Optimal. Leaf size=786 \[ -\frac {B g i^3 n (b c-a d)^5 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A+B n\right )}{10 b^4 d^2}-\frac {B g i^3 n (a+b x) (b c-a d)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^4 d}+\frac {g i^3 (a+b x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{20 b^4}-\frac {B g i^3 n (a+b x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^4}+\frac {g i^3 (a+b x)^2 (c+d x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{10 b^3}+\frac {3 B g i^3 n (c+d x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 d^2}+\frac {3 g i^3 (a+b x)^2 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{20 b^2}+\frac {B g i^3 n (c+d x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{30 b d^2}-\frac {B g i^3 n (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{10 d^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b}-\frac {B^2 g i^3 n^2 (b c-a d)^5 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{10 b^4 d^2}-\frac {B^2 g i^3 n^2 (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac {11 B^2 g i^3 n^2 (b c-a d)^5 \log (c+d x)}{60 b^4 d^2}+\frac {B^2 g i^3 n^2 x (b c-a d)^4}{60 b^3 d}+\frac {B^2 g i^3 n^2 (c+d x)^2 (b c-a d)^3}{30 b^2 d^2}+\frac {B^2 g i^3 n^2 (c+d x)^3 (b c-a d)^2}{30 b d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.93, antiderivative size = 706, normalized size of antiderivative = 0.90, 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, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 43} \[ \frac {B^2 g i^3 n^2 (b c-a d)^5 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{10 b^4 d^2}+\frac {B g i^3 n (b c-a d)^5 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^4 d^2}+\frac {B g i^3 n (c+d x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 d^2}+\frac {A B g i^3 n x (b c-a d)^4}{10 b^3 d}+\frac {B g i^3 n (c+d x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{30 b d^2}-\frac {g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d^2}-\frac {B g i^3 n (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{10 d^2}+\frac {b g i^3 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 d^2}+\frac {B^2 g i^3 n^2 (c+d x)^2 (b c-a d)^3}{30 b^2 d^2}-\frac {B^2 g i^3 n^2 (b c-a d)^5 \log ^2(a+b x)}{20 b^4 d^2}+\frac {B^2 g i^3 n^2 (b c-a d)^5 \log (a+b x)}{60 b^4 d^2}-\frac {B^2 g i^3 n^2 (b c-a d)^5 \log (c+d x)}{10 b^4 d^2}+\frac {B^2 g i^3 n^2 (b c-a d)^5 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{10 b^4 d^2}+\frac {B^2 g i^3 n (a+b x) (b c-a d)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{10 b^4 d}+\frac {B^2 g i^3 n^2 x (b c-a d)^4}{60 b^3 d}+\frac {B^2 g i^3 n^2 (c+d x)^3 (b c-a d)^2}{30 b d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 43
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2486
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int (180 c+180 d x)^3 (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 (180 c+180 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac {b g (180 c+180 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{180 d}\right ) \, dx\\ &=\frac {(b g) \int (180 c+180 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{180 d}+\frac {((-b c+a d) g) \int (180 c+180 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{d}\\ &=-\frac {1458000 (b c-a d) 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 {1166400 b g (c+d x)^5 \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 {188956800000 (b c-a d) (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{81000 d^2}+\frac {(B (b c-a d) g n) \int \frac {1049760000 (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}{360 d^2}\\ &=-\frac {1458000 (b c-a d) 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 {1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac {(2332800 b B (b c-a d) g n) \int \frac {(c+d x)^4 \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 (2916000 B (b c-a d)^2 g n\right ) \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 {1458000 (b c-a d) 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 {1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac {(2332800 b B (b c-a d) g n) \int \left (\frac {d (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4}+\frac {(b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3}+\frac {d (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 )}{b^2}+\frac {d (c+d x)^3 \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 (2916000 B (b c-a d)^2 g n\right ) \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 {1458000 (b c-a d) 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 {1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac {(2332800 B (b c-a d) g n) \int (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d}-\frac {\left (2332800 B (b c-a d)^2 g n\right ) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d}+\frac {\left (2916000 B (b c-a d)^2 g n\right ) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d}-\frac {\left (2332800 B (b c-a d)^3 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^2 d}+\frac {\left (2916000 B (b c-a d)^3 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^2 d}-\frac {\left (2332800 B (b c-a d)^4 