 4 a^{2} b d g^{3} x - 4 a b^{2} c g^{3} x + 2 a b^{2} d g^{3} x^{2} - 2 b^{3} c g^{3} x^{2}} + \frac {- 2 A^{2} a d i - 2 A^{2} b c i - 2 A B a d i - 2 A B b c i - B^{2} a d i - B^{2} b c i + x \left (- 4 A^{2} b d i - 4 A B b d i - 2 B^{2} b d i\right )}{4 a^{2} b^{2} g^{3} + 8 a b^{3} g^{3} x + 4 b^{4} g^{3} x^{2}} + \frac {\left (- 2 A B a d i - 2 A B b c i - 4 A B b d i x - B^{2} a d i - B^{2} b c i - 2 B^{2} b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} b^{2} g^{3} + 4 a b^{3} g^{3} x + 2 b^{4} g^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________