Optimal. Leaf size=1172 \[ \frac {5 B^2 (b c-a d)^6 g^3 i^3 n^2 x}{84 b^3 d^3}+\frac {B^2 (b c-a d)^3 g^3 i^3 n^2 (a+b x)^4}{140 b^4}-\frac {29 B^2 (b c-a d)^5 g^3 i^3 n^2 (c+d x)^2}{840 b^2 d^4}+\frac {47 B^2 (b c-a d)^4 g^3 i^3 n^2 (c+d x)^3}{1260 b d^4}-\frac {13 B^2 (b c-a d)^3 g^3 i^3 n^2 (c+d x)^4}{420 d^4}+\frac {b B^2 (b c-a d)^2 g^3 i^3 n^2 (c+d x)^5}{105 d^4}-\frac {B (b c-a d)^4 g^3 i^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{210 b^4 d}-\frac {3 B (b c-a d)^3 g^3 i^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{140 b^4}-\frac {B (b c-a d)^2 g^3 i^3 n (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^3}+\frac {2 B (b c-a d)^4 g^3 i^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 b d^4}-\frac {3 B (b c-a d)^3 g^3 i^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{14 d^4}+\frac {6 b B (b c-a d)^2 g^3 i^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}-\frac {b^2 B (b c-a d) g^3 i^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {(b c-a d)^3 g^3 i^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{140 b^4}+\frac {(b c-a d)^2 g^3 i^3 (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{35 b^3}+\frac {(b c-a d) g^3 i^3 (a+b x)^4 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{14 b^2}+\frac {g^3 i^3 (a+b x)^4 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 b}+\frac {B (b c-a d)^5 g^3 i^3 n (a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{420 b^4 d^2}-\frac {B (b c-a d)^6 g^3 i^3 n (a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{420 b^4 d^3}-\frac {B (b c-a d)^7 g^3 i^3 n \left (6 A+11 B n+6 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 )}{420 b^4 d^4}-\frac {B^2 (b c-a d)^7 g^3 i^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{210 b^4 d^4}-\frac {11 B^2 (b c-a d)^7 g^3 i^3 n^2 \log (c+d x)}{420 b^4 d^4}-\frac {B^2 (b c-a d)^7 g^3 i^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{70 b^4 d^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.05, antiderivative size = 1172, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 14, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.311, Rules used = {2561, 2383,
2381, 2384, 2354, 2438, 2373, 45, 47, 37, 2382, 12, 79, 1634} \begin {gather*} -\frac {B g^3 i^3 n \left (6 A+11 B n+6 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)^7}{420 b^4 d^4}-\frac {B^2 g^3 i^3 n^2 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^7}{210 b^4 d^4}-\frac {11 B^2 g^3 i^3 n^2 \log (c+d x) (b c-a d)^7}{420 b^4 d^4}-\frac {B^2 g^3 i^3 n^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)^7}{70 b^4 d^4}+\frac {5 B^2 g^3 i^3 n^2 x (b c-a d)^6}{84 b^3 d^3}-\frac {B g^3 i^3 n (a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^6}{420 b^4 d^3}-\frac {29 B^2 g^3 i^3 n^2 (c+d x)^2 (b c-a d)^5}{840 b^2 d^4}+\frac {B g^3 i^3 n (a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^5}{420 b^4 d^2}+\frac {47 B^2 g^3 i^3 n^2 (c+d x)^3 (b c-a d)^4}{1260 b d^4}-\frac {B g^3 i^3 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)^4}{210 b^4 d}+\frac {2 B g^3 i^3 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)^4}{21 b d^4}+\frac {B^2 g^3 i^3 n^2 (a+b x)^4 (b c-a d)^3}{140 b^4}-\frac {13 B^2 g^3 i^3 n^2 (c+d x)^4 (b c-a d)^3}{420 d^4}+\frac {g^3 i^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^3}{140 b^4}-\frac {3 B g^3 i^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{140 b^4}-\frac {3 B g^3 i^3 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)^3}{14 d^4}+\frac {b B^2 g^3 i^3 n^2 (c+d x)^5 (b c-a d)^2}{105 d^4}+\frac {g^3 i^3 (a+b x)^4 (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)^2}{35 b^3}+\frac {6 b B g^3 i^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{35 d^4}-\frac {B g^3 i^3 n (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{35 b^3}+\frac {g^3 i^3 (a+b x)^4 (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)}{14 b^2}-\frac {b^2 B g^3 i^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{21 d^4}+\frac {g^3 i^3 (a+b x)^4 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 37
Rule 45
Rule 47
Rule 79
Rule 1634
Rule 2354
Rule 2373
Rule 2381
Rule 2382
Rule 2383
Rule 2384
Rule 2438
Rule 2561
Rubi steps
\begin {align*} \int (178 c+178 d x)^3 (a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (\frac {(-b c+a d)^3 g^3 (178 c+178 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^3}+\frac {3 b (b c-a d)^2 g^3 (178 c+178 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{178 d^3}-\frac {3 b^2 (b c-a d) g^3 (178 c+178 d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{31684 d^3}+\frac {b^3 g^3 (178 c+178 d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5639752 d^3}\right ) \, dx\\ &=\frac {\left (b^3 g^3\right ) \int (178 c+178 d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{5639752 d^3}-\frac {\left (3 b^2 (b c-a d) g^3\right ) \int (178 c+178 d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{31684 d^3}+\frac {\left (3 b (b c-a d)^2 g^3\right ) \int (178 c+178 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{178 d^3}-\frac {\left ((b c-a d)^3 g^3\right ) \int (178 c+178 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{d^3}\\ &=-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {\left (b^3 B g^3 n\right ) \int \frac {5661610866627712 (b c-a d) (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{3513565496 d^4}+\frac {\left (b^2 B (b c-a d) g^3 n\right ) \int \frac {31806802621504 (b c-a d) (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{5639752 d^4}-\frac {\left (3 b B (b c-a d)^2 g^3 n\right ) \int \frac {178689902368 (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}{79210 d^4}+\frac {\left (B (b c-a d)^3 g^3 n\right ) \int \frac {1003875856 (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}{356 d^4}\\ &=-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {\left (11279504 b^3 B (b c-a d) g^3 n\right ) \int \frac {(c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{7 d^4}+\frac {\left (5639752 b^2 B (b c-a d)^2 g^3 n\right ) \int \frac {(c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{d^4}-\frac {\left (33838512 b B (b c-a d)^3 g^3 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^4}+\frac {\left (2819876 B (b c-a d)^4 g^3 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^4}\\ &=-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {\left (11279504 b^3 B (b c-a d) g^3 n\right ) \int \left (\frac {d (b c-a d)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^6}+\frac {(b c-a d)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^6 (a+b x)}+\frac {d (b c-a d)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^5}+\frac {d (b c-a d)^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4}+\frac {d (b c-a d)^2 (c+d x)^3 \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)^4 \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)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{7 d^4}+\frac {\left (5639752 b^2 B (b c-a d)^2 g^3 n\right ) \int \left (\frac {d (b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^5}+\frac {(b c-a d)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^5 (a+b x)}+\frac {d (b c-a d)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4}+\frac {d (b c-a d)^2 (c+d x)^2 \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)^3 \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)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{d^4}-\frac {\left (33838512 b B (b c-a d)^3 g^3 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^4}+\frac {\left (2819876 B (b c-a d)^4 g^3 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^4}\\ &=-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {\left (11279504 b^2 B (b c-a d) g^3 n\right ) \int (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{7 d^3}-\frac {\left (11279504 b B (b c-a d)^2 g^3 n\right ) \int (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{7 d^3}+\frac {\left (5639752 b B (b c-a d)^2 g^3 n\right ) \int (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d^3}-\frac {\left (11279504 B (b c-a d)^3 g^3 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}{7 d^3}+\frac {\left (5639752 B (b c-a d)^3 g^3 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}{d^3}-\frac {\left (33838512 B (b c-a d)^3 g^3 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^3}-\frac {\left (11279504 B (b c-a d)^4 g^3 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}{7 b d^3}+\frac {\left (2819876 B (b c-a d)^4 g^3 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^3}+\frac {\left (5639752 B (b c-a d)^4 g^3 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^3}-\frac {\left (33838512 B (b c-a d)^4 g^3 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 b d^3}-\frac {\left (11279504 B (b c-a d)^5 g^3 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}{7 b^2 d^3}+\frac {\left (2819876 B (b c-a d)^5 g^3 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^3}+\frac {\left (5639752 B (b c-a d)^5 g^3 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^3}-\frac {\left (33838512 B (b c-a d)^5 g^3 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^2 d^3}-\frac {\left (11279504 B (b c-a d)^6 g^3 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{7 b^3 d^3}+\frac {\left (2819876 B (b c-a d)^6 g^3 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^3}+\frac {\left (5639752 B (b c-a d)^6 g^3 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^3}-\frac {\left (33838512 B (b c-a d)^6 g^3 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^3 d^3}-\frac {\left (11279504 B (b c-a d)^7 g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{7 b^3 d^4}+\frac {\left (2819876 B (b c-a d)^7 g^3 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^4}+\frac {\left (5639752 B (b c-a d)^7 g^3 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^4}-\frac {\left (33838512 B (b c-a d)^7 g^3 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^3 d^4}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {\left (11279504 B^2 (b c-a d)^6 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{7 b^3 d^3}+\frac {\left (2819876 B^2 (b c-a d)^6 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 d^3}+\frac {\left (5639752 B^2 (b c-a d)^6 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 d^3}-\frac {\left (33838512 B^2 (b c-a