Optimal. Leaf size=819 \[ -\frac {B^2 (b c-a d)^4 g^2 i^2 n^2 x}{10 b^2 d^2}-\frac {B^2 (b c-a d)^3 g^2 i^2 n^2 (c+d x)^2}{20 b d^3}+\frac {B^2 (b c-a d)^2 g^2 i^2 n^2 (c+d x)^3}{30 d^3}-\frac {B (b c-a d)^3 g^2 i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^3 d}-\frac {B (b c-a d)^2 g^2 i^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^3}-\frac {B (b c-a d)^3 g^2 i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^2 i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 d^3}-\frac {b B (b c-a d) g^2 i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}+\frac {(b c-a d)^2 g^2 i^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{30 b^3}+\frac {(b c-a d) g^2 i^2 (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{10 b^2}+\frac {g^2 i^2 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b}+\frac {B (b c-a d)^4 g^2 i^2 n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^3 d^2}+\frac {B (b c-a d)^5 g^2 i^2 n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{30 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{30 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 n^2 \log (c+d x)}{10 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{15 b^3 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.67, antiderivative size = 819, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 11, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {2561, 2383,
2381, 2384, 2354, 2438, 2373, 45, 2382, 12, 907} \begin {gather*} \frac {B g^2 i^2 n \left (2 A+3 B n+2 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 ) (b c-a d)^5}{30 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^5}{30 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 \log (c+d x) (b c-a d)^5}{10 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)^5}{15 b^3 d^3}-\frac {B^2 g^2 i^2 n^2 x (b c-a d)^4}{10 b^2 d^2}+\frac {B g^2 i^2 n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4}{30 b^3 d^2}-\frac {B^2 g^2 i^2 n^2 (c+d x)^2 (b c-a d)^3}{20 b d^3}-\frac {B g^2 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 ) (b c-a d)^3}{30 b^3 d}-\frac {B g^2 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 ) (b c-a d)^3}{5 b d^3}+\frac {B^2 g^2 i^2 n^2 (c+d x)^3 (b c-a d)^2}{30 d^3}+\frac {g^2 i^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^2}{30 b^3}-\frac {B g^2 i^2 n (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)^2}{15 b^3}+\frac {4 B g^2 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 ) (b c-a d)^2}{15 d^3}+\frac {g^2 i^2 (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)}{10 b^2}-\frac {b B g^2 i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{10 d^3}+\frac {g^2 i^2 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 907
Rule 2354
Rule 2373
Rule 2381
Rule 2382
Rule 2383
Rule 2384
Rule 2438
Rule 2561
Rubi steps
\begin {align*} \int (169 c+169 d x)^2 (a g+b g x)^2 \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)^2 g^2 (169 c+169 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac {2 b (b c-a d) g^2 (169 c+169 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{169 d^2}+\frac {b^2 g^2 (169 c+169 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{28561 d^2}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int (169 c+169 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{28561 d^2}-\frac {\left (2 b (b c-a d) g^2\right ) \int (169 c+169 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{169 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int (169 c+169 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{d^2}\\ &=\frac {28561 (b c-a d)^2 g^2 (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^3}-\frac {28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac {28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}-\frac {\left (2 b^2 B g^2 n\right ) \int \frac {137858491849 (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}{24134045 d^3}+\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {815730721 (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}{28561 d^3}-\frac {\left (2 B (b c-a d)^2 g^2 n\right ) \int \frac {4826809 (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}{507 d^3}\\ &=\frac {28561 (b c-a d)^2 g^2 (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^3}-\frac {28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac {28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}-\frac {\left (57122 b^2 B (b c-a d) g^2 n\right ) \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}{5 d^3}+\frac {\left (28561 b B (b c-a d)^2 g^2 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^3}-\frac {\left (57122 B (b c-a d)^3 g^2 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^3}\\ &=\frac {28561 (b c-a d)^2 g^2 (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^3}-\frac {28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac {28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}-\frac {\left (57122 b^2 B (b c-a d) g^2 n\right ) \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}{5 d^3}+\frac {\left (28561 b B (b c-a d)^2 g^2 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^3}-\frac {\left (57122 B (b c-a d)^3 g^2 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^3}\\ &=\frac {28561 (b c-a d)^2 g^2 (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^3}-\frac {28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac {28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}-\frac {\left (57122 b B (b c-a d) g^2 n\right ) \int (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 d^2}-\frac {\left (57122 B (b c-a d)^2 g^2 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}{5 d^2}+\frac {\left (28561 B (b c-a d)^2 g^2 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}{d^2}-\frac {\left (57122 B (b c-a d)^3 g^2 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}{5 b d^2}-\frac {\left (57122 B (b c-a d)^3 g^2 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^2}+\frac {\left (28561 B (b c-a d)^3 g^2 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^2}-\frac {\left (57122 B (b c-a d)^4 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^2 d^2}-\frac {\left (57122 B (b c-a d)^4 g^2 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^2}+\frac {\left (28561 B (b c-a d)^4 g^2 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^2}-\frac {\left (57122 B (b c-a d)^5 g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{5 b^2 d^3}-\frac {\left (57122 B (b c-a d)^5 g^2 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^3}+\frac {\left (28561 B (b c-a d)^5 g^2 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^3}\\ &=-\frac {28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac {28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac {28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac {28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac {28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac {28561 (b c-a d)^2 g^2 (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^3}-\frac {28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac {28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}-\frac {\left (57122 B^2 (b c-a d)^4 g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{5 b^2 d^2}-\frac {\left (57122 B^2 (b c-a d)^4 g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{3 b^2 d^2}+\frac {\left (28561 B^2 (b c-a d)^4 g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 d^2}+\frac {\left (28561 b B^2 (b c-a d) g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{10 d^3}+\frac {\left (57122 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{15 d^3}-\frac {\left (28561 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 d^3}+\frac {\left (28561 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{5 b d^3}+\frac {\left (28561 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{3 b d^3}-\frac {\left (28561 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{2 b d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 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}{5 b^3 d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 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^3}-\frac {\left (28561 B^2 (b c-a d)^5 g^2 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^3}\\ &=-\frac {28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac {28561 B^2 (b c-a d)^4 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac {28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac {28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac {28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac {28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac {28561 (b c-a d)^2 g^2 (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^3}-\frac {28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac {28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {\left (28561 b B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{10 d^3}+\frac {\left (57122 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{15 d^3}-\frac {\left (28561 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 d^3}+\frac {\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{5 b d^3}+\frac {\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{3 b d^3}-\frac {\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{2 b d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 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}{5 b^3 d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 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^3}-\frac {\left (28561 B^2 (b c-a d)^5 g^2 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^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{5 b^3 d^2}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{3 b^3 d^2}-\frac {\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^3 d^2}\\ &=-\frac {28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac {28561 B^2 (b c-a d)^4 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac {28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac {28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac {28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac {28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac {28561 (b c-a d)^2 g^2 (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^3}-\frac {28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac {28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log (c+d x)}{15 b^3 d^3}+\frac {\left (28561 b B^2 (b c-a d)^2 g^2 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}{10 d^3}+\frac {\left (57122 B^2 (b c-a d)^3 g^2 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}{15 d^3}-\frac {\left (28561 B^2 (b c-a d)^3 g^2 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^3}+\frac {\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{5 b d^3}+\frac {\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{3 b d^3}-\frac {\left (28561 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{2 b d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{5 b^2 d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b^2 d^3}-\frac {\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 d^3}-\frac {\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{5 b^3 d^2}-\frac {\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^3 d^2}+\frac {\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 d^2}\\ &=-\frac {28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac {28561 B^2 (b c-a d)^4 g^2 n^2 x}{15 b^2 d^2}-\frac {28561 B^2 (b c-a