Optimal. Leaf size=429 \[ \frac{2 b^3 B n \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d g^4 (b c-a d)^3}-\frac{2 b^2 B n (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 (c+d x) (b c-a d)^3}-\frac{2 B d^2 n (a+b x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^4 (c+d x)^3 (b c-a d)^3}+\frac{b B d n (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 (c+d x)^2 (b c-a d)^3}-\frac{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d g^4 (c+d x)^3}+\frac{2 b^2 B^2 n^2 (a+b x)}{g^4 (c+d x) (b c-a d)^3}-\frac{b^3 B^2 n^2 \log ^2\left (\frac{a+b x}{c+d x}\right )}{3 d g^4 (b c-a d)^3}+\frac{2 B^2 d^2 n^2 (a+b x)^3}{27 g^4 (c+d x)^3 (b c-a d)^3}-\frac{b B^2 d n^2 (a+b x)^2}{2 g^4 (c+d x)^2 (b c-a d)^3} \]
[Out]
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Rubi [C] time = 1.098, antiderivative size = 736, normalized size of antiderivative = 1.72, number of steps used = 32, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac{2 b^3 B^2 n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{3 d g^4 (b c-a d)^3}+\frac{2 b^3 B^2 n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{3 d g^4 (b c-a d)^3}+\frac{2 b^3 B n \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d g^4 (b c-a d)^3}-\frac{2 b^3 B n \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d g^4 (b c-a d)^3}+\frac{2 b^2 B n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d g^4 (c+d x) (b c-a d)^2}+\frac{b B n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d g^4 (c+d x)^2 (b c-a d)}-\frac{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d g^4 (c+d x)^3}+\frac{2 B n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{9 d g^4 (c+d x)^3}-\frac{11 b^2 B^2 n^2}{9 d g^4 (c+d x) (b c-a d)^2}-\frac{b^3 B^2 n^2 \log ^2(a+b x)}{3 d g^4 (b c-a d)^3}-\frac{b^3 B^2 n^2 \log ^2(c+d x)}{3 d g^4 (b c-a d)^3}-\frac{11 b^3 B^2 n^2 \log (a+b x)}{9 d g^4 (b c-a d)^3}+\frac{11 b^3 B^2 n^2 \log (c+d x)}{9 d g^4 (b c-a d)^3}+\frac{2 b^3 B^2 n^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{3 d g^4 (b c-a d)^3}+\frac{2 b^3 B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 d g^4 (b c-a d)^3}-\frac{5 b B^2 n^2}{18 d g^4 (c+d x)^2 (b c-a d)}-\frac{2 B^2 n^2}{27 d g^4 (c+d x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2525
Rule 12
Rule 2528
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 44
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^4} \, dx &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac{(2 B n) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{g^3 (a+b x) (c+d x)^4} \, dx}{3 d g}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac{(2 B (b c-a d) n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^4} \, dx}{3 d g^4}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac{(2 B (b c-a d) n) \int \left (\frac{b^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^4}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)^3}-\frac{b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)^2}-\frac{b^3 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 d g^4}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}-\frac{(2 B n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^4} \, dx}{3 g^4}-\frac{\left (2 b^3 B n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3 (b c-a d)^3 g^4}+\frac{\left (2 b^4 B n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{3 d (b c-a d)^3 g^4}-\frac{\left (2 b^2 B n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{3 (b c-a d)^2 g^4}-\frac{(2 b B n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3 (b c-a d) g^4}\\ &=\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac{2 b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac{2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}-\frac{2 b^3 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac{\left (2 B^2 n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^4} \, dx}{9 d g^4}-\frac{\left (2 b^3 B^2 n^2\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}{3 d (b c-a d)^3 g^4}+\frac{\left (2 b^3 B^2 n^2\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}{3 d (b c-a d)^3 g^4}-\frac{\left (2 b^2 B^2 n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{3 d (b c-a d)^2 g^4}-\frac{\left (b B^2 n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{3 d (b c-a d) g^4}\\ &=\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac{2 b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac{2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}-\frac{2 b^3 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac{\left (b B^2 n^2\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{3 d g^4}-\frac{\left (2 b^3 B^2 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{3 d (b c-a d)^3 g^4}+\frac{\left (2 b^3 B^2 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{3 d (b c-a d)^3 g^4}-\frac{\left (2 b^2 B^2 n^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{3 d (b c-a d) g^4}-\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x) (c+d x)^4} \, dx}{9 d g^4}\\ &=\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac{2 b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac{2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}-\frac{2 b^3 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac{\left (b B^2 