d)^6 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{5 b^3 d^3}+\frac {\left (5639752 b^2 B^2 (b c-a d) g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^5}{a+b x} \, dx}{21 d^4}+\frac {\left (11279504 b B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^4}{a+b x} \, dx}{35 d^4}-\frac {\left (5639752 b B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^4}{a+b x} \, dx}{5 d^4}+\frac {\left (2819876 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{7 d^4}-\frac {\left (1409938 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{d^4}+\frac {\left (8459628 B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{5 d^4}+\frac {\left (11279504 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{21 b d^4}-\frac {\left (2819876 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 b d^4}-\frac {\left (5639752 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 b d^4}+\frac {\left (11279504 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{5 b d^4}+\frac {\left (5639752 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{7 b^2 d^4}-\frac {\left (1409938 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^4}-\frac {\left (2819876 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^4}+\frac {\left (16919256 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{5 b^2 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{7 b^4 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{5 b^4 d^4}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {2819876 B^2 (b c-a d)^6 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{35 b^4 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}+\frac {\left (5639752 b^2 B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {(c+d x)^5}{a+b x} \, dx}{21 d^4}+\frac {\left (11279504 b B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{35 d^4}-\frac {\left (5639752 b B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{5 d^4}+\frac {\left (2819876 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{7 d^4}-\frac {\left (1409938 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{d^4}+\frac {\left (8459628 B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{5 d^4}+\frac {\left (11279504 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{21 b d^4}-\frac {\left (2819876 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 b d^4}-\frac {\left (5639752 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 b d^4}+\frac {\left (11279504 B^2 (b c-a d)^5 g^3 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{5 b d^4}+\frac {\left (5639752 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{7 b^2 d^4}-\frac {\left (1409938 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{b^2 d^4}-\frac {\left (2819876 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{b^2 d^4}+\frac {\left (16919256 B^2 (b c-a d)^6 g^3 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{5 b^2 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 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}{7 b^4 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 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^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 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^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 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^4 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{7 b^4 d^3}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^4 d^3}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^4 d^3}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{5 b^4 d^3}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {2819876 B^2 (b c-a d)^6 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{35 b^4 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (c+d x)}{35 b^4 d^4}+\frac {\left (5639752 b^2 B^2 (b c-a d)^2 g^3 n^2\right ) \int \left (\frac {d (b c-a d)^4}{b^5}+\frac {(b c-a d)^5}{b^5 (a+b x)}+\frac {d (b c-a d)^3 (c+d x)}{b^4}+\frac {d (b c-a d)^2 (c+d x)^2}{b^3}+\frac {d (b c-a d) (c+d x)^3}{b^2}+\frac {d (c+d x)^4}{b}\right ) \, dx}{21 d^4}+\frac {\left (11279504 b B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (\frac {d (b c-a d)^3}{b^4}+\frac {(b c-a d)^4}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x)}{b^3}+\frac {d (b c-a d) (c+d x)^2}{b^2}+\frac {d (c+d x)^3}{b}\right ) \, dx}{35 d^4}-\frac {\left (5639752 b B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (\frac {d (b c-a d)^3}{b^4}+\frac {(b c-a d)^4}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x)}{b^3}+\frac {d (b c-a d) (c+d x)^2}{b^2}+\frac {d (c+d x)^3}{b}\right ) \, dx}{5 d^4}+\frac {\left (2819876 B^2 (b c-a d)^4 g^3 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}{7 d^4}-\frac {\left (1409938 B^2 (b c-a d)^4 g^3 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^4}+\frac {\left (8459628 B^2 (b c-a d)^4 g^3 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}{5 d^4}+\frac {\left (11279504 B^2 (b c-a d)^5 g^3 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}{21 b d^4}-\frac {\left (2819876 B^2 (b c-a d)^5 g^3 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 b d^4}-\frac {\left (5639752 B^2 (b c-a d)^5 g^3 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 b d^4}+\frac {\left (11279504 B^2 (b c-a d)^5 g^3 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}{5 b d^4}+\frac {\left (5639752 B^2 (b c-a d)^6 g^3 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{7 b^2 d^4}-\frac {\left (1409938 B^2 (b c-a d)^6 g^3 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^4}-\frac {\left (2819876 B^2 (b c-a d)^6 g^3 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^4}+\frac {\left (16919256 B^2 (b c-a d)^6 g^3 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{5 b^2 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{7 b^3 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{5 b^3 d^4}-\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{7 b^4 d^3}+\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 d^3}+\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 d^3}-\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{5 b^4 d^3}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {2819876 B^2 (b c-a d)^6 g^3 n^2 x}{35 b^3 d^3}+\frac {2114907 B^2 (b c-a d)^5 g^3 n^2 (c+d x)^2}{35 b^2 d^4}+\frac {15509318 B^2 (b c-a d)^4 g^3 n^2 (c+d x)^3}{315 b d^4}-\frac {2819876 B^2 (b c-a d)^3 g^3 n^2 (c+d x)^4}{21 d^4}+\frac {5639752 b B^2 (b c-a d)^2 g^3 n^2 (c+d x)^5}{105 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^6 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{35 b^4 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (c+d x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{35 b^4 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{7 b^4 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{5 b^4 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{7 b^3 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^3 d^4}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {2819876 B^2 (b c-a d)^6 g^3 n^2 x}{35 b^3 d^3}+\frac {2114907 B^2 (b c-a d)^5 g^3 n^2 (c+d x)^2}{35 b^2 d^4}+\frac {15509318 B^2 (b c-a d)^4 g^3 n^2 (c+d x)^3}{315 b d^4}-\frac {2819876 B^2 (b c-a d)^3 g^3 n^2 (c+d x)^4}{21 d^4}+\frac {5639752 b B^2 (b c-a d)^2 g^3 n^2 (c+d x)^5}{105 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x)}{35 b^4 d^4}-\frac {1409938 B^2 (b c-a d)^7 g^3 n^2 \log ^2(a+b x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^6 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{35 b^4 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (c+d x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{35 b^4 d^4}+\frac {\left (11279504 B^2 (b c-a d)^7 g^3 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 )}{7 b^4 d^4}-\frac {\left (2819876 B^2 (b c-a d)^7 g^3 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^4 d^4}-\frac {\left (5639752 B^2 (b c-a d)^7 g^3 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^4 d^4}+\frac {\left (33838512 B^2 (b c-a d)^7 g^3 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^4 d^4}\\ &=\frac {2819876 A B (b c-a d)^6 g^3 n x}{35 b^3 d^3}+\frac {2819876 B^2 (b c-a d)^6 g^3 n^2 x}{35 b^3 d^3}+\frac {2114907 B^2 (b c-a d)^5 g^3 n^2 (c+d x)^2}{35 b^2 d^4}+\frac {15509318 B^2 (b c-a d)^4 g^3 n^2 (c+d x)^3}{315 b d^4}-\frac {2819876 B^2 (b c-a d)^3 g^3 n^2 (c+d x)^4}{21 d^4}+\frac {5639752 b B^2 (b c-a d)^2 g^3 n^2 (c+d x)^5}{105 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x)}{35 b^4 d^4}-\frac {1409938 B^2 (b c-a d)^7 g^3 n^2 \log ^2(a+b x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^6 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{35 b^4 d^3}+\frac {1409938 B (b c-a d)^5 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^2 d^4}+\frac {2819876 B (b c-a d)^4 g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{105 b d^4}-\frac {23968946 B (b c-a d)^3 g^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}+\frac {5639752 b B (b c-a d)^2 g^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac {5639752 b^2 B (b c-a d) g^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {2819876 B (b c-a d)^7 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^4 d^4}-\frac {1409938 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {16919256 b (b c-a d)^2 g^3 (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^4}-\frac {2819876 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^4}+\frac {5639752 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 d^4}-\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (c+d x)}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{35 b^4 d^4}+\frac {2819876 B^2 (b c-a d)^7 g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{35 b^4 d^4}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2448\) vs. \(2(1172)=2344\).
time = 2.20, size = 2448, normalized size = 2.09 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.37, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{3} \left (d i x +c i \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 6587 vs. \(2 (1070) = 2140\).
time = 1.02, size = 6587, normalized size = 5.62 \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 )}^3\,{\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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