Optimal. Leaf size=282 \[ -\frac {2 b B g n \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i^2}-\frac {b g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^2 i^2}-\frac {g (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i^2 (c+d x)}+\frac {2 A B g n (a+b x)}{d i^2 (c+d x)}+\frac {2 b B^2 g n^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i^2}+\frac {2 B^2 g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d i^2 (c+d x)}-\frac {2 B^2 g n^2 (a+b x)}{d i^2 (c+d x)} \]
[Out]
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Rubi [B] time = 4.17, antiderivative size = 1157, normalized size of antiderivative = 4.10, number of steps used = 69, number of rules used = 25, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, 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} \[ \frac {b B^2 g n^2 \log ^3(c+d x)}{3 d^2 i^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{d^2 i^2}+\frac {A b B g n \log ^2(c+d x)}{d^2 i^2}-\frac {b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{d^2 i^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{d^2 i^2}-\frac {2 b B^2 g n^2 \log (c+d x)}{d^2 i^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{d^2 i^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{d^2 i^2}-\frac {2 b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 i^2}-\frac {2 A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 i^2}+\frac {2 b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d^2 i^2}-\frac {2 b B^2 g n \log (a+b x) \log \left ((c+d x)^{-n}\right ) \log (c+d x)}{d^2 i^2}+\frac {2 b B^2 g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \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 ) \log (c+d x)}{d^2 i^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{d^2 i^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2 i^2 (c+d x)}-\frac {b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d^2 i^2}+\frac {b B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{d^2 i^2}+\frac {2 b B^2 g n^2 \log (a+b x)}{d^2 i^2}-\frac {2 b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2 i^2}-\frac {2 B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2 i^2 (c+d x)}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^2}-\frac {2 b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^2}-\frac {2 b B^2 g n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i^2}+\frac {2 b B^2 g n \log \left ((a+b x)^n\right ) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i^2}-\frac {2 b B^2 g n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^2}-\frac {2 A b B g n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^2}-\frac {2 b B^2 g n \log \left ((c+d x)^{-n}\right ) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^2}+\frac {2 b B^2 g 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 {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^2}-\frac {2 b B^2 g n^2 \text {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i^2}-\frac {2 b B^2 g n^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^2}+\frac {2 B^2 (b c-a d) g n^2}{d^2 i^2 (c+d x)} \]
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) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(196 c+196 d x)^2} \, dx &=\int \left (\frac {(-b c+a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d (c+d x)^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d (c+d x)}\right ) \, dx\\ &=\frac {(b g) \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}{38416 d}-\frac {((b c-a d) g) \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}{38416 d}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {(b B g n) \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}{19208 d^2}-\frac {(B (b c-a d) g n) \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}{19208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {(b B g n) \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}{19208 d^2}-\frac {\left (B (b c-a d)^2 g 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}{19208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {(b B (b c-a d) g n) \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}{19208 d^2}-\frac {\left (B (b c-a d)^2 g 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}{19208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {\left (b^2 B g n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{19208 d^2}+\frac {(b B g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{19208 d}-\frac {(b B (b c-a d) g n) \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}{19208 d^2}+\frac {(B (b c-a d) g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{19208 d}\\ &=-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {\left (b^2 B g 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}{19208 d^2}+\frac {(b B g n) \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}{19208 d}+\frac {\left (b B^2 g n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{19208 d^2}-\frac {\left (b B^2 g n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{19208 d^2}+\frac {\left (B^2 (b c-a d) g n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{19208 d^2}\\ &=-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {\left (b^2 B g 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}{19208 d^2}+\frac {(b B g n) \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}{19208 d}+\frac {\left (b B^2 g n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{19208 d^2}-\frac {\left (b B^2 g 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}{19208 d^2}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{19208 d^2}\\ &=-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {\left (A b^2 B g n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{19208 d^2}-\frac {\left (b^2 B^2 g 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}{19208 d^2}+\frac {(A b B g n) \int \frac {\log (c+d x)}{c+d x} \, dx}{19208 d}+\frac {\left (b B^2 g 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}{19208 d}+\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{19208 d^2}-\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{19208 d^2}-\frac {\left (b B^2 g n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{19208 d}+\frac {\left (b B^2 g n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{19208 d}+\frac {\left (B^2 (b c-a d)^2 g 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}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {(A b B g n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{19208 d^2}-\frac {\left (b^2 B^2 g n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{19208 d^2}-\frac {\left (b^2 B^2 g n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{19208 d^2}+\frac {(A b B g n) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19208 d}+\frac {\left (b B^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{19208 d^2}+\frac {\left (b B^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{19208 d^2}-\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{38416 d^2}+\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{19208 d^2}+\frac {\left (b B^2 g n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{38416 d}+\frac {\left (b B^2 g n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19208 d}-\frac {\left (b^2 B^2 g 