Optimal. Leaf size=635 \[ -\frac{B^2 g i^2 n^2 (b c-a d)^4 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{6 b^3 d^2}-\frac{B g i^2 n (b c-a d)^4 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A+B n\right )}{6 b^3 d^2}-\frac{B g i^2 n (a+b x) (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^3 d}+\frac{g i^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{12 b^3}-\frac{B g i^2 n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^3}+\frac{g i^2 (a+b x)^2 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b^2}+\frac{B g i^2 n (c+d x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 b d^2}-\frac{B g i^2 n (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^2}+\frac{g i^2 (a+b x)^2 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b}-\frac{B^2 g i^2 n^2 (b c-a d)^4 \log \left (\frac{a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac{B^2 g i^2 n^2 (b c-a d)^4 \log (c+d x)}{4 b^3 d^2}+\frac{B^2 g i^2 n^2 x (b c-a d)^3}{12 b^2 d}+\frac{B^2 g i^2 n^2 (c+d x)^2 (b c-a d)^2}{12 b d^2} \]
[Out]
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Rubi [A] time = 1.65681, antiderivative size = 614, normalized size of antiderivative = 0.97, number of steps used = 44, number of rules used = 13, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.302, Rules used = {2528, 2525, 12, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 43} \[ \frac{B^2 g i^2 n^2 (b c-a d)^4 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{6 b^3 d^2}+\frac{B g i^2 n (b c-a d)^4 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^3 d^2}+\frac{A B g i^2 n x (b c-a d)^3}{6 b^2 d}+\frac{B g i^2 n (c+d x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{12 b d^2}-\frac{g i^2 (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d^2}-\frac{B g i^2 n (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^2}+\frac{b g i^2 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d^2}-\frac{B^2 g i^2 n^2 (b c-a d)^4 \log ^2(a+b x)}{12 b^3 d^2}+\frac{B^2 g i^2 n^2 (b c-a d)^4 \log (a+b x)}{12 b^3 d^2}-\frac{B^2 g i^2 n^2 (b c-a d)^4 \log (c+d x)}{6 b^3 d^2}+\frac{B^2 g i^2 n^2 (b c-a d)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{6 b^3 d^2}+\frac{B^2 g i^2 n (a+b x) (b c-a d)^3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{6 b^3 d}+\frac{B^2 g i^2 n^2 x (b c-a d)^3}{12 b^2 d}+\frac{B^2 g i^2 n^2 (c+d x)^2 (b c-a d)^2}{12 b d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 2486
Rule 31
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 43
Rubi steps
\begin{align*} \int (170 c+170 d x)^2 (a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (\frac{(-b c+a d) g (170 c+170 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{b g (170 c+170 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{170 d}\right ) \, dx\\ &=\frac{(b g) \int (170 c+170 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{170 d}+\frac{((-b c+a d) g) \int (170 c+170 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{d}\\ &=-\frac{28900 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^2}+\frac{7225 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{(b B g n) \int \frac{835210000 (b c-a d) (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{57800 d^2}+\frac{(B (b c-a d) g n) \int \frac{4913000 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{255 d^2}\\ &=-\frac{28900 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^2}+\frac{7225 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{(14450 b B (b c-a d) g n) \int \frac{(c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{d^2}+\frac{\left (57800 B (b c-a d)^2 g n\right ) \int \frac{(c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{3 d^2}\\ &=-\frac{28900 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^2}+\frac{7225 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{(14450 b B (b c-a d) g n) \int \left (\frac{d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3}+\frac{(b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 (a+b x)}+\frac{d (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 )}{b^2}+\frac{d (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{d^2}+\frac{\left (57800 B (b c-a d)^2 g n\right ) \int \left (\frac{d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{(b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 (a+b x)}+\frac{d (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{3 d^2}\\ &=-\frac{28900 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^2}+\frac{7225 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{(14450 B (b c-a d) g n) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d}-\frac{\left (14450 B (b c-a d)^2 g n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d}+\frac{\left (57800 B (b c-a d)^2 g