Optimal. Leaf size=441 \[ -\frac {2 b^2 B g^2 n \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i^3}-\frac {b^2 g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^3 i^3}-\frac {b g^2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^2 i^3 (c+d x)}+\frac {2 A b B g^2 n (a+b x)}{d^2 i^3 (c+d x)}-\frac {g^2 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d i^3 (c+d x)^2}+\frac {B g^2 n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d i^3 (c+d x)^2}+\frac {2 b^2 B^2 g^2 n^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^3}+\frac {2 b B^2 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^2 i^3 (c+d x)}-\frac {2 b B^2 g^2 n^2 (a+b x)}{d^2 i^3 (c+d x)}-\frac {B^2 g^2 n^2 (a+b x)^2}{4 d i^3 (c+d x)^2} \]
[Out]
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Rubi [B] time = 5.12, antiderivative size = 1435, normalized size of antiderivative = 3.25, number of steps used = 97, number of rules used = 25, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2528, 2525, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44, 6688, 6742, 2500, 2433, 2375, 2317, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 44
Rule 2301
Rule 2302
Rule 2317
Rule 2374
Rule 2375
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2418
Rule 2433
Rule 2434
Rule 2440
Rule 2499
Rule 2500
Rule 2524
Rule 2525
Rule 2528
Rule 6589
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int \frac {(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}{(203 c+203 d x)^3} \, dx &=\int \left (\frac {(-b c+a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^2 (c+d x)^3}-\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^2 (c+d x)^2}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^2 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x} \, dx}{8365427 d^2}-\frac {\left (2 b (b c-a d) g^2\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2} \, dx}{8365427 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3} \, dx}{8365427 d^2}\\ &=-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}-\frac {\left (2 b^2 B g^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{8365427 d^3}-\frac {\left (4 b B (b c-a d) g^2 n\right ) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^2} \, dx}{8365427 d^3}+\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^3} \, dx}{8365427 d^3}\\ &=-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}-\frac {\left (2 b^2 B g^2 n\right ) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{8365427 d^3}-\frac {\left (4 b B (b c-a d)^2 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) (c+d x)^2} \, dx}{8365427 d^3}+\frac {\left (B (b c-a d)^3 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) (c+d x)^3} \, dx}{8365427 d^3}\\ &=-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}-\frac {\left (2 b^2 B (b c-a d) g^2 n\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{8365427 d^3}-\frac {\left (4 b B (b c-a d)^2 g^2 n\right ) \int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{8365427 d^3}+\frac {\left (B (b c-a d)^3 g^2 n\right ) \int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{8365427 d^3}\\ &=-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}+\frac {\left (b^3 B 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}{8365427 d^3}-\frac {\left (4 b^3 B 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}{8365427 d^3}-\frac {\left (b^2 B g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{8365427 d^2}+\frac {\left (4 b^2 B g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{8365427 d^2}-\frac {\left (2 b^2 B (b c-a d) g^2 n\right ) \int \left (\frac {b \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) (a+b x)}-\frac {d \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) (c+d x)}\right ) \, dx}{8365427 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{8365427 d^2}+\frac {\left (4 b B (b c-a d) g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{8365427 d^2}-\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{8365427 d^2}\\ &=\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}-\frac {\left (2 b^3 B g^2 n\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{8365427 d^3}+\frac {\left (2 b^2 B g^2 n\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{c+d x} \, dx}{8365427 d^2}-\frac {\left (b^2 B^2 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}{8365427 d^3}+\frac {\left (b^2 B^2 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 (c+d x)}{a+b x} \, dx}{8365427 d^3}+\frac {\left (4 b^2 B^2 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}{8365427 d^3}-\frac {\left (4 b^2 B^2 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 (c+d x)}{a+b x} \, dx}{8365427 d^3}-\frac {\left (b B^2 (b c-a