Optimal. Leaf size=498 \[ \frac {b^3 B (c+d x)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^4}-\frac {2 b^2 B d (c+d x)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^4}+\frac {B d^4 \log \left (\frac {c+d x}{a+b x}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{2 b g^5 (b c-a d)^4}-\frac {2 B d^3 (c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{g^5 (a+b x) (b c-a d)^4}+\frac {3 b B d^2 (c+d x)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^4}-\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{4 b g^5 (a+b x)^4}-\frac {b^3 B^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^4}+\frac {2 b^2 B^2 d (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^4}-\frac {B^2 d^4 \log ^2\left (\frac {c+d x}{a+b x}\right )}{4 b g^5 (b c-a d)^4}+\frac {2 B^2 d^3 (c+d x)}{g^5 (a+b x) (b c-a d)^4}-\frac {3 b B^2 d^2 (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^4} \]
[Out]
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Rubi [C] time = 1.26, antiderivative size = 763, normalized size of antiderivative = 1.53, number of steps used = 38, number of rules used = 11, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B^2 d^4 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{2 b g^5 (b c-a d)^4}+\frac {B^2 d^4 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{2 b g^5 (b c-a d)^4}-\frac {B d^4 \log (a+b x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{2 b g^5 (b c-a d)^4}+\frac {B d^4 \log (c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{2 b g^5 (b c-a d)^4}-\frac {B d^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{2 b g^5 (a+b x) (b c-a d)^3}+\frac {B d^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{4 b g^5 (a+b x)^2 (b c-a d)^2}-\frac {B d \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{6 b g^5 (a+b x)^3 (b c-a d)}-\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{4 b g^5 (a+b x)^4}+\frac {B \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{8 b g^5 (a+b x)^4}+\frac {25 B^2 d^3}{24 b g^5 (a+b x) (b c-a d)^3}-\frac {13 B^2 d^2}{48 b g^5 (a+b x)^2 (b c-a d)^2}-\frac {B^2 d^4 \log ^2(a+b x)}{4 b g^5 (b c-a d)^4}-\frac {B^2 d^4 \log ^2(c+d x)}{4 b g^5 (b c-a d)^4}+\frac {25 B^2 d^4 \log (a+b x)}{24 b g^5 (b c-a d)^4}+\frac {B^2 d^4 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{2 b g^5 (b c-a d)^4}-\frac {25 B^2 d^4 \log (c+d x)}{24 b g^5 (b c-a d)^4}+\frac {B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b g^5 (b c-a d)^4}+\frac {7 B^2 d}{72 b g^5 (a+b x)^3 (b c-a d)}-\frac {B^2}{32 b g^5 (a+b x)^4} \]
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 {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^5} \, dx &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {B \int \frac {(b c-a d) \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{g^4 (a+b x)^5 (c+d x)} \, dx}{2 b g}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {(B (b c-a d)) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b g^5}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {(B (b c-a d)) \int \left (\frac {b \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d) (a+b x)^5}-\frac {b d \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^5 (a+b x)}-\frac {d^5 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b g^5}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {B \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^5} \, dx}{2 g^5}+\frac {\left (B d^4\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{a+b x} \, dx}{2 (b c-a d)^4 g^5}-\frac {\left (B d^5\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}-\frac {\left (B d^3\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3 g^5}+\frac {\left (B d^2\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^3} \, dx}{2 (b c-a d)^2 g^5}-\frac {(B d) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^4} \, dx}{2 (b c-a d) g^5}\\ &=\frac {B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {B^2 \int \frac {-b c+a d}{(a+b x)^5 (c+d x)} \, dx}{8 b g^5}+\frac {\left (B^2 d^4\right ) \int \frac {(a+b x) \left (\frac {d e}{a+b x}-\frac {b e (c+d x)}{(a+b x)^2}\right ) \log (a+b x)}{e (c+d x)} \, dx}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4\right ) \int \frac {(a+b x) \left (\frac {d e}{a+b x}-\frac {b e (c+d x)}{(a+b x)^2}\right ) \log (c+d x)}{e (c+d x)} \, dx}{2 b (b c-a d)^4 g^5}+\frac {\left (B^2 d^3\right ) \int \frac {-b c+a d}{(a+b x)^2 (c+d x)} \, dx}{2 b (b c-a d)^3 g^5}-\frac {\left (B^2 d^2\right ) \int \frac {-b c+a d}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d)^2 g^5}+\frac {\left (B^2 d\right ) \int \frac {-b c+a d}{(a+b x)^4 (c+d x)} \, dx}{6 b (b c-a d) g^5}\\ &=\frac {B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {\left (B^2 d\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{6 