Optimal. Leaf size=819 \[ \frac {B g^2 i^2 n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right ) (b c-a d)^5}{30 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^5}{30 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 \log (c+d x) (b c-a d)^5}{10 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)^5}{15 b^3 d^3}-\frac {B^2 g^2 i^2 n^2 x (b c-a d)^4}{10 b^2 d^2}+\frac {B g^2 i^2 n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4}{30 b^3 d^2}-\frac {B^2 g^2 i^2 n^2 (c+d x)^2 (b c-a d)^3}{20 b d^3}-\frac {B g^2 i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{30 b^3 d}-\frac {B g^2 i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{5 b d^3}+\frac {B^2 g^2 i^2 n^2 (c+d x)^3 (b c-a d)^2}{30 d^3}+\frac {g^2 i^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^2}{30 b^3}-\frac {B g^2 i^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{15 b^3}+\frac {4 B g^2 i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{15 d^3}+\frac {g^2 i^2 (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)}{10 b^2}-\frac {b B g^2 i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{10 d^3}+\frac {g^2 i^2 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.61, antiderivative size = 714, normalized size of antiderivative = 0.87, number of steps used = 71, number of rules used = 13, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2528, 2525, 12, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {B^2 g^2 i^2 n^2 (b c-a d)^5 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{15 b^3 d^3}-\frac {B g^2 i^2 n (b c-a d)^5 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{15 b^3 d^3}+\frac {d^2 g^2 i^2 (a+b x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b^3}+\frac {A B g^2 i^2 n x (b c-a d)^4}{15 b^2 d^2}-\frac {B g^2 i^2 n (a+b x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{30 b^3 d}+\frac {g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b^3}-\frac {B g^2 i^2 n (a+b x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^3}+\frac {d g^2 i^2 (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b^3}-\frac {B d g^2 i^2 n (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^3}+\frac {B^2 g^2 i^2 n (a+b x) (b c-a d)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{15 b^3 d^2}-\frac {B^2 g^2 i^2 n^2 x (b c-a d)^4}{15 b^2 d^2}-\frac {B^2 g^2 i^2 n^2 (b c-a d)^5 \log ^2(c+d x)}{30 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 (b c-a d)^5 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{15 b^3 d^3}+\frac {B^2 g^2 i^2 n^2 (a+b x)^2 (b c-a d)^3}{20 b^3 d}+\frac {B^2 g^2 i^2 n^2 (a+b x)^3 (b c-a d)^2}{30 b^3} \]
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 (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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b^3 d^3}-\frac {\left (28561 B^2 (b c-a d)^5 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 d^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*}
________________________________________________________________________________________
Mathematica [A] time = 1.02, size = 1254, normalized size = 1.53 \[ \frac {g^2 i^2 \left (12 d^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^5+30 d^4 (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+20 d^3 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3+20 B (b c-a d)^3 n \left (-2 B n \log (c+d x) (b c-a d)^2-2 \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)^2+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 {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right ) (b c-a d)^2+2 A b d x (b c-a d)+2 B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) (b c-a d)+B n (b d x+(a d-b c) \log (c+d x)) (b c-a d)-d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )-10 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 {Li}_2\left (\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 )+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 {Li}_2\left (\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 )\right )}{60 b^3 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b^{2} d^{2} g^{2} i^{2} x^{4} + A^{2} a^{2} c^{2} g^{2} i^{2} + 2 \, {\left (A^{2} b^{2} c d + A^{2} a b d^{2}\right )} g^{2} i^{2} x^{3} + {\left (A^{2} b^{2} c^{2} + 4 \, A^{2} a b c d + A^{2} a^{2} d^{2}\right )} g^{2} i^{2} x^{2} + 2 \, {\left (A^{2} a b c^{2} + A^{2} a^{2} c d\right )} g^{2} i^{2} x + {\left (B^{2} b^{2} d^{2} g^{2} i^{2} x^{4} + B^{2} a^{2} c^{2} g^{2} i^{2} + 2 \, {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} g^{2} i^{2} x^{3} + {\left (B^{2} b^{2} c^{2} + 4 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} g^{2} i^{2} x^{2} + 2 \, {\left (B^{2} a b c^{2} + B^{2} a^{2} c d\right )} g^{2} i^{2} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b^{2} d^{2} g^{2} i^{2} x^{4} + A B a^{2} c^{2} g^{2} i^{2} + 2 \, {\left (A B b^{2} c d + A B a b d^{2}\right )} g^{2} i^{2} x^{3} + {\left (A B b^{2} c^{2} + 4 \, A B a b c d + A B a^{2} d^{2}\right )} g^{2} i^{2} x^{2} + 2 \, {\left (A B a b c^{2} + A B a^{2} c d\right )} g^{2} i^{2} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 7.58, size = 4247, normalized size = 5.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a\,g+b\,g\,x\right )}^2\,{\left (c\,i+d\,i\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________