n^2\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}{3 d g^4}+\frac{\left (2 b^3 B^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 (b c-a d)^3 g^4}-\frac{\left (2 b^3 B^2 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 (b c-a d)^3 g^4}-\frac{\left (2 b^4 B^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3 d (b c-a d)^3 g^4}+\frac{\left (2 b^4 B^2 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 d (b c-a d)^3 g^4}-\frac{\left (2 b^2 B^2 n^2\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}{3 d (b c-a d) g^4}-\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \left (\frac{b^4}{(b c-a d)^4 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^4}-\frac{b d}{(b c-a d)^2 (c+d x)^3}-\frac{b^2 d}{(b c-a d)^3 (c+d x)^2}-\frac{b^3 d}{(b c-a d)^4 (c+d x)}\right ) \, dx}{9 d g^4}\\ &=-\frac{2 B^2 n^2}{27 d g^4 (c+d x)^3}-\frac{5 b B^2 n^2}{18 d (b c-a d) g^4 (c+d x)^2}-\frac{11 b^2 B^2 n^2}{9 d (b c-a d)^2 g^4 (c+d x)}-\frac{11 b^3 B^2 n^2 \log (a+b x)}{9 d (b c-a d)^3 g^4}+\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac{2 b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac{2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac{11 b^3 B^2 n^2 \log (c+d x)}{9 d (b c-a d)^3 g^4}+\frac{2 b^3 B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac{2 b^3 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}+\frac{2 b^3 B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 d (b c-a d)^3 g^4}-\frac{\left (2 b^3 B^2 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 (b c-a d)^3 g^4}-\frac{\left (2 b^3 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 d (b c-a d)^3 g^4}-\frac{\left (2 b^3 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 d (b c-a d)^3 g^4}-\frac{\left (2 b^4 B^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 d (b c-a d)^3 g^4}\\ &=-\frac{2 B^2 n^2}{27 d g^4 (c+d x)^3}-\frac{5 b B^2 n^2}{18 d (b c-a d) g^4 (c+d x)^2}-\frac{11 b^2 B^2 n^2}{9 d (b c-a d)^2 g^4 (c+d x)}-\frac{11 b^3 B^2 n^2 \log (a+b x)}{9 d (b c-a d)^3 g^4}-\frac{b^3 B^2 n^2 \log ^2(a+b x)}{3 d (b c-a d)^3 g^4}+\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac{2 b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac{2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac{11 b^3 B^2 n^2 \log (c+d x)}{9 d (b c-a d)^3 g^4}+\frac{2 b^3 B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac{2 b^3 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac{b^3 B^2 n^2 \log ^2(c+d x)}{3 d (b c-a d)^3 g^4}+\frac{2 b^3 B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 d (b c-a d)^3 g^4}-\frac{\left (2 b^3 B^2 n^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 )}{3 d (b c-a d)^3 g^4}-\frac{\left (2 b^3 B^2 n^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 )}{3 d (b c-a d)^3 g^4}\\ &=-\frac{2 B^2 n^2}{27 d g^4 (c+d x)^3}-\frac{5 b B^2 n^2}{18 d (b c-a d) g^4 (c+d x)^2}-\frac{11 b^2 B^2 n^2}{9 d (b c-a d)^2 g^4 (c+d x)}-\frac{11 b^3 B^2 n^2 \log (a+b x)}{9 d (b c-a d)^3 g^4}-\frac{b^3 B^2 n^2 \log ^2(a+b x)}{3 d (b c-a d)^3 g^4}+\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac{2 b^2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac{2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac{11 b^3 B^2 n^2 \log (c+d x)}{9 d (b c-a d)^3 g^4}+\frac{2 b^3 B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac{2 b^3 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac{b^3 B^2 n^2 \log ^2(c+d x)}{3 d (b c-a d)^3 g^4}+\frac{2 b^3 B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 d (b c-a d)^3 g^4}+\frac{2 b^3 B^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 d (b c-a d)^3 g^4}+\frac{2 b^3 B^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 d (b c-a d)^3 g^4}\\ \end{align*}
Mathematica [C] time = 0.774851, size = 609, normalized size = 1.42 \[ \frac{\frac{B n \left (-18 b^3 B n (c+d x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+18 b^3 B n (c+d x)^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+36 b^2 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+36 b^3 (c+d x)^3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-36 b^3 (c+d x)^3 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+12 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+18 b (c+d x) (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-36 b^2 B n (c+d x)^2 (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)-9 b B n (c+d x) \left (2 b^2 (c+d x)^2 \log (a+b x)+2 b (c+d x) (b c-a d)+(b c-a d)^2-2 b^2 (c+d x)^2 \log (c+d x)\right )-2 B n \left (6 b^2 (c+d x)^2 (b c-a d)+6 b^3 (c+d x)^3 \log (a+b x)+3 b (c+d x) (b c-a d)^2+2 (b c-a d)^3-6 b^3 (c+d x)^3 \log (c+d x)\right )\right )}{(b c-a d)^3}-18 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{54 d g^4 (c+d x)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.441, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dgx+cg \right ) ^{4}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61998, size = 1937, normalized size = 4.52 \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 = 1.06222, size = 2383, normalized size = 5.55 \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 (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (d g x + c g\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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