b g^5}-\frac {\left (B^2 d^3\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{2 b (b c-a d)^2 g^5}+\frac {\left (B^2 d^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d) g^5}+\frac {\left (B^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{8 b g^5}+\frac {\left (B^2 d^4\right ) \int \frac {(a+b x) \left (\frac {d e}{a+b x}-\frac {b e (c+d x)}{(a+b x)^2}\right ) \log (a+b x)}{c+d x} \, dx}{2 b (b c-a d)^4 e g^5}-\frac {\left (B^2 d^4\right ) \int \frac {(a+b x) \left (\frac {d e}{a+b x}-\frac {b e (c+d x)}{(a+b x)^2}\right ) \log (c+d x)}{c+d x} \, dx}{2 b (b c-a d)^4 e g^5}\\ &=\frac {B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {\left (B^2 d\right ) \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}{6 b g^5}-\frac {\left (B^2 d^3\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}{2 b (b c-a d)^2 g^5}+\frac {\left (B^2 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}{4 b (b c-a d) g^5}+\frac {\left (B^2 (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{8 b g^5}+\frac {\left (B^2 d^4\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}{2 b (b c-a d)^4 e g^5}-\frac {\left (B^2 d^4\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}{2 b (b c-a d)^4 e g^5}\\ &=-\frac {B^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {\left (B^2 d^4\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{2 (b c-a d)^4 g^5}+\frac {\left (B^2 d^4\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2 (b c-a d)^4 g^5}+\frac {\left (B^2 d^5\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^5\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}\\ &=-\frac {B^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac {B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {\left (B^2 d^4\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 (b c-a d)^4 g^5}-\frac {\left (B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^5\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}\\ &=-\frac {B^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(a+b x)}{4 b (b c-a d)^4 g^5}-\frac {25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(c+d x)}{4 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac {B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {\left (B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2 b (b c-a d)^4 g^5}\\ &=-\frac {B^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(a+b x)}{4 b (b c-a d)^4 g^5}-\frac {25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(c+d x)}{4 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac {B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {B^2 d^4 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}\\ \end {align*}
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Mathematica [C] time = 0.92, size = 748, normalized size = 1.50 \[ \frac {\frac {B \left (-144 d^4 (a+b x)^4 \log (a+b x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )+144 d^4 (a+b x)^4 \log (c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )+144 d^3 (a+b x)^3 (a d-b c) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )+72 d^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )+36 (b c-a d)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )+48 d (a+b x) (a d-b c)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )-72 B d^4 (a+b x)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+72 B d^4 (a+b x)^4 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+144 B d^3 (a+b x)^3 (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)-36 B d^2 (a+b x)^2 \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+8 B d (a+b x) \left (-6 d^3 (a+b x)^3 \log (c+d x)+6 d^2 (a+b x)^2 (b c-a d)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )-3 B \left (12 d^4 (a+b x)^4 \log (c+d x)+12 d^3 (a+b x)^3 (a d-b c)+6 d^2 (a+b x)^2 (b c-a d)^2+4 d (a+b x) (a d-b c)^3+3 (b c-a d)^4-12 d^4 (a+b x)^4 \log (a+b x)\right )\right )}{(b c-a d)^4}-72 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{288 b g^5 (a+b x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 1045, normalized size = 2.10 \[ -\frac {9 \, {\left (8 \, A^{2} - 4 \, A B + B^{2}\right )} b^{4} c^{4} - 32 \, {\left (9 \, A^{2} - 6 \, A B + 2 \, B^{2}\right )} a b^{3} c^{3} d + 216 \, {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a^{2} b^{2} c^{2} d^{2} - 288 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a^{3} b c d^{3} + {\left (72 \, A^{2} - 300 \, A B + 415 \, B^{2}\right )} a^{4} d^{4} + 12 \, {\left ({\left (12 \, A B - 25 \, B^{2}\right )} b^{4} c d^{3} - {\left (12 \, A B - 25 \, B^{2}\right )} a b^{3} d^{4}\right )} x^{3} - 6 \, {\left ({\left (12 \, A B - 13 \, B^{2}\right )} b^{4} c^{2} d^{2} - 16 \, {\left (6 \, A B - 11 \, B^{2}\right )} a b^{3} c d^{3} + {\left (84 \, A B - 163 