Optimal. Leaf size=483 \[ \frac{6 b^2 d^2 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (b c-a d)^5}-\frac{b^4 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac{4 b^3 d (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (a+b x) (b c-a d)^5}-\frac{4 b d^3 (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (c+d x) (b c-a d)^5}+\frac{d^4 (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (c+d x)^2 (b c-a d)^5}-\frac{3 b^2 B d^2 n \log ^2\left (\frac{a+b x}{c+d x}\right )}{g^3 i^3 (b c-a d)^5}-\frac{b^4 B n (c+d x)^2}{4 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac{4 b^3 B d n (c+d x)}{g^3 i^3 (a+b x) (b c-a d)^5}+\frac{4 b B d^3 n (a+b x)}{g^3 i^3 (c+d x) (b c-a d)^5}-\frac{B d^4 n (a+b x)^2}{4 g^3 i^3 (c+d x)^2 (b c-a d)^5} \]
[Out]
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Rubi [C] time = 1.38898, antiderivative size = 701, normalized size of antiderivative = 1.45, number of steps used = 34, number of rules used = 11, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{6 b^2 B d^2 n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac{6 b^2 B d^2 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac{6 b^2 d^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (b c-a d)^5}-\frac{6 b^2 d^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (b c-a d)^5}+\frac{3 b^2 d \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (a+b x) (b c-a d)^4}-\frac{b^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (a+b x)^2 (b c-a d)^3}+\frac{3 b d^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (c+d x) (b c-a d)^4}+\frac{d^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (c+d x)^2 (b c-a d)^3}-\frac{3 b^2 B d^2 n \log ^2(a+b x)}{g^3 i^3 (b c-a d)^5}-\frac{3 b^2 B d^2 n \log ^2(c+d x)}{g^3 i^3 (b c-a d)^5}+\frac{6 b^2 B d^2 n \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac{6 b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac{7 b^2 B d n}{2 g^3 i^3 (a+b x) (b c-a d)^4}-\frac{b^2 B n}{4 g^3 i^3 (a+b x)^2 (b c-a d)^3}-\frac{7 b B d^2 n}{2 g^3 i^3 (c+d x) (b c-a d)^4}-\frac{B d^2 n}{4 g^3 i^3 (c+d x)^2 (b c-a d)^3} \]
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{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(157 c+157 d x)^3 (a g+b g x)^3} \, dx &=\int \left (\frac{b^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^3 g^3 (a+b x)^3}-\frac{3 b^3 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)^2}+\frac{6 b^3 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^3 g^3 (c+d x)^3}-\frac{3 b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)^2}-\frac{6 b^2 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3 (c+d x)}\right ) \, dx\\ &=\frac{\left (6 b^3 d^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^2 d^3\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac{\left (3 b^3 d\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{3869893 (b c-a d)^4 g^3}-\frac{\left (3 b d^3\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{3869893 (b c-a d)^4 g^3}+\frac{b^3 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3869893 (b c-a d)^3 g^3}-\frac{d^3 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3869893 (b c-a d)^3 g^3}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2 n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac{\left (6 b^2 B d^2 n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d n\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3869893 (b c-a d)^4 g^3}-\frac{\left (3 b B d^2 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{3869893 (b c-a d)^4 g^3}+\frac{\left (b^2 B n\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{7739786 (b c-a d)^3 g^3}-\frac{\left (B d^2 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{7739786 (b c-a d)^3 g^3}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2 n\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{3869893 (b c-a d)^5 g^3}+\frac{\left (6 b^2 B d^2 n\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{3869893 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{3869893 (b c-a d)^3 g^3}-\frac{\left (3 b B d^2 n\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{3869893 (b c-a d)^3 g^3}+\frac{\left (b^2 B n\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{7739786 (b c-a d)^2 g^3}-\frac{\left (B d^2 n\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{7739786 (b c-a d)^2 g^3}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^3 B d^2 n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac{\left (6 b^3 B d^2 n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac{\left (6 b^2 B d^3 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^3 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d n\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}{3869893 (b c-a d)^3 g^3}-\frac{\left (3 b B d^2 n\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3869893 (b c-a d)^3 g^3}+\frac{\left (b^2 B n\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}{7739786 (b c-a d)^2 g^3}-\frac{\left (B d^2 n\right ) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7739786 (b c-a d)^2 g^3}\\ &=-\frac{b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^3 B d^2 n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^3 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac{3 b^2 B d^2 n \log ^2(a+b x)}{3869893 (b c-a d)^5 g^3}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac{3 b^2 B d^2 n \log ^2(c+d x)}{3869893 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3869893 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3869893 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac{3 b^2 B d^2 n \log ^2(a+b x)}{3869893 (b c-a d)^5 g^3}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac{3 b^2 B d^2 n \log ^2(c+d x)}{3869893 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}\\ \end{align*}
Mathematica [C] time = 1.35937, size = 561, normalized size = 1.16 \[ -\frac{12 b^2 B d^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-12 b^2 B d^2 n \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-24 b^2 d^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+24 b^2 d^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{12 b^2 d (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+\frac{2 b^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}-\frac{12 b d^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-\frac{2 d^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(c+d x)^2}-\frac{12 b^3 B c d n}{a+b x}-\frac{2 b^2 B d n (b c-a d)}{a+b x}+\frac{b^2 B n (b c-a d)^2}{(a+b x)^2}+\frac{12 a b^2 B d^2 n}{a+b x}-\frac{12 a b B d^3 n}{c+d x}+\frac{2 b B d^2 n (b c-a d)}{c+d x}+\frac{B d^2 n (b c-a d)^2}{(c+d x)^2}+\frac{12 b^2 B c d^2 n}{c+d x}}{4 g^3 i^3 (b c-a d)^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.745, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{3} \left ( dix+ci \right ) ^{3}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.90219, size = 3217, normalized size = 6.66 \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 = 0.684744, size = 2859, normalized size = 5.92 \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{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}{{\left (b g x + a g\right )}^{3}{\left (d i x + c i\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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