g n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^3 d}+\frac {\left (2916000 B (b c-a d)^4 g n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^3 d}-\frac {\left (2332800 B (b c-a d)^5 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^3 d^2}+\frac {\left (2916000 B (b c-a d)^5 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^3 d^2}\\ &=\frac {583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac {291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac {194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac {583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac {1458000 (b c-a d) 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 {1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac {\left (2332800 B^2 (b c-a d)^4 g n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 d}+\frac {\left (2916000 B^2 (b c-a d)^4 g n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 d}+\frac {\left (583200 B^2 (b c-a d) g n^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{d^2}+\frac {\left (777600 B^2 (b c-a d)^2 g n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{b d^2}-\frac {\left (972000 B^2 (b c-a d)^2 g n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{b d^2}+\frac {\left (1166400 B^2 (b c-a d)^3 g n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^2}-\frac {\left (1458000 B^2 (b c-a d)^3 g n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^2}+\frac {\left (2332800 B^2 (b c-a d)^5 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^4 d^2}-\frac {\left (2916000 B^2 (b c-a d)^5 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^4 d^2}\\ &=\frac {583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac {583200 B^2 (b c-a d)^4 g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 d}+\frac {291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac {194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac {583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac {1458000 (b c-a d) 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 {1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac {\left (583200 B^2 (b c-a d)^2 g n^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{d^2}+\frac {\left (777600 B^2 (b c-a d)^3 g n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{b d^2}-\frac {\left (972000 B^2 (b c-a d)^3 g n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{b d^2}+\frac {\left (1166400 B^2 (b c-a d)^4 g n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{b^2 d^2}-\frac {\left (1458000 B^2 (b c-a d)^4 g n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{b^2 d^2}+\frac {\left (2332800 B^2 (b c-a d)^5 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^4 d^2}-\frac {\left (2916000 B^2 (b c-a d)^5 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^4 d^2}+\frac {\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \int \frac {1}{c+d x} \, dx}{b^4 d}-\frac {\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \int \frac {1}{c+d x} \, dx}{b^4 d}\\ &=\frac {583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac {583200 B^2 (b c-a d)^4 g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 d}+\frac {291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac {194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac {583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac {1458000 (b c-a d) 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 {1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac {583200 B^2 (b c-a d)^5 g n^2 \log (c+d x)}{b^4 d^2}+\frac {\left (583200 B^2 (b c-a d)^2 g n^2\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{d^2}+\frac {\left (777600 B^2 (b c-a d)^3 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}{b d^2}-\frac {\left (972000 B^2 (b c-a d)^3 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}{b d^2}+\frac {\left (1166400 B^2 (b c-a d)^4 g n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^2}-\frac {\left (1458000 B^2 (b c-a d)^4 g n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^2}+\frac {\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 d^2}-\frac {\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 d^2}-\frac {\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 d}+\frac {\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 d}\\ &=\frac {583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac {97200 B^2 (b c-a d)^4 g n^2 x}{b^3 d}+\frac {194400 B^2 (b c-a d)^3 g n^2 (c+d x)^2}{b^2 d^2}+\frac {194400 B^2 (b c-a d)^2 g n^2 (c+d x)^3}{b d^2}+\frac {97200 B^2 (b c-a d)^5 g n^2 \log (a+b x)}{b^4 d^2}+\frac {583200 B^2 (b c-a d)^4 g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 d}+\frac {291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac {194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac {583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac {1458000 (b c-a d) 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 {1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac {583200 B^2 (b c-a d)^5 g n^2 \log (c+d x)}{b^4 d^2}+\frac {583200 B^2 (b c-a d)^5 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 d^2}+\frac {\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^2}-\frac {\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^2}+\frac {\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^2}-\frac {\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^2}\\ &=\frac {583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac {97200 B^2 (b c-a d)^4 g n^2 x}{b^3 d}+\frac {194400 B^2 (b c-a d)^3 g n^2 (c+d x)^2}{b^2 d^2}+\frac {194400 B^2 (b c-a d)^2 g n^2 (c+d x)^3}{b d^2}+\frac {97200 B^2 (b c-a d)^5 g n^2 \log (a+b x)}{b^4 d^2}-\frac {291600 B^2 (b c-a d)^5 g n^2 \log ^2(a+b x)}{b^4 d^2}+\frac {583200 B^2 (b c-a d)^4 g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 d}+\frac {291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac {194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac {583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac {1458000 (b c-a d) 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 {1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac {583200 B^2 (b c-a d)^5 g n^2 \log (c+d x)}{b^4 d^2}+\frac {583200 B^2 (b c-a d)^5 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 d^2}+\frac {\left (2332800 B^2 (b c-a d)^5 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^4 d^2}-\frac {\left (2916000 B^2 (b c-a d)^5 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^4 d^2}\\ &=\frac {583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac {97200 B^2 (b c-a d)^4 g n^2 x}{b^3 d}+\frac {194400 B^2 (b c-a d)^3 g n^2 (c+d x)^2}{b^2 d^2}+\frac {194400 B^2 (b c-a d)^2 g n^2 (c+d x)^3}{b d^2}+\frac {97200 B^2 (b c-a d)^5 g n^2 \log (a+b x)}{b^4 d^2}-\frac {291600 B^2 (b c-a d)^5 g n^2 \log ^2(a+b x)}{b^4 d^2}+\frac {583200 B^2 (b c-a d)^4 g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^4 d}+\frac {291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac {194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac {583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac {583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac {1458000 (b c-a d) 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 {1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac {583200 B^2 (b c-a d)^5 g n^2 \log (c+d x)}{b^4 d^2}+\frac {583200 B^2 (b c-a d)^5 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 d^2}+\frac {583200 B^2 (b c-a d)^5 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 d^2}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 945, normalized size = 1.20 \[ \frac {g i^3 \left (4 b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (c+d 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 (c+d x)^4+\frac {5 B (b c-a d)^2 n \left (6 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3-6 B n \log (c+d x) (b c-a d)^3-3 B 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 {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right ) (b c-a d)^3+6 A b d x (b c-a d)^2-3 B n (b d x+(b c-a d) \log (a+b 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-B n \left (2 \log (a+b x) (b c-a d)^2+2 b d x (b c-a d)+b^2 (c+d x)^2\right ) (b c-a d)+3 b^2 (c+d 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 b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{3 b^4}-\frac {B (b c-a d) n \left (24 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4-24 B n \log (c+d x) (b c-a d)^4-12 B 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 {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right ) (b c-a d)^4+24 A b d x (b c-a d)^3-12 B n (b d x+(b c-a d) \log (a+b 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)^3-4 B n \left (2 \log (a+b x) (b c-a d)^2+2 b d x (b c-a d)+b^2 (c+d x)^2\right ) (b c-a d)^2+12 b^2 (c+d 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-B n \left (6 \log (a+b x) (b c-a d)^3+6 b d x (b c-a d)^2+3 b^2 (c+d x)^2 (b c-a d)+2 b^3 (c+d x)^3\right ) (b c-a d)+8 b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)+6 b^4 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{3 b^4}\right )}{20 d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b d^{3} g i^{3} x^{4} + A^{2} a c^{3} g i^{3} + {\left (3 \, A^{2} b c d^{2} + A^{2} a d^{3}\right )} g i^{3} x^{3} + 3 \, {\left (A^{2} b c^{2} d + A^{2} a c d^{2}\right )} g i^{3} x^{2} + {\left (A^{2} b c^{3} + 3 \, A^{2} a c^{2} d\right )} g i^{3} x + {\left (B^{2} b d^{3} g i^{3} x^{4} + B^{2} a c^{3} g i^{3} + {\left (3 \, B^{2} b c d^{2} + B^{2} a d^{3}\right )} g i^{3} x^{3} + 3 \, {\left (B^{2} b c^{2} d + B^{2} a c d^{2}\right )} g i^{3} x^{2} + {\left (B^{2} b c^{3} + 3 \, B^{2} a c^{2} d\right )} g i^{3} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b d^{3} g i^{3} x^{4} + A B a c^{3} g i^{3} + {\left (3 \, A B b c d^{2} + A B a d^{3}\right )} g i^{3} x^{3} + 3 \, {\left (A B b c^{2} d + A B a c d^{2}\right )} g i^{3} x^{2} + {\left (A B b c^{3} + 3 \, A B a c^{2} d\right )} g i^{3} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right ) \left (d i x +c i \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 5.49, size = 3724, normalized size = 4.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (a\,g+b\,g\,x\right )\,{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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