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} x^{2} - 72 \, {\left (B^{2} b^{4} d^{4} x^{4} + 4 \, B^{2} a b^{3} d^{4} x^{3} + 6 \, B^{2} a^{2} b^{2} d^{4} x^{2} + 4 \, B^{2} a^{3} b d^{4} x - B^{2} b^{4} c^{4} + 4 \, B^{2} a b^{3} c^{3} d - 6 \, B^{2} a^{2} b^{2} c^{2} d^{2} + 4 \, B^{2} a^{3} b c d^{3}\right )} \log \left (\frac {d e x + c e}{b x + a}\right )^{2} + 4 \, {\left ({\left (12 \, A B - 7 \, B^{2}\right )} b^{4} c^{3} d - 12 \, {\left (6 \, A B - 5 \, B^{2}\right )} a b^{3} c^{2} d^{2} + 108 \, {\left (2 \, A B - 3 \, B^{2}\right )} a^{2} b^{2} c d^{3} - {\left (156 \, A B - 271 \, B^{2}\right )} a^{3} b d^{4}\right )} x - 12 \, {\left ({\left (12 \, A B - 25 \, B^{2}\right )} b^{4} d^{4} x^{4} - 3 \, {\left (4 \, A B - B^{2}\right )} b^{4} c^{4} + 16 \, {\left (3 \, A B - B^{2}\right )} a b^{3} c^{3} d - 36 \, {\left (2 \, A B - B^{2}\right )} a^{2} b^{2} c^{2} d^{2} + 48 \, {\left (A B - B^{2}\right )} a^{3} b c d^{3} - 4 \, {\left (3 \, B^{2} b^{4} c d^{3} - 2 \, {\left (6 \, A B - 11 \, B^{2}\right )} a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (B^{2} b^{4} c^{2} d^{2} - 8 \, B^{2} a b^{3} c d^{3} + 6 \, {\left (2 \, A B - 3 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (B^{2} b^{4} c^{3} d - 6 \, B^{2} a b^{3} c^{2} d^{2} + 18 \, B^{2} a^{2} b^{2} c d^{3} - 12 \, {\left (A B - B^{2}\right )} a^{3} b d^{4}\right )} x\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{288 \, {\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.34, size = 1029, normalized size = 2.07 \[ \frac {{\left (\frac {288 \, {\left (d x e + c e\right )} B^{2} d^{3} e^{3} \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{b x + a} - \frac {432 \, {\left (d x e + c e\right )}^{2} B^{2} b d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{{\left (b x + a\right )}^{2}} + \frac {288 \, {\left (d x e + c e\right )}^{3} B^{2} b^{2} d e \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{{\left (b x + a\right )}^{3}} + \frac {576 \, {\left (d x e + c e\right )} A B d^{3} e^{3} \log \left (\frac {d x e + c e}{b x + a}\right )}{b x + a} - \frac {576 \, {\left (d x e + c e\right )} B^{2} d^{3} e^{3} \log \left (\frac {d x e + c e}{b x + a}\right )}{b x + a} - \frac {864 \, {\left (d x e + c e\right )}^{2} A B b d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{2}} + \frac {432 \, {\left (d x e + c e\right )}^{2} B^{2} b d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{2}} + \frac {576 \, {\left (d x e + c e\right )}^{3} A B b^{2} d e \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{3}} - \frac {192 \, {\left (d x e + c e\right )}^{3} B^{2} b^{2} d e \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{3}} - \frac {72 \, {\left (d x e + c e\right )}^{4} B^{2} b^{3} \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{{\left (b x + a\right )}^{4}} + \frac {288 \, {\left (d x e + c e\right )} A^{2} d^{3} e^{3}}{b x + a} - \frac {576 \, {\left (d x e + c e\right )} A B d^{3} e^{3}}{b x + a} + \frac {576 \, {\left (d x e + c e\right )} B^{2} d^{3} e^{3}}{b x + a} - \frac {432 \, {\left (d x e + c e\right )}^{2} A^{2} b d^{2} e^{2}}{{\left (b x + a\right )}^{2}} + \frac {432 \, {\left (d x e + c e\right )}^{2} A B b d^{2} e^{2}}{{\left (b x + a\right )}^{2}} - \frac {216 \, {\left (d x e + c e\right )}^{2} B^{2} b d^{2} e^{2}}{{\left (b x + a\right )}^{2}} + \frac {288 \, {\left (d x e + c e\right )}^{3} A^{2} b^{2} d e}{{\left (b x + a\right )}^{3}} - \frac {192 \, {\left (d x e + c e\right )}^{3} A B b^{2} d e}{{\left (b x + a\right )}^{3}} + \frac {64 \, {\left (d x e + c e\right )}^{3} B^{2} b^{2} d e}{{\left (b x + a\right )}^{3}} - \frac {144 \, {\left (d x e + c e\right )}^{4} A B b^{3} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{4}} + \frac {36 \, {\left (d x e + c e\right )}^{4} B^{2} b^{3} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{4}} - \frac {72 \, {\left (d x e + c e\right )}^{4} A^{2} b^{3}}{{\left (b x + a\right )}^{4}} + \frac {36 \, {\left (d x e + c e\right )}^{4} A B b^{3}}{{\left (b x + a\right )}^{4}} - \frac {9 \, {\left (d x e + c e\right )}^{4} B^{2} b^{3}}{{\left (b x + a\right )}^{4}}\right )} {\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 )}}{288 \, {\left (b^{3} c^{3} g^{5} e^{3} - 3 \, a b^{2} c^{2} d g^{5} e^{3} + 3 \, a^{2} b c d^{2} g^{5} e^{3} - a^{3} d^{3} g^{5} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 3717, normalized size = 7.46 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.91, size = 2122, normalized size = 4.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.94, size = 1880, normalized size = 3.78 \[ \text {result too large to display} \]
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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