Optimal. Leaf size=287 \[ -\frac{b i (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)^2}-\frac{2 b B i (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)^2}+\frac{d i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^4 (a+b x)^2 (b c-a d)^2}+\frac{B d i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 (a+b x)^2 (b c-a d)^2}-\frac{2 b B^2 i (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)^2}+\frac{B^2 d i (c+d x)^2}{4 g^4 (a+b x)^2 (b c-a d)^2} \]
[Out]
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Rubi [C] time = 2.2905, antiderivative size = 741, normalized size of antiderivative = 2.58, number of steps used = 66, 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} \[ \frac{B^2 d^3 i \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{3 b^2 g^4 (b c-a d)^2}+\frac{B^2 d^3 i \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{3 b^2 g^4 (b c-a d)^2}+\frac{B d^3 i \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^4 (b c-a d)^2}-\frac{B d^3 i \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^4 (b c-a d)^2}+\frac{B d^2 i \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^4 (a+b x) (b c-a d)}-\frac{d i \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 b^2 g^4 (a+b x)^2}-\frac{B d i \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{6 b^2 g^4 (a+b x)^2}-\frac{i (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{2 B i (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{9 b^2 g^4 (a+b x)^3}+\frac{5 B^2 d^2 i}{18 b^2 g^4 (a+b x) (b c-a d)}-\frac{B^2 d^3 i \log ^2(a+b x)}{6 b^2 g^4 (b c-a d)^2}-\frac{B^2 d^3 i \log ^2(c+d x)}{6 b^2 g^4 (b c-a d)^2}+\frac{5 B^2 d^3 i \log (a+b x)}{18 b^2 g^4 (b c-a d)^2}-\frac{5 B^2 d^3 i \log (c+d x)}{18 b^2 g^4 (b c-a d)^2}+\frac{B^2 d^3 i \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b^2 g^4 (b c-a d)^2}+\frac{B^2 d^3 i \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^2 g^4 (b c-a d)^2}-\frac{2 B^2 i (b c-a d)}{27 b^2 g^4 (a+b x)^3}+\frac{B^2 d i}{36 b^2 g^4 (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{(62 c+62 d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx &=\int \left (\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g^4 (a+b x)^4}+\frac{62 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g^4 (a+b x)^3}\right ) \, dx\\ &=\frac{(62 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^4}+\frac{(62 (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)^4} \, dx}{b g^4}\\ &=-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}+\frac{(62 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)^3 (c+d x)} \, dx}{b^2 g^4}+\frac{(124 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)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}+\frac{(62 B d (b c-a d)) \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^4}+\frac{\left (124 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)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}+\frac{(62 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)^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^4}+\frac{\left (124 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)^4}-\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)^3}+\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)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^2 g^4}\\ &=-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac{(124 B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{3 b g^4}+\frac{(62 B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^4}-\frac{\left (124 B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{3 b (b c-a d)^2 g^4}+\frac{\left (62 B d^3\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)^2 g^4}+\frac{\left (124 B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{3 b^2 (b c-a d)^2 g^4}-\frac{\left (62 B d^4\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)^2 g^4}+\frac{\left (124 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{3 b (b c-a d) g^4}-\frac{\left (62 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b (b c-a d) g^4}+\frac{(124 B (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{3 b g^4}\\ &=-\frac{124 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac{31 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac{62 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac{62 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac{62 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac{\left (62 B^2 d\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^2 g^4}+\frac{\left (31 B^2 d\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac{\left (124 B^2 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}{3 b^2 (b c-a d)^2 g^4}-\frac{\left (124 B^2 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}{3 b^2 (b c-a d)^2 g^4}-\frac{\left (62 B^2 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}{b^2 (b c-a d)^2 g^4}+\frac{\left (62 B^2 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}{b^2 (b c-a d)^2 g^4}+\frac{\left (124 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^2 (b c-a d) g^4}-\frac{\left (62 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 (b c-a d) g^4}+\frac{\left (124 B^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^2 g^4}\\ &=-\frac{124 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac{31 