Optimal. Leaf size=151 \[ \frac{g (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 i^3 (c+d x)^2 (b c-a d)}-\frac{B g n (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)}+\frac{B^2 g n^2 (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)} \]
[Out]
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Rubi [C] time = 1.99646, antiderivative size = 686, normalized size of antiderivative = 4.54, number of steps used = 54, number of rules used = 11, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac{b^2 B^2 g n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac{b^2 B^2 g n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac{b^2 B g n \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i^3 (b c-a d)}-\frac{b^2 B g n \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i^3 (b c-a d)}-\frac{b g \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^2 i^3 (c+d x)}+\frac{b B g n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i^3 (c+d x)}+\frac{g (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d^2 i^3 (c+d x)^2}-\frac{B g n (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^2 i^3 (c+d x)^2}-\frac{b^2 B^2 g n^2 \log ^2(a+b x)}{2 d^2 i^3 (b c-a d)}-\frac{b^2 B^2 g n^2 \log ^2(c+d x)}{2 d^2 i^3 (b c-a d)}-\frac{b^2 B^2 g n^2 \log (a+b x)}{2 d^2 i^3 (b c-a d)}+\frac{b^2 B^2 g n^2 \log (c+d x)}{2 d^2 i^3 (b c-a d)}+\frac{b^2 B^2 g n^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac{b^2 B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac{B^2 g n^2 (b c-a d)}{4 d^2 i^3 (c+d x)^2}-\frac{b B^2 g n^2}{2 d^2 i^3 (c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 44
Rubi steps
\begin{align*} \int \frac{(a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(204 c+204 d x)^3} \, dx &=\int \left (\frac{(-b c+a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d (c+d x)^3}+\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d (c+d x)^2}\right ) \, dx\\ &=\frac{(b g) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2} \, dx}{8489664 d}-\frac{((b c-a d) g) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3} \, dx}{8489664 d}\\ &=\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac{(b B g 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 )}{(a+b x) (c+d x)^2} \, dx}{4244832 d^2}-\frac{(B (b c-a d) g 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 )}{(a+b x) (c+d x)^3} \, dx}{8489664 d^2}\\ &=\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac{(b B (b c-a d) g 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)^2} \, dx}{4244832 d^2}-\frac{\left (B (b c-a d)^2 g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^3} \, dx}{8489664 d^2}\\ &=\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac{(b B (b c-a d) g n) \int \left (\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (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)^2}-\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)}\right ) \, dx}{4244832 d^2}-\frac{\left (B (b c-a d)^2 g n\right ) \int \left (\frac{b^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (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)^3}-\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)^2}-\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)}\right ) \, dx}{8489664 d^2}\\ &=\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac{(b B g n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{8489664 d}-\frac{(b B g n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{4244832 d}-\frac{\left (b^3 B g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8489664 d^2 (b c-a d)}+\frac{\left (b^3 B g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{4244832 d^2 (b c-a d)}+\frac{\left (b^2 B g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{8489664 d (b c-a d)}-\frac{\left (b^2 B g n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{4244832 d (b c-a d)}+\frac{(B (b c-a d) g n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{8489664 d}\\ &=-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac{b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}-\frac{b^2 B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}+\frac{\left (b B^2 g n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{8489664 d^2}-\frac{\left (b B^2 g n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{4244832 d^2}+\frac{\left (b^2 B^2 g 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}{8489664 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g 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}{8489664 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g 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}{4244832 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g 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}{4244832 d^2 (b c-a d)}+\frac{\left (B^2 (b c-a d) g n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{16979328 d^2}\\ &=-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac{b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}-\frac{b^2 B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g 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}{8489664 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g 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}{8489664 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g 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}{4244832 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g 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}{4244832 d^2 (b c-a d)}+\frac{\left (b B^2 (b c-a d) g n^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{8489664 d^2}-\frac{\left (b B^2 (b c-a d) g n^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{4244832 d^2}+\frac{\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{16979328 d^2}\\ &=-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac{b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}-\frac{b^2 B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}+\frac{\left (b^3 B^2 g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{8489664 d^2 (b c-a d)}-\frac{\left (b^3 B^2 g n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{8489664 d^2 (b c-a d)}-\frac{\left (b^3 B^2 g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{4244832 d^2 (b c-a d)}+\frac{\left (b^3 B^2 g n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{4244832 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{8489664 d (b c-a d)}+\frac{\left (b^2 B^2 g n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{8489664 d (b c-a d)}+\frac{\left (b^2 B^2 g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{4244832 d (b c-a d)}-\frac{\left (b^2 B^2 g n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{4244832 d (b c-a d)}+\frac{\left (b B^2 (b c-a d) g 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}{8489664 d^2}-\frac{\left (b B^2 (b c-a d) g 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}{4244832 