d) g^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8365427 d^3}+\frac {\left (4 b B^2 (b c-a d) g^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8365427 d^3}-\frac {\left (B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{16730854 d^3}\\ &=\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}-\frac {\left (2 b^3 B g^2 n\right ) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x}\right ) \, dx}{8365427 d^3}+\frac {\left (2 b^2 B g^2 n\right ) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x}\right ) \, dx}{8365427 d^2}-\frac {\left (b^2 B^2 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}{8365427 d^3}+\frac {\left (b^2 B^2 g^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{8365427 d^3}+\frac {\left (4 b^2 B^2 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}{8365427 d^3}-\frac {\left (4 b^2 B^2 g^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{8365427 d^3}-\frac {\left (b B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8365427 d^3}+\frac {\left (4 b B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8365427 d^3}-\frac {\left (B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{16730854 d^3}\\ &=\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}-\frac {\left (2 A b^3 B g^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8365427 d^3}-\frac {\left (2 b^3 B^2 g^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x} \, dx}{8365427 d^3}+\frac {\left (2 A b^2 B g^2 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8365427 d^2}+\frac {\left (2 b^2 B^2 g^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{8365427 d^2}-\frac {\left (b^3 B^2 g^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8365427 d^3}+\frac {\left (b^3 B^2 g^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8365427 d^3}+\frac {\left (4 b^3 B^2 g^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8365427 d^3}-\frac {\left (4 b^3 B^2 g^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8365427 d^3}+\frac {\left (b^2 B^2 g^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8365427 d^2}-\frac {\left (b^2 B^2 g^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8365427 d^2}-\frac {\left (4 b^2 B^2 g^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8365427 d^2}+\frac {\left (4 b^2 B^2 g^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8365427 d^2}-\frac {\left (b B^2 (b c-a d)^2 g^2 n^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{8365427 d^3}+\frac {\left (4 b B^2 (b c-a d)^2 g^2 n^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{8365427 d^3}-\frac {\left (B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{16730854 d^3}\\ &=-\frac {B^2 (b c-a d)^2 g^2 n^2}{33461708 d^3 (c+d x)^2}+\frac {5 b B^2 (b c-a d) g^2 n^2}{16730854 d^3 (c+d x)}+\frac {5 b^2 B^2 g^2 n^2 \log (a+b x)}{16730854 d^3}+\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}-\frac {5 b^2 B^2 g^2 n^2 \log (c+d x)}{16730854 d^3}-\frac {2 A b^2 B g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {\left (2 A b^2 B g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8365427 d^3}-\frac {\left (2 b^3 B^2 g^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{8365427 d^3}-\frac {\left (2 b^3 B^2 g^2 n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{8365427 d^3}+\frac {\left (2 A b^2 B g^2 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8365427 d^2}-\frac {\left (b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8365427 d^3}-\frac {\left (b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8365427 d^3}+\frac {\left (4 b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8365427 d^3}+\frac {\left (4 b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8365427 d^3}-\frac {\left (b^3 B^2 g^2 n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{8365427 d^3}-\frac {\left (b^3 B^2 g^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8365427 d^3}+\frac {\left (4 b^3 B^2 g^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8365427 d^3}-\frac {\left (b^2 B^2 g^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8365427 d^2}+\frac {\left (b^2 B^2 g^2 n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{8365427 d^2}+\frac {\left (4 b^2 B^2 g^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8365427 d^2}-\frac {\left (2 b^3 B^2 g^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8365427 d^3}\\ &=-\frac {B^2 (b c-a d)^2 g^2 n^2}{33461708 d^3 (c+d x)^2}+\frac {5 b B^2 (b c-a d) g^2 n^2}{16730854 d^3 (c+d x)}+\frac {5 b^2 B^2 g^2 n^2 \log (a+b x)}{16730854 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(a+b x)}{16730854 d^3}+\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}-\frac {5 b^2 B^2 g^2 n^2 \log (c+d x)}{16730854 d^3}-\frac {2 A b^2 B g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}+\frac {A b^2 B g^2 n \log ^2(c+d x)}{8365427 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(c+d x)}{16730854 d^3}-\frac {b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{8365427 d^3}+\frac {\left (2 A b^2 B g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}-\frac {\left (2 b^2 B^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{8365427 d^3}-\frac {\left (2 b^2 B^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{8365427 d^3}+\frac {\left (b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{8365427 d^3}-\frac {\left (b^2 B^2 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 )}{8365427 d^3}-\frac {\left (b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}+\frac {\left (4 b^2 B^2 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 )}{8365427 d^3}+\frac {\left (4 b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}+\frac {\left (2 b^2 B^2 g^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{8365427 d^2}+\frac {\left (2 b^2 B^2 g^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8365427 d^2}\\ &=-\frac {B^2 (b c-a d)^2 g^2 n^2}{33461708 d^3 (c+d x)^2}+\frac {5 b B^2 (b c-a d) g^2 n^2}{16730854 d^3 (c+d x)}+\frac {5 b^2 B^2 g^2 n^2 \log (a+b x)}{16730854 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(a+b x)}{16730854 d^3}+\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}-\frac {5 b^2 B^2 g^2 n^2 \log (c+d x)}{16730854 d^3}-\frac {2 A b^2 B g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8365427 d^3}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}+\frac {A b^2 B g^2 n \log ^2(c+d x)}{8365427 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(c+d x)}{16730854 d^3}-\frac {b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 A b^2 B g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {\left (b B^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{8365427 d^2}+\frac {\left (2 b B^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{8365427 d^2}+\frac {\left (b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{8365427 d^3}+\frac {\left (2 b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}-\frac {\left (2 b B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{8365427 d^2}+\frac {\left (2 b^2 B^2 g^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}\\ &=-\frac {B^2 (b c-a d)^2 g^2 n^2}{33461708 d^3 (c+d x)^2}+\frac {5 b B^2 (b c-a d) g^2 n^2}{16730854 d^3 (c+d x)}+\frac {5 b^2 B^2 g^2 n^2 \log (a+b x)}{16730854 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(a+b x)}{16730854 d^3}+\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}-\frac {5 b^2 B^2 g^2 n^2 \log (c+d x)}{16730854 d^3}-\frac {2 A b^2 B g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8365427 d^3}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}+\frac {A b^2 B g^2 n \log ^2(c+d x)}{8365427 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(c+d x)}{16730854 d^3}-\frac {b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n^2 \log ^3(c+d x)}{25096281 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 A b^2 B g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {\left (2 b^2 B^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}-\frac {\left (2 b^2 B^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{8365427 d^3}-\frac {\left (2 b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}+\frac {\left (2 b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}\\ &=-\frac {B^2 (b c-a d)^2 g^2 n^2}{33461708 d^3 (c+d x)^2}+\frac {5 b B^2 (b c-a d) g^2 n^2}{16730854 d^3 (c+d x)}+\frac {5 b^2 B^2 g^2 n^2 \log (a+b x)}{16730854 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(a+b x)}{16730854 d^3}+\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}-\frac {5 b^2 B^2 g^2 n^2 \log (c+d x)}{16730854 d^3}-\frac {2 A b^2 B g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8365427 d^3}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}+\frac {A b^2 B g^2 n \log ^2(c+d x)}{8365427 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(c+d x)}{16730854 d^3}-\frac {b^2 B^2 g^2 n^2 \log (a+b x) \log ^2(c+d x)}{8365427 d^3}-\frac {b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n^2 \log ^3(c+d x)}{25096281 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{8365427 d^3}-\frac {b^2 B^2 g^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 A b^2 B g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {\left (b^3 B^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{8365427 d^4}+\frac {\left (b^3 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{8365427 d^4}-\frac {\left (2 b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{8365427 d^3}\\ &=-\frac {B^2 (b c-a d)^2 g^2 n^2}{33461708 d^3 (c+d x)^2}+\frac {5 b B^2 (b c-a d) g^2 n^2}{16730854 d^3 (c+d x)}+\frac {5 b^2 B^2 g^2 n^2 \log (a+b x)}{16730854 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(a+b x)}{16730854 d^3}+\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}-\frac {5 b^2 B^2 g^2 n^2 \log (c+d x)}{16730854 d^3}-\frac {2 A b^2 B g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8365427 d^3}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}+\frac {A b^2 B g^2 n \log ^2(c+d x)}{8365427 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(c+d x)}{16730854 d^3}-\frac {b^2 B^2 g^2 n^2 \log (a+b x) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n^2 \log ^3(c+d x)}{25096281 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{8365427 d^3}-\frac {b^2 B^2 g^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8365427 d^3}+\frac {b^2 B^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 A b^2 B g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {\left (2 b^2 B^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}-\frac {\left (2 b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}\\ &=-\frac {B^2 (b c-a d)^2 g^2 n^2}{33461708 d^3 (c+d x)^2}+\frac {5 b B^2 (b c-a d) g^2 n^2}{16730854 d^3 (c+d x)}+\frac {5 b^2 B^2 g^2 n^2 \log (a+b x)}{16730854 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(a+b x)}{16730854 d^3}+\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}-\frac {5 b^2 B^2 g^2 n^2 \log (c+d x)}{16730854 d^3}-\frac {2 A b^2 B g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8365427 d^3}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}+\frac {A b^2 B g^2 n \log ^2(c+d x)}{8365427 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(c+d x)}{16730854 d^3}-\frac {b^2 B^2 g^2 n^2 \log (a+b x) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n^2 \log ^3(c+d x)}{25096281 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{8365427 d^3}-\frac {b^2 B^2 g^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8365427 d^3}+\frac {b^2 B^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 A b^2 B g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-2 \frac {\left (2 b^2 B^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8365427 d^3}\\ &=-\frac {B^2 (b c-a d)^2 g^2 n^2}{33461708 d^3 (c+d x)^2}+\frac {5 b B^2 (b c-a d) g^2 n^2}{16730854 d^3 (c+d x)}+\frac {5 b^2 B^2 g^2 n^2 \log (a+b x)}{16730854 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(a+b x)}{16730854 d^3}+\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16730854 d^3 (c+d x)^2}-\frac {3 b B (b c-a d) g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8365427 d^3 (c+d x)}-\frac {3 b^2 B 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 )}{8365427 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16730854 d^3 (c+d x)^2}+\frac {2 b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8365427 d^3 (c+d x)}-\frac {5 b^2 B^2 g^2 n^2 \log (c+d x)}{16730854 d^3}-\frac {2 A b^2 B g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8365427 d^3}-\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8365427 d^3}+\frac {3 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8365427 d^3}+\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8365427 d^3}+\frac {A b^2 B g^2 n \log ^2(c+d x)}{8365427 d^3}+\frac {3 b^2 B^2 g^2 n^2 \log ^2(c+d x)}{16730854 d^3}-\frac {b^2 B^2 g^2 n^2 \log (a+b x) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8365427 d^3}+\frac {b^2 B^2 g^2 n^2 \log ^3(c+d x)}{25096281 d^3}-\frac {3 b^2 B^2 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {b^2 B^2 g^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{8365427 d^3}-\frac {b^2 B^2 g^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8365427 d^3}+\frac {b^2 B^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 A b^2 B g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {3 b^2 B^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}+\frac {2 b^2 B^2 g^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{8365427 d^3}-\frac {2 b^2 B^2 g^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{8365427 d^3}\\ \end {align*}
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Mathematica [B] time = 6.59, size = 3172, normalized size = 7.19 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} b^{2} g^{2} x^{2} + 2 \, A^{2} a b g^{2} x + A^{2} a^{2} g^{2} + {\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b^{2} g^{2} x^{2} + 2 \, A B a b g^{2} x + A B a^{2} g^{2}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{d^{3} i^{3} x^{3} + 3 \, c d^{2} i^{3} x^{2} + 3 \, c^{2} d i^{3} x + c^{3} i^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (d i x +c i \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (c\,i+d\,i\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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