Optimal. Leaf size=584 \[ -\frac{B^2 g^3 i n^2 (b c-a d)^5 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{10 b^2 d^4}-\frac{B g^3 i n (b c-a d)^5 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+6 A+11 B n\right )}{60 b^2 d^4}-\frac{B g^3 i n (a+b x) (b c-a d)^4 \left (6 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{60 b^2 d^3}+\frac{B g^3 i n (a+b x)^2 (b c-a d)^3 \left (3 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{60 b^2 d^2}-\frac{B g^3 i n (a+b x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{30 b^2 d}+\frac{g^3 i (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{20 b^2}-\frac{B g^3 i n (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^2}+\frac{g^3 i (a+b x)^4 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b}-\frac{B^2 g^3 i n^2 (b c-a d)^5 \log (c+d x)}{10 b^2 d^4}+\frac{3 B^2 g^3 i n^2 x (b c-a d)^4}{10 b d^3}-\frac{3 B^2 g^3 i n^2 (c+d x)^2 (b c-a d)^3}{20 d^4}+\frac{b B^2 g^3 i n^2 (c+d x)^3 (b c-a d)^2}{30 d^4} \]
[Out]
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Rubi [A] time = 1.90112, antiderivative size = 670, normalized size of antiderivative = 1.15, number of steps used = 52, 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, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{B^2 g^3 i n^2 (b c-a d)^5 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{10 b^2 d^4}+\frac{B g^3 i n (b c-a d)^5 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^2 d^4}+\frac{B g^3 i n (a+b x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 d^2}-\frac{B g^3 i n (a+b x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{30 b^2 d}+\frac{g^3 i (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b^2}-\frac{B g^3 i n (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^2}+\frac{d g^3 i (a+b x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b^2}-\frac{A B g^3 i n x (b c-a d)^4}{10 b d^3}-\frac{B^2 g^3 i n (a+b x) (b c-a d)^4 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{10 b^2 d^3}-\frac{B^2 g^3 i n^2 (a+b x)^2 (b c-a d)^3}{30 b^2 d^2}+\frac{B^2 g^3 i n^2 (b c-a d)^5 \log ^2(c+d x)}{20 b^2 d^4}+\frac{B^2 g^3 i n^2 (b c-a d)^5 \log (c+d x)}{12 b^2 d^4}-\frac{B^2 g^3 i n^2 (b c-a d)^5 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{10 b^2 d^4}+\frac{B^2 g^3 i n^2 (a+b x)^3 (b c-a d)^2}{30 b^2 d}+\frac{B^2 g^3 i n^2 x (b c-a d)^4}{60 b d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 2486
Rule 31
Rule 43
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int (159 c+159 d x) (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (\frac{159 (b c-a d) (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b}+\frac{159 d (a g+b g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}\right ) \, dx\\ &=\frac{(159 (b c-a d)) \int (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b}+\frac{(159 d) \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b g}\\ &=\frac{159 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2}+\frac{159 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^2}-\frac{(318 B d n) \int \frac{(b c-a d) g^5 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{5 b^2 g^2}-\frac{(159 B (b c-a d) n) \int \frac{(b c-a d) g^4 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{2 b^2 g}\\ &=\frac{159 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2}+\frac{159 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^2}-\frac{\left (318 B d (b c-a d) g^3 n\right ) \int \frac{(a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{5 b^2}-\frac{\left (159 B (b c-a d)^2 g^3 n\right ) \int \frac{(a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{2 b^2}\\ &=\frac{159 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2}+\frac{159 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^2}-\frac{\left (318 B d (b c-a d) g^3 n\right ) \int \left (-\frac{b (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^4}+\frac{b (b c-a d)^2 (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{b (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac{(-b c+a