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}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}+\frac {(A b B g n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19208 d^2}-\frac {\left (b B^2 g 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 )}{19208 d^2}-\frac {\left (b B^2 g 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 )}{19208 d^2}+\frac {\left (b B^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{38416 d^2}+\frac {\left (b B^2 g 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 )}{19208 d^2}+\frac {\left (b B^2 g 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 )}{19208 d^2}+\frac {\left (b B^2 g 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}{19208 d}+\frac {\left (b B^2 g 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}{19208 d}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {\left (B^2 g\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 )}{38416 d}+\frac {\left (B^2 g 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 )}{19208 d}+\frac {\left (b B^2 g n^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{38416 d^2}+\frac {\left (b B^2 g 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 )}{19208 d^2}-\frac {\left (B^2 g 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 )}{19208 d}+\frac {\left (b B^2 g 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 )}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}+\frac {\left (b B^2 g 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 )}{19208 d^2}-\frac {\left (b B^2 g 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 )}{19208 d^2}-\frac {\left (b B^2 g 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 )}{19208 d^2}+\frac {\left (b B^2 g 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 )}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac {b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}+\frac {b B^2 g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {\left (b^2 B^2 g\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 )}{38416 d^3}+\frac {\left (b^2 B^2 g 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 )}{38416 d^3}-\frac {\left (b B^2 g 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 )}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac {b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {\left (b B^2 g 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 )}{19208 d^2}-\frac {\left (b B^2 g 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 )}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac {b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-2 \frac {\left (b B^2 g 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 )}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac {b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}\\ \end {align*}
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Mathematica [B] time = 1.99, size = 1261, normalized size = 4.47 \[ \frac {g \left (-\frac {a B^2 d \left ((b c-a d) \log ^2\left (\frac {a+b x}{c+d x}\right )-2 (b c-a d) \log \left (\frac {a+b x}{c+d x}\right )-2 b (c+d x) \log (a+b x) \log \left (\frac {a+b x}{c+d x}\right )-2 b (c+d x) \log \left (\frac {b c-a d}{b c+b d x}\right ) \log \left (\frac {a+b x}{c+d x}\right )+2 b c-2 a d+2 b (c+d x) \log (a+b x)-2 b (c+d x) \log (c+d x)+b (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+b (c+d x) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right ) n^2}{(b c-a d) (c+d x)}+b B^2 \left (-\log \left (\frac {b c-a d}{b c+b d x}\right ) \log ^2\left (\frac {a+b x}{c+d x}\right )+\frac {c \log ^2\left (\frac {a+b x}{c+d x}\right )}{c+d x}-2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (\frac {a+b x}{c+d x}\right )+\frac {c \left (2 b c-2 a d+2 b (c+d x) \log (a+b x)-2 (b c-a d) \log \left (\frac {a+b x}{c+d x}\right )-2 b (c+d x) \log (a+b x) \log \left (\frac {a+b x}{c+d x}\right )-2 b (c+d x) \log (c+d x)-2 b (c+d x) \log \left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+b (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+b (c+d x) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d) (c+d x)}+2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right ) n^2+\frac {2 a B d \left (-A-B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n \log \left (\frac {a+b x}{c+d x}\right )\right ) \left (b c-b \log \left (\frac {b (c+d x)}{b c-a d}\right ) c-a d+b (c+d x) \log \left (\frac {a}{b}+x\right )+(a d-b c) \log \left (\frac {a+b x}{c+d x}\right )-b d x \log \left (\frac {b (c+d x)}{b c-a d}\right )\right ) n}{(a d-b c) (c+d x)}+b B \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right ) \left (-\log ^2\left (\frac {c}{d}+x\right )+2 \log (c+d x) \log \left (\frac {c}{d}+x\right )+2 \left (\frac {b \log (a+b x) c}{a d-b c}+\frac {b \log (c+d x) c}{b c-a d}-\frac {c}{c+d x}-\log \left (\frac {a}{b}+x\right ) \log (c+d x)+\log \left (\frac {a+b x}{c+d x}\right ) \left (\frac {c}{c+d x}+\log (c+d x)\right )+\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right ) n+\frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right )^2}{c+d x}+b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right )^2 \log (c+d x)\right )}{d^2 i^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} b g x + A^{2} a g + {\left (B^{2} b g x + B^{2} a g\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b g x + A B a g\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{d^{2} i^{2} x^{2} + 2 \, c d i^{2} x + c^{2} i^{2}}, x\right ) \]
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 = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (d i x +c i \right )^{2}}\, dx \]
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 \[ 2 \, A B a g n {\left (\frac {1}{d^{2} i^{2} x + c d i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} + A^{2} b g {\left (\frac {c}{d^{3} i^{2} x + c d^{2} i^{2}} + \frac {\log \left (d x + c\right )}{d^{2} i^{2}}\right )} - \frac {2 \, A B a g \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac {A^{2} a g}{d^{2} i^{2} x + c d i^{2}} + \frac {{\left ({\left (b c g - a d g\right )} B^{2} + {\left (B^{2} b d g x + B^{2} b c g\right )} \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{d^{3} i^{2} x + c d^{2} i^{2}} - \int -\frac {B^{2} a d g \log \relax (e)^{2} + {\left (B^{2} b d g x + B^{2} a d g\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (B^{2} b d g \log \relax (e)^{2} + 2 \, A B b d g \log \relax (e)\right )} x + 2 \, {\left (B^{2} a d g \log \relax (e) + {\left (B^{2} b d g \log \relax (e) + A B b d g\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left ({\left (b c g n - {\left (g n - g \log \relax (e)\right )} a d\right )} B^{2} + {\left (B^{2} b d g \log \relax (e) + A B b d g\right )} x + {\left (B^{2} b d g n x + B^{2} b c g n\right )} \log \left (d x + c\right ) + {\left (B^{2} b d g x + B^{2} a d g\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{d^{3} i^{2} x^{2} + 2 \, c d^{2} i^{2} x + c^{2} d i^{2}}\,{d x} \]
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 \[ \int \frac {\left (a\,g+b\,g\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (c\,i+d\,i\,x\right )}^2} \,d x \]
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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