n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b d}-\frac{\left (14450 B (b c-a d)^3 g n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d}+\frac{\left (57800 B (b c-a d)^3 g n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b^2 d}-\frac{\left (14450 B (b c-a d)^4 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}{b^2 d^2}+\frac{\left (57800 B (b c-a d)^4 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}{3 b^2 d^2}\\ &=\frac{14450 A B (b c-a d)^3 g n x}{3 b^2 d}+\frac{7225 B (b c-a d)^2 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b d^2}-\frac{14450 B (b c-a d) g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac{14450 B (b c-a d)^4 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac{28900 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^2}+\frac{7225 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{\left (14450 B^2 (b c-a d)^3 g n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 d}+\frac{\left (57800 B^2 (b c-a d)^3 g n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{3 b^2 d}+\frac{\left (14450 B^2 (b c-a d) g n^2\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 d^2}+\frac{\left (7225 B^2 (b c-a d)^2 g n^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{b d^2}-\frac{\left (28900 B^2 (b c-a d)^2 g n^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{3 b d^2}+\frac{\left (14450 B^2 (b c-a d)^4 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}{b^3 d^2}-\frac{\left (57800 B^2 (b c-a d)^4 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}{3 b^3 d^2}\\ &=\frac{14450 A B (b c-a d)^3 g n x}{3 b^2 d}+\frac{14450 B^2 (b c-a d)^3 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^3 d}+\frac{7225 B (b c-a d)^2 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b d^2}-\frac{14450 B (b c-a d) g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac{14450 B (b c-a d)^4 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac{28900 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^2}+\frac{7225 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{\left (14450 B^2 (b c-a d)^2 g n^2\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{3 d^2}+\frac{\left (7225 B^2 (b c-a d)^3 g n^2\right ) \int \frac{c+d x}{a+b x} \, dx}{b d^2}-\frac{\left (28900 B^2 (b c-a d)^3 g n^2\right ) \int \frac{c+d x}{a+b x} \, dx}{3 b d^2}+\frac{\left (14450 B^2 (b c-a d)^4 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}{b^3 d^2}-\frac{\left (57800 B^2 (b c-a d)^4 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}{3 b^3 d^2}+\frac{\left (14450 B^2 (b c-a d)^4 g n^2\right ) \int \frac{1}{c+d x} \, dx}{b^3 d}-\frac{\left (57800 B^2 (b c-a d)^4 g n^2\right ) \int \frac{1}{c+d x} \, dx}{3 b^3 d}\\ &=\frac{14450 A B (b c-a d)^3 g n x}{3 b^2 d}+\frac{14450 B^2 (b c-a d)^3 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^3 d}+\frac{7225 B (b c-a d)^2 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b d^2}-\frac{14450 B (b c-a d) g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac{14450 B (b c-a d)^4 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac{28900 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^2}+\frac{7225 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{14450 B^2 (b c-a d)^4 g n^2 \log (c+d x)}{3 b^3 d^2}+\frac{\left (14450 B^2 (b c-a d)^2 g n^2\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{3 d^2}+\frac{\left (7225 B^2 (b c-a d)^3 g n^2\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{b d^2}-\frac{\left (28900 B^2 (b c-a d)^3 g n^2\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{3 b d^2}+\frac{\left (14450 B^2 (b c-a d)^4 g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 d^2}-\frac{\left (57800 B^2 (b c-a d)^4 g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3 b^2 d^2}-\frac{\left (14450 B^2 (b c-a d)^4 g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 d}+\frac{\left (57800 B^2 (b c-a d)^4 g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 b^3 d}\\ &=\frac{14450 A B (b c-a d)^3 g n x}{3 b^2 d}+\frac{7225 B^2 (b c-a d)^3 g n^2 x}{3 b^2 d}+\frac{7225 B^2 (b c-a d)^2 g n^2 (c+d x)^2}{3 b d^2}+\frac{7225 B^2 (b c-a d)^4 g n^2 \log (a+b x)}{3 b^3 d^2}+\frac{14450 B^2 (b c-a d)^3 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^3 d}+\frac{7225 B (b c-a d)^2 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b d^2}-\frac{14450 B (b c-a d) g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac{14450 B (b c-a d)^4 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac{28900 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^2}+\frac{7225 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{14450 B^2 (b c-a d)^4 g n^2 \log (c+d x)}{3 b^3 d^2}+\frac{14450 B^2 (b c-a d)^4 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 