d)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^4 (c+d x)}\right ) \, dx}{5 b^2}-\frac{\left (159 B (b c-a d)^2 g^3 n\right ) \int \left (\frac{b (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{b (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac{(-b c+a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3 (c+d x)}\right ) \, dx}{2 b^2}\\ &=\frac{159 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2}+\frac{159 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^2}-\frac{\left (318 B (b c-a d) g^3 n\right ) \int (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b}+\frac{\left (318 B (b c-a d)^2 g^3 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b d}-\frac{\left (159 B (b c-a d)^2 g^3 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{2 b d}-\frac{\left (318 B (b c-a d)^3 g^3 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b d^2}+\frac{\left (159 B (b c-a d)^3 g^3 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{2 b d^2}+\frac{\left (318 B (b c-a d)^4 g^3 n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b d^3}-\frac{\left (159 B (b c-a d)^4 g^3 n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{2 b d^3}-\frac{\left (318 B (b c-a d)^5 g^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{5 b^2 d^3}+\frac{\left (159 B (b c-a d)^5 g^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{2 b^2 d^3}\\ &=-\frac{159 A B (b c-a d)^4 g^3 n x}{10 b d^3}+\frac{159 B (b c-a d)^3 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^2 d^2}-\frac{53 B (b c-a d)^2 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2 d}-\frac{159 B (b c-a d) g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2}+\frac{159 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2}+\frac{159 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^2}+\frac{159 B (b c-a d)^5 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10 b^2 d^4}+\frac{\left (318 B^2 (b c-a d)^4 g^3 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{5 b d^3}-\frac{\left (159 B^2 (b c-a d)^4 g^3 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{2 b d^3}+\frac{\left (159 B^2 (b c-a d) g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)^3}{c+d x} \, dx}{10 b^2}-\frac{\left (106 B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)^2}{c+d x} \, dx}{5 b^2 d}+\frac{\left (53 B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)^2}{c+d x} \, dx}{2 b^2 d}+\frac{\left (159 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{5 b^2 d^2}-\frac{\left (159 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{4 b^2 d^2}+\frac{\left (318 B^2 (b c-a d)^5 g^3 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}{5 b^2 d^4}-\frac{\left (159 B^2 (b c-a d)^5 g^3 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}{2 b^2 d^4}\\ &=-\frac{159 A B (b c-a d)^4 g^3 n x}{10 b d^3}-\frac{159 B^2 (b c-a d)^4 g^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{10 b^2 d^3}+\frac{159 B (b c-a d)^3 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^2 d^2}-\frac{53 B (b c-a d)^2 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2 d}-\frac{159 B (b c-a d) g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2}+\frac{159 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2}+\frac{159 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^2}+\frac{159 B (b c-a d)^5 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10 b^2 d^4}+\frac{\left (159 B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{10 b^2}-\frac{\left (106 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{5 b^2 d}+\frac{\left (53 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{2 b^2 d}+\frac{\left (159 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac{a+b x}{c+d x} \, dx}{5 b^2 d^2}-\frac{\left (159 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac{a+b x}{c+d x} \, dx}{4 b^2 d^2}+\frac{\left (318 B^2 (b c-a d)^5 g^3 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}{5 b^2 d^4}-\frac{\left (159 B^2 (b c-a d)^5 g^3 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}{2 b^2 d^4}-\frac{\left (318 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{1}{c+d x} \, dx}{5 b^2 d^3}+\frac{\left (159 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{1}{c+d x} \, dx}{2 b^2 d^3}\\ &=-\frac{159 A B (b c-a d)^4 g^3 n x}{10 b d^3}-\frac{159 B^2 (b c-a d)^4 g^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{10 b^2 d^3}+\frac{159 B (b c-a d)^3 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^2 d^2}-\frac{53 B (b c-a d)^2 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2 d}-\frac{159 B (b c-a d) g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2}+\frac{159 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2}+\frac{159 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^2}+\frac{159 B^2 (b c-a d)^5 g^3 n^2 \log (c+d x)}{10 b^2 d^4}+\frac{159 B (b c-a d)^5 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10 b^2 d^4}+\frac{\left (159 B^2 (b c-a d)^2 g^3 n^2\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{10 b^2}-\frac{\left (106 B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{5 b^2 d}+\frac{\left (53 B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{2 b^2 d}+\frac{\left (159 B^2 (b c-a d)^4 g^3 n^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{5 b^2 d^2}-\frac{\left (159 B^2 (b c-a d)^4 g^3 n^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{4 b^2 d^2}+\frac{\left (318 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{5 b d^4}-\frac{\left (159 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{2 b d^4}-\frac{\left (318 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{5 b^2 d^3}+\frac{\left (159 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{2 b^2 d^3}\\ &=-\frac{159 A B (b c-a d)^4 g^3 n x}{10 b d^3}+\frac{53 B^2 (b c-a d)^4 g^3 n^2 x}{20 b d^3}-\frac{53 B^2 (b c-a d)^3 g^3 n^2 (a+b x)^2}{10 b^2 d^2}+\frac{53 B^2 (b c-a d)^2 g^3 n^2 (a+b x)^3}{10 b^2 d}-\frac{159 B^2 (b c-a d)^4 g^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{10 b^2 d^3}+\frac{159 B (b c-a d)^3 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^2 d^2}-\frac{53 B (b c-a d)^2 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2 d}-\frac{159 B (b c-a d) g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2}+\frac{159 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2}+\frac{159 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^2}+\frac{53 B^2 (b c-a d)^5 g^3 n^2 \log (c+d x)}{4 b^2 d^4}-\frac{159 B^2 (b c-a d)^5 g^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{10 b^2 d^4}+\frac{159 B (b c-a d)^5 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10 b^2 d^4}-\frac{\left (318 B^2 (b c-a d)^5 g^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{5 b^2 d^4}+\frac{\left (159 B^2 (b c-a d)^5 g^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{2 b^2 d^4}-\frac{\left (318 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b^2 d^3}+\frac{\left (159 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b^2 d^3}\\ &=-\frac{159 A B (b c-a d)^4 g^3 n x}{10 b d^3}+\frac{53 B^2 (b c-a d)^4 g^3 n^2 x}{20 b d^3}-\frac{53 B^2 (b c-a d)^3 g^3 n^2 (a+b x)^2}{10 b^2 d^2}+\frac{53 B^2 (b c-a d)^2 g^3 n^2 (a+b x)^3}{10 b^2 d}-\frac{159 B^2 (b c-a d)^4 g^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{10 b^2 d^3}+\frac{159 B (b c-a d)^3 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^2 d^2}-\frac{53 B (b c-a d)^2 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2 d}-\frac{159 B (b c-a d) g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2}+\frac{159 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2}+\frac{159 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^2}+\frac{53 B^2 (b c-a d)^5 g^3 n^2 \log (c+d x)}{4 b^2 d^4}-\frac{159 B^2 (b c-a d)^5 g^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{10 b^2 d^4}+\frac{159 B (b c-a d)^5 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10 b^2 d^4}+\frac{159 B^2 (b c-a d)^5 g^3 n^2 \log ^2(c+d x)}{20 b^2 d^4}-\frac{\left (318 B^2 (b c-a d)^5 g^3 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 )}{5 b^2 d^4}+\frac{\left (159 B^2 (b c-a d)^5 g^3 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 )}{2 b^2 d^4}\\ &=-\frac{159 A B (b c-a d)^4 g^3 n x}{10 b d^3}+\frac{53 B^2 (b c-a d)^4 g^3 n^2 x}{20 b d^3}-\frac{53 B^2 (b c-a d)^3 g^3 n^2 (a+b x)^2}{10 b^2 d^2}+\frac{53 B^2 (b c-a d)^2 g^3 n^2 (a+b x)^3}{10 b^2 d}-\frac{159 B^2 (b c-a d)^4 g^3 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{10 b^2 d^3}+\frac{159 B (b c-a d)^3 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20 b^2 d^2}-\frac{53 B (b c-a d)^2 g^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2 d}-\frac{159 B (b c-a d) g^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b^2}+\frac{159 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^2}+\frac{159 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^2}+\frac{53 B^2 (b c-a d)^5 g^3 n^2 \log (c+d x)}{4 b^2 d^4}-\frac{159 B^2 (b c-a d)^5 g^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{10 b^2 d^4}+\frac{159 B (b c-a d)^5 g^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10 b^2 d^4}+\frac{159 B^2 (b c-a d)^5 g^3 n^2 \log ^2(c+d x)}{20 b^2 d^4}-\frac{159 B^2 (b c-a d)^5 g^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{10 b^2 d^4}\\ \end{align*}
Mathematica [A] time = 0.785927, size = 949, normalized size = 1.62 \[ \frac{g^3 i \left (4 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^5+5 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4-\frac{5 B (b c-a d)^2 n \left (-6 B n \log (c+d x) (b c-a d)^3-6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) (b c-a d)^3+3 B n \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 ) (b c-a d)^3+6 A b d x (b c-a d)^2+6 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) (b c-a d)^2+3 B n (b d x+(a d-b c) \log (c+d x)) (b c-a d)^2+B n \left (-2 \log (c+d x) (b c-a d)^2+2 b d x (b c-a d)-d^2 (a+b x)^2\right ) (b c-a d)+2 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+3 d^2 (a d-b c) (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )\right )}{3 d^4}+\frac{B (b c-a d) n \left (-24 B n \log (c+d x) (b c-a d)^4-24 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) (b c-a d)^4+12 B n \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 ) (b c-a d)^4+24 A b d x (b c-a d)^3+24 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) (b c-a d)^3+12 B n (b d x+(a d-b c) \log (c+d x)) (b c-a d)^3-12 d^2 (a+b x)^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 n \left (-2 \log (c+d x) (b c-a d)^2+2 b d x (b c-a d)-d^2 (a+b x)^2\right ) (b c-a d)^2+8 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)+B n \left (-6 \log (c+d x) (b c-a d)^3+6 b d x (b c-a d)^2+2 d^3 (a+b x)^3+3 d^2 (a d-b c) (a+b x)^2\right ) (b c-a d)-6 d^4 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )\right )}{3 d^4}\right )}{20 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.526, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{3} \left ( dix+ci \right ) \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 = 3.89072, size = 5081, normalized size = 8.7 \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^{3} d g^{3} i x^{4} + A^{2} a^{3} c g^{3} i +{\left (A^{2} b^{3} c + 3 \, A^{2} a b^{2} d\right )} g^{3} i x^{3} + 3 \,{\left (A^{2} a b^{2} c + A^{2} a^{2} b d\right )} g^{3} i x^{2} +{\left (3 \, A^{2} a^{2} b c + A^{2} a^{3} d\right )} g^{3} i x +{\left (B^{2} b^{3} d g^{3} i x^{4} + B^{2} a^{3} c g^{3} i +{\left (B^{2} b^{3} c + 3 \, B^{2} a b^{2} d\right )} g^{3} i x^{3} + 3 \,{\left (B^{2} a b^{2} c + B^{2} a^{2} b d\right )} g^{3} i x^{2} +{\left (3 \, B^{2} a^{2} b c + B^{2} a^{3} d\right )} g^{3} i x\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B b^{3} d g^{3} i x^{4} + A B a^{3} c g^{3} i +{\left (A B b^{3} c + 3 \, A B a b^{2} d\right )} g^{3} i x^{3} + 3 \,{\left (A B a b^{2} c + A B a^{2} b d\right )} g^{3} i x^{2} +{\left (3 \, A B a^{2} b c + A B a^{3} d\right )} g^{3} i 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 )}^{3}{\left (d i x + c i\right )}{\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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