d)^3 g^2 n^2 (c+d x)^2}{20 b d^3}+\frac {28561 B^2 (b c-a d)^2 g^2 n^2 (c+d x)^3}{30 d^3}-\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x)}{15 b^3 d^3}-\frac {28561 B^2 (b c-a d)^4 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac {28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac {28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac {28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac {28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac {28561 (b c-a d)^2 g^2 (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^3}-\frac {28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac {28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log (c+d x)}{15 b^3 d^3}-\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{15 b^3 d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{5 b^3 d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 d^3}-\frac {\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^2 d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 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^3}-\frac {\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 d^3}\\ &=-\frac {28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac {28561 B^2 (b c-a d)^4 g^2 n^2 x}{15 b^2 d^2}-\frac {28561 B^2 (b c-a d)^3 g^2 n^2 (c+d x)^2}{20 b d^3}+\frac {28561 B^2 (b c-a d)^2 g^2 n^2 (c+d x)^3}{30 d^3}-\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x)}{15 b^3 d^3}+\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log ^2(a+b x)}{30 b^3 d^3}-\frac {28561 B^2 (b c-a d)^4 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac {28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac {28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac {28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac {28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac {28561 (b c-a d)^2 g^2 (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^3}-\frac {28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac {28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log (c+d x)}{15 b^3 d^3}-\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{15 b^3 d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 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 )}{5 b^3 d^3}+\frac {\left (57122 B^2 (b c-a d)^5 g^2 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^3}-\frac {\left (28561 B^2 (b c-a d)^5 g^2 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^3}\\ &=-\frac {28561 A B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac {28561 B^2 (b c-a d)^4 g^2 n^2 x}{15 b^2 d^2}-\frac {28561 B^2 (b c-a d)^3 g^2 n^2 (c+d x)^2}{20 b d^3}+\frac {28561 B^2 (b c-a d)^2 g^2 n^2 (c+d x)^3}{30 d^3}-\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x)}{15 b^3 d^3}+\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log ^2(a+b x)}{30 b^3 d^3}-\frac {28561 B^2 (b c-a d)^4 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac {28561 B (b c-a d)^3 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b d^3}+\frac {28561 B (b c-a d)^2 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac {28561 b B (b c-a d) g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}-\frac {28561 B (b c-a d)^5 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^3 d^3}+\frac {28561 (b c-a d)^2 g^2 (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^3}-\frac {28561 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3}+\frac {28561 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log (c+d x)}{15 b^3 d^3}-\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{15 b^3 d^3}-\frac {28561 B^2 (b c-a d)^5 g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{15 b^3 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 1254, normalized size = 1.53 \begin {gather*} \frac {g^2 i^2 \left (20 d^3 (b c-a d)^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+30 d^4 (b c-a d) (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+12 d^5 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+20 B (b c-a d)^3 n \left (2 A b d (b c-a d) 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 )-d^2 (a+b x)^2 \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)-2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n (b d x+(-b c+a d) \log (c+d x))+B (b c-a d)^2 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )-10 B (b c-a d)^2 n \left (6 A b d (b c-a d)^2 x+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 d^2 (-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 )+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 )-6 B (b c-a d)^3 n \log (c+d x)-6 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 n (b d x+(-b c+a d) \log (c+d x))+3 B (b c-a d)^3 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+B (b c-a d) n \left (24 A b d (b c-a d)^3 x+24 B d (b c-a d)^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-12 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+8 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-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 )-24 B (b c-a d)^4 n \log (c+d x)-24 (b c-a d)^4 \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 c-a d)^2 n \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+B (b c-a d) n \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+12 B (b c-a d)^3 n (b d x+(-b c+a d) \log (c+d x))+12 B (b c-a d)^4 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{60 b^3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}\, dx\]
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. 3499 vs.
\(2 (748) = 1496\).
time = 0.81, size = 3499, normalized size = 4.27 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 )}^2\,{\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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