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac{62 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac{62 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac{62 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac{\left (124 B^2 d^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{3 b^2 g^4}-\frac{\left (62 B^2 d^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^4}-\frac{\left (62 B^2 d (b c-a d)\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{3 b^2 g^4}+\frac{\left (31 B^2 d (b c-a d)\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac{\left (124 B^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{9 b^2 g^4}+\frac{\left (124 B^2 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}{3 b^2 (b c-a d)^2 e g^4}-\frac{\left (124 B^2 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}{3 b^2 (b c-a d)^2 e g^4}-\frac{\left (62 B^2 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}{b^2 (b c-a d)^2 e g^4}+\frac{\left (62 B^2 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}{b^2 (b c-a d)^2 e g^4}\\ &=-\frac{124 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac{31 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac{62 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac{62 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac{62 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac{\left (124 B^2 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}{3 b^2 g^4}-\frac{\left (62 B^2 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}{b^2 g^4}-\frac{\left (62 B^2 d (b c-a d)\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{3 b^2 g^4}+\frac{\left (31 B^2 d (b c-a d)\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^4}+\frac{\left (124 B^2 (b c-a d)^2\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}{9 b^2 g^4}+\frac{\left (124 B^2 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}{3 b^2 (b c-a d)^2 e g^4}-\frac{\left (124 B^2 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}{3 b^2 (b c-a d)^2 e g^4}-\frac{\left (62 B^2 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}{b^2 (b c-a d)^2 e g^4}+\frac{\left (62 B^2 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}{b^2 (b c-a d)^2 e g^4}\\ &=-\frac{124 B^2 (b c-a d)}{27 b^2 g^4 (a+b x)^3}+\frac{31 B^2 d}{18 b^2 g^4 (a+b x)^2}+\frac{155 B^2 d^2}{9 b^2 (b c-a d) g^4 (a+b x)}+\frac{155 B^2 d^3 \log (a+b x)}{9 b^2 (b c-a d)^2 g^4}-\frac{124 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac{31 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac{62 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac{62 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac{155 B^2 d^3 \log (c+d x)}{9 b^2 (b c-a d)^2 g^4}-\frac{62 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac{\left (124 B^2 d^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3 b (b c-a d)^2 g^4}-\frac{\left (124 B^2 d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 b (b c-a d)^2 g^4}-\frac{\left (62 B^2 d^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b (b c-a d)^2 g^4}+\frac{\left (62 B^2 d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b (b c-a d)^2 g^4}-\frac{\left (124 B^2 d^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 b^2 (b c-a d)^2 g^4}+\frac{\left (124 B^2 d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 b^2 (b c-a d)^2 g^4}+\frac{\left (62 B^2 d^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d)^2 g^4}-\frac{\left (62 B^2 d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d)^2 g^4}\\ &=-\frac{124 B^2 (b c-a d)}{27 b^2 g^4 (a+b x)^3}+\frac{31 B^2 d}{18 b^2 g^4 (a+b x)^2}+\frac{155 B^2 d^2}{9 b^2 (b c-a d) g^4 (a+b x)}+\frac{155 B^2 d^3 \log (a+b x)}{9 b^2 (b c-a d)^2 g^4}-\frac{124 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac{31 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac{62 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac{62 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac{155 B^2 d^3 \log (c+d x)}{9 b^2 (b c-a d)^2 g^4}+\frac{62 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac{62 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac{62 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^2 (b c-a d)^2 g^4}+\frac{\left (124 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 b^2 (b c-a d)^2 g^4}+\frac{\left (124 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 b^2 (b c-a d)^2 g^4}-\frac{\left (62 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d)^2 g^4}-\frac{\left (62 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d)^2 g^4}+\frac{\left (124 B^2 d^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b (b c-a d)^2 g^4}-\frac{\left (62 B^2 d^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d)^2 g^4}+\frac{\left (124 B^2 d^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^2 (b c-a d)^2 g^4}-\frac{\left (62 B^2 d^4\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)^2 g^4}\\ &=-\frac{124 B^2 (b c-a d)}{27 b^2 g^4 (a+b x)^3}+\frac{31 B^2 d}{18 b^2 g^4 (a+b x)^2}+\frac{155 B^2 d^2}{9 b^2 (b c-a d) g^4 (a+b x)}+\frac{155 B^2 d^3 \log (a+b x)}{9 b^2 (b c-a d)^2 g^4}-\frac{31 B^2 d^3 \log ^2(a+b x)}{3 b^2 (b c-a d)^2 g^4}-\frac{124 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac{31 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac{62 