Optimal. Leaf size=635 \[ \frac {B^2 (b c-a d)^3 g i^2 n^2 x}{12 b^2 d}+\frac {B^2 (b c-a d)^2 g i^2 n^2 (c+d x)^2}{12 b d^2}-\frac {B (b c-a d)^3 g i^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3 d}-\frac {B (b c-a d)^2 g i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3}+\frac {B (b c-a d)^2 g i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b d^2}-\frac {B (b c-a d) g i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^2}+\frac {(b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^3}+\frac {(b c-a d) g i^2 (a+b x)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}-\frac {B (b c-a d)^4 g i^2 n \left (A+B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{6 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \log (c+d x)}{4 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^3 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.44, antiderivative size = 635, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2561, 2383,
2381, 2384, 2354, 2438, 2373, 45, 2382, 12, 78} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 78
Rule 2354
Rule 2373
Rule 2381
Rule 2382
Rule 2383
Rule 2384
Rule 2438
Rule 2561
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.42, size = 713, normalized size = 1.12 \begin {gather*} \frac {g i^2 \left (-4 (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 )^2+3 b (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {4 B (b c-a d)^2 n \left (2 A b d (b c-a d) x-B (b c-a d) n (b d x+(b c-a d) \log (a+b x))+2 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 n \log (c+d x)-B (b c-a d)^2 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)}{-b c+a d}\right )\right )\right )}{b^3}-\frac {B (b c-a d) n \left (6 A b d (b c-a d)^2 x-3 B (b c-a d)^2 n (b d x+(b c-a d) \log (a+b x))-B (b c-a d) n \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )+6 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 b^2 (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 )+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 )+6 (b c-a d)^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^3 n \log (c+d x)-3 B (b c-a d)^3 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)}{-b c+a d}\right )\right )\right )}{b^3}\right )}{12 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right ) \left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2166 vs.
\(2 (577) = 1154\).
time = 0.80, size = 2166, normalized size = 3.41 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} g i^{2} \left (\int A^{2} a c^{2}\, dx + \int A^{2} a d^{2} x^{2}\, dx + \int A^{2} b c^{2} x\, dx + \int A^{2} b d^{2} x^{3}\, dx + \int B^{2} a c^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 A B a c^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 A^{2} a c d x\, dx + \int 2 A^{2} b c d x^{2}\, dx + \int B^{2} a d^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int B^{2} b c^{2} x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int B^{2} b d^{2} x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 A B a d^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 A B b c^{2} x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 A B b d^{2} x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 2 B^{2} a c d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 B^{2} b c d x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 4 A B a c d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int 4 A B b c d x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a\,g+b\,g\,x\right )\,{\left (c\,i+d\,i\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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