d^2}+\frac{\left (14450 B^2 (b c-a d)^4 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 d^2}-\frac{\left (57800 B^2 (b c-a d)^4 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 d^2}+\frac{\left (14450 B^2 (b c-a d)^4 g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 d^2}-\frac{\left (57800 B^2 (b c-a d)^4 g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b^2 d^2}\\ &=\frac{14450 A B (b c-a d)^3 g n x}{3 b^2 d}+\frac{7225 B^2 (b c-a d)^3 g n^2 x}{3 b^2 d}+\frac{7225 B^2 (b c-a d)^2 g n^2 (c+d x)^2}{3 b d^2}+\frac{7225 B^2 (b c-a d)^4 g n^2 \log (a+b x)}{3 b^3 d^2}-\frac{7225 B^2 (b c-a d)^4 g n^2 \log ^2(a+b x)}{3 b^3 d^2}+\frac{14450 B^2 (b c-a d)^3 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^3 d}+\frac{7225 B (b c-a d)^2 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b d^2}-\frac{14450 B (b c-a d) g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac{14450 B (b c-a d)^4 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac{28900 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^2}+\frac{7225 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{14450 B^2 (b c-a d)^4 g n^2 \log (c+d x)}{3 b^3 d^2}+\frac{14450 B^2 (b c-a d)^4 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 d^2}+\frac{\left (14450 B^2 (b c-a d)^4 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 )}{b^3 d^2}-\frac{\left (57800 B^2 (b c-a d)^4 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 )}{3 b^3 d^2}\\ &=\frac{14450 A B (b c-a d)^3 g n x}{3 b^2 d}+\frac{7225 B^2 (b c-a d)^3 g n^2 x}{3 b^2 d}+\frac{7225 B^2 (b c-a d)^2 g n^2 (c+d x)^2}{3 b d^2}+\frac{7225 B^2 (b c-a d)^4 g n^2 \log (a+b x)}{3 b^3 d^2}-\frac{7225 B^2 (b c-a d)^4 g n^2 \log ^2(a+b x)}{3 b^3 d^2}+\frac{14450 B^2 (b c-a d)^3 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^3 d}+\frac{7225 B (b c-a d)^2 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b d^2}-\frac{14450 B (b c-a d) g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac{14450 B (b c-a d)^4 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac{28900 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^2}+\frac{7225 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{14450 B^2 (b c-a d)^4 g n^2 \log (c+d x)}{3 b^3 d^2}+\frac{14450 B^2 (b c-a d)^4 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 d^2}+\frac{14450 B^2 (b c-a d)^4 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b^3 d^2}\\ \end{align*}
Mathematica [A] time = 0.629466, size = 713, normalized size = 1.12 \[ \frac{g i^2 \left (\frac{4 B n (b c-a d)^2 \left (-B n (b c-a d)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+b^2 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 (b c-a d)^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 A b d x (b c-a d)+2 B d (a+b x) (b c-a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 B n (b c-a d)^2 \log (c+d x)-B n (b c-a d) ((b c-a d) \log (a+b x)+b d x)\right )}{b^3}-\frac{B n (b c-a d) \left (-3 B n (b c-a d)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+3 b^2 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 b^3 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+6 (b c-a d)^3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+6 A b d x (b c-a d)^2-B n (b c-a d) \left (2 b d x (b c-a d)+2 (b c-a d)^2 \log (a+b x)+b^2 (c+d x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-6 B n (b c-a d)^3 \log (c+d x)-3 B n (b c-a d)^2 ((b c-a d) \log (a+b x)+b d x)\right )}{b^3}+3 b (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2-4 (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2\right )}{12 d^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.48, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) \left ( dix+ci \right ) ^{2} \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 = 3.88865, size = 3594, normalized size = 5.66 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} b d^{2} g i^{2} x^{3} + A^{2} a c^{2} g i^{2} +{\left (2 \, A^{2} b c d + A^{2} a d^{2}\right )} g i^{2} x^{2} +{\left (A^{2} b c^{2} + 2 \, A^{2} a c d\right )} g i^{2} x +{\left (B^{2} b d^{2} g i^{2} x^{3} + B^{2} a c^{2} g i^{2} +{\left (2 \, B^{2} b c d + B^{2} a d^{2}\right )} g i^{2} x^{2} +{\left (B^{2} b c^{2} + 2 \, B^{2} a c d\right )} g i^{2} x\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B b d^{2} g i^{2} x^{3} + A B a c^{2} g i^{2} +{\left (2 \, A B b c d + A B a d^{2}\right )} g i^{2} x^{2} +{\left (A B b c^{2} + 2 \, A B a c d\right )} g i^{2} x\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ), x\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{\left (b g x + a g\right )}{\left (d i x + c i\right )}^{2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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