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac{62 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac{155 B^2 d^3 \log (c+d x)}{9 b^2 (b c-a d)^2 g^4}+\frac{62 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac{62 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac{31 B^2 d^3 \log ^2(c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac{62 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^2 (b c-a d)^2 g^4}+\frac{\left (124 B^2 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 )}{3 b^2 (b c-a d)^2 g^4}+\frac{\left (124 B^2 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 )}{3 b^2 (b c-a d)^2 g^4}-\frac{\left (62 B^2 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 )}{b^2 (b c-a d)^2 g^4}-\frac{\left (62 B^2 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 )}{b^2 (b c-a d)^2 g^4}\\ &=-\frac{124 B^2 (b c-a d)}{27 b^2 g^4 (a+b x)^3}+\frac{31 B^2 d}{18 b^2 g^4 (a+b x)^2}+\frac{155 B^2 d^2}{9 b^2 (b c-a d) g^4 (a+b x)}+\frac{155 B^2 d^3 \log (a+b x)}{9 b^2 (b c-a d)^2 g^4}-\frac{31 B^2 d^3 \log ^2(a+b x)}{3 b^2 (b c-a d)^2 g^4}-\frac{124 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac{31 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac{62 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac{62 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac{62 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac{31 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac{155 B^2 d^3 \log (c+d x)}{9 b^2 (b c-a d)^2 g^4}+\frac{62 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac{62 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac{31 B^2 d^3 \log ^2(c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac{62 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^2 (b c-a d)^2 g^4}+\frac{62 B^2 d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b^2 (b c-a d)^2 g^4}+\frac{62 B^2 d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^2 (b c-a d)^2 g^4}\\ \end{align*}
Mathematica [C] time = 1.10025, size = 1035, normalized size = 3.61 \[ -\frac{i \left (36 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^3+54 d (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^2+27 B d (a+b x) \left (2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2+4 d (a d-b c) (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\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{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )+2 B \left (12 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^3-18 d (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2+36 d^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)+36 d^3 (a+b x)^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-36 d^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+36 B d^2 (a+b x)^2 (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-9 B d (a+b x) \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B \left (2 (b c-a d)^3-3 d (a+b x) (b c-a d)^2+6 d^2 (a+b x)^2 (b c-a d)+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )-18 B d^3 (a+b x)^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{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+18 B d^3 (a+b x)^3 \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )\right )}{108 b^2 (b c-a d)^2 g^4 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 1765, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.51904, size = 4431, normalized size = 15.44 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.560472, size = 1249, normalized size = 4.35 \begin{align*} \frac{6 \,{\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} c d^{2} -{\left (6 \, A B + 5 \, B^{2}\right )} a b^{2} d^{3}\right )} i x^{2} - 3 \,{\left ({\left (18 \, A^{2} + 6 \, A B - B^{2}\right )} b^{3} c^{2} d - 18 \,{\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b^{2} c d^{2} +{\left (18 \, A^{2} + 30 \, A B + 19 \, B^{2}\right )} a^{2} b d^{3}\right )} i x + 18 \,{\left (B^{2} b^{3} d^{3} i x^{3} + 3 \, B^{2} a b^{2} d^{3} i x^{2} - 3 \,{\left (B^{2} b^{3} c^{2} d - 2 \, B^{2} a b^{2} c d^{2}\right )} i x -{\left (2 \, B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d\right )} i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} -{\left (4 \,{\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - 27 \,{\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b^{2} c^{2} d +{\left (18 \, A^{2} + 30 \, A B + 19 \, B^{2}\right )} a^{3} d^{3}\right )} i + 6 \,{\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} d^{3} i x^{3} + 3 \,{\left (2 \, B^{2} b^{3} c d^{2} + 3 \,{\left (2 \, A B + B^{2}\right )} a b^{2} d^{3}\right )} i x^{2} - 3 \,{\left ({\left (6 \, A B + B^{2}\right )} b^{3} c^{2} d - 6 \,{\left (2 \, A B + B^{2}\right )} a b^{2} c d^{2}\right )} i x -{\left (4 \,{\left (3 \, A B + B^{2}\right )} b^{3} c^{3} - 9 \,{\left (2 \, A B + B^{2}\right )} a b^{2} c^{2} d\right )} i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{108 \,{\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \,{\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x +{\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 27.9882, size = 1386, normalized size = 4.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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