d^2}+\frac{\left (B^2 (b c-a d)^2 g 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}{16979328 d^2}\\ &=\frac{B^2 (b c-a d) g n^2}{33958656 d^2 (c+d x)^2}-\frac{b B^2 g n^2}{16979328 d^2 (c+d x)}-\frac{b^2 B^2 g n^2 \log (a+b x)}{16979328 d^2 (b c-a d)}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac{b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac{b^2 B^2 g n^2 \log (c+d x)}{16979328 d^2 (b c-a d)}+\frac{b^2 B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}-\frac{b^2 B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}+\frac{b^2 B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{8489664 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{8489664 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{8489664 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{4244832 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{4244832 d^2 (b c-a d)}+\frac{\left (b^3 B^2 g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8489664 d^2 (b c-a d)}-\frac{\left (b^3 B^2 g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{4244832 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8489664 d (b c-a d)}-\frac{\left (b^2 B^2 g n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4244832 d (b c-a d)}\\ &=\frac{B^2 (b c-a d) g n^2}{33958656 d^2 (c+d x)^2}-\frac{b B^2 g n^2}{16979328 d^2 (c+d x)}-\frac{b^2 B^2 g n^2 \log (a+b x)}{16979328 d^2 (b c-a d)}-\frac{b^2 B^2 g n^2 \log ^2(a+b x)}{16979328 d^2 (b c-a d)}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac{b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac{b^2 B^2 g n^2 \log (c+d x)}{16979328 d^2 (b c-a d)}+\frac{b^2 B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}-\frac{b^2 B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}-\frac{b^2 B^2 g n^2 \log ^2(c+d x)}{16979328 d^2 (b c-a d)}+\frac{b^2 B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{8489664 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g 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 )}{8489664 d^2 (b c-a d)}+\frac{\left (b^2 B^2 g 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 )}{8489664 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g 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 )}{4244832 d^2 (b c-a d)}-\frac{\left (b^2 B^2 g 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 )}{4244832 d^2 (b c-a d)}\\ &=\frac{B^2 (b c-a d) g n^2}{33958656 d^2 (c+d x)^2}-\frac{b B^2 g n^2}{16979328 d^2 (c+d x)}-\frac{b^2 B^2 g n^2 \log (a+b x)}{16979328 d^2 (b c-a d)}-\frac{b^2 B^2 g n^2 \log ^2(a+b x)}{16979328 d^2 (b c-a d)}-\frac{B (b c-a d) g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac{b B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac{b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac{(b c-a d) g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac{b g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac{b^2 B^2 g n^2 \log (c+d x)}{16979328 d^2 (b c-a d)}+\frac{b^2 B^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}-\frac{b^2 B g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}-\frac{b^2 B^2 g n^2 \log ^2(c+d x)}{16979328 d^2 (b c-a d)}+\frac{b^2 B^2 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{8489664 d^2 (b c-a d)}+\frac{b^2 B^2 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{8489664 d^2 (b c-a d)}+\frac{b^2 B^2 g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{8489664 d^2 (b c-a d)}\\ \end{align*}
Mathematica [C] time = 0.949506, size = 803, normalized size = 5.32 \[ \frac{g \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2-4 b (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2+4 b B n (c+d x) \left (2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+2 b (c+d x) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-2 b (c+d x) \log (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-2 B n (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-b B n (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+b B n (c+d x) \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )-B n \left (2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2+4 b (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)+4 b^2 (c+d x)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-4 b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-4 b B n (c+d x) (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-B n \left ((b c-a d)^2+2 b (c+d x) (b c-a d)+2 b^2 (c+d x)^2 \log (a+b x)-2 b^2 (c+d x)^2 \log (c+d x)\right )-2 b^2 B n (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+2 b^2 B n (c+d x)^2 \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )\right )}{4 d^2 (b c-a d) i^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.527, size = 0, normalized size = 0. \begin{align*} \int{\frac{bgx+ag}{ \left ( dix+ci \right ) ^{3}} \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 = 1.75734, size = 2693, normalized size = 17.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.569616, size = 1214, normalized size = 8.04 \begin{align*} -\frac{{\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} g n^{2} - 2 \,{\left (A B b^{2} c^{2} - A B a^{2} d^{2}\right )} g n + 2 \,{\left (2 \,{\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} g x +{\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} g\right )} \log \left (e\right )^{2} - 2 \,{\left (B^{2} b^{2} d^{2} g n^{2} x^{2} + 2 \, B^{2} a b d^{2} g n^{2} x + B^{2} a^{2} d^{2} g n^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )^{2} + 2 \,{\left (A^{2} b^{2} c^{2} - A^{2} a^{2} d^{2}\right )} g + 2 \,{\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} g n^{2} - 2 \,{\left (A B b^{2} c d - A B a b d^{2}\right )} g n + 2 \,{\left (A^{2} b^{2} c d - A^{2} a b d^{2}\right )} g\right )} x - 2 \,{\left ({\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} g n - 2 \,{\left (A B b^{2} c^{2} - A B a^{2} d^{2}\right )} g + 2 \,{\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} g n - 2 \,{\left (A B b^{2} c d - A B a b d^{2}\right )} g\right )} x + 2 \,{\left (B^{2} b^{2} d^{2} g n x^{2} + 2 \, B^{2} a b d^{2} g n x + B^{2} a^{2} d^{2} g n\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 2 \,{\left (B^{2} a^{2} d^{2} g n^{2} - 2 \, A B a^{2} d^{2} g n +{\left (B^{2} b^{2} d^{2} g n^{2} - 2 \, A B b^{2} d^{2} g n\right )} x^{2} + 2 \,{\left (B^{2} a b d^{2} g n^{2} - 2 \, A B a b d^{2} g n\right )} x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{4 \,{\left ({\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \,{\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x +{\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}\right )}} \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 g x + a g\right )}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\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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