Optimal. Leaf size=498 \[ -\frac{2 b^2 B d (c+d x)^3 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^4}+\frac{b^3 B (c+d x)^4 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^4}+\frac{B d^4 \log \left (\frac{c+d x}{a+b x}\right ) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{2 b g^5 (b c-a d)^4}-\frac{2 B d^3 (c+d x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{g^5 (a+b x) (b c-a d)^4}+\frac{3 b B d^2 (c+d x)^2 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^4}-\frac{\left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )^2}{4 b g^5 (a+b x)^4}+\frac{2 b^2 B^2 d (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^4}-\frac{b^3 B^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^4}+\frac{2 B^2 d^3 (c+d x)}{g^5 (a+b x) (b c-a d)^4}-\frac{3 b B^2 d^2 (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^4}-\frac{B^2 d^4 \log ^2\left (\frac{c+d x}{a+b x}\right )}{4 b g^5 (b c-a d)^4} \]
[Out]
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Rubi [C] time = 1.26372, 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 2525
Rule 12
Rule 2528
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
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*}
Mathematica [C] time = 1.06083, size = 748, normalized size = 1.5 \[ \frac{\frac{B \left (-72 B d^4 (a+b x)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+72 B d^4 (a+b x)^4 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+72 d^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )+144 d^3 (a+b x)^3 (a d-b c) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )-144 d^4 (a+b x)^4 \log (a+b x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )+144 d^4 (a+b x)^4 \log (c+d x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )+36 (b c-a d)^4 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )+48 d (a+b x) (a d-b c)^3 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )+144 B d^3 (a+b x)^3 (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)-36 B d^2 (a+b x)^2 \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+8 B d (a+b x) \left (6 d^2 (a+b x)^2 (b c-a d)-6 d^3 (a+b x)^3 \log (c+d x)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )-3 B \left (6 d^2 (a+b x)^2 (b c-a d)^2+12 d^3 (a+b x)^3 (a d-b c)+12 d^4 (a+b x)^4 \log (c+d x)+4 d (a+b x) (a d-b c)^3+3 (b c-a d)^4-12 d^4 (a+b x)^4 \log (a+b x)\right )\right )}{(b c-a d)^4}-72 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )^2}{288 b g^5 (a+b x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 3717, normalized size = 7.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.29015, size = 2865, normalized size = 5.75 \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 = 1.38743, size = 2152, normalized size = 4.32 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (B \log \left (\frac{{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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