Optimal. Leaf size=472 \[ \frac {d^2 i^2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^3 g^2}+\frac {4 B d i^2 n (b c-a d) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2}+\frac {2 B d i^2 n (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2}-\frac {2 d i^2 (b c-a d) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^3 g^2}-\frac {i^2 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 g^2 (a+b x)}-\frac {2 B i^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2 (a+b x)}+\frac {2 B^2 d i^2 n^2 (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 g^2}+\frac {4 B^2 d i^2 n^2 (b c-a d) \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}-\frac {2 B^2 i^2 n^2 (c+d x) (b c-a d)}{b^2 g^2 (a+b x)} \]
[Out]
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Rubi [B] time = 3.76, antiderivative size = 1309, normalized size of antiderivative = 2.77, number of steps used = 60, number of rules used = 21, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 44, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ -\frac {a B^2 d^2 n^2 \log ^2(a+b x) i^2}{b^3 g^2}+\frac {B^2 d (b c-a d) n^2 \log ^2(a+b x) i^2}{b^3 g^2}-\frac {2 A B d (b c-a d) n \log ^2(a+b x) i^2}{b^3 g^2}-\frac {2 B^2 d (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) i^2}{b^3 g^2}-\frac {2 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) i^2}{b^3 g^2}+\frac {d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 i^2}{b^2 g^2}+\frac {2 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 i^2}{b^3 g^2}-\frac {(b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 i^2}{b^3 g^2 (a+b x)}-\frac {B^2 c d n^2 \log ^2(c+d x) i^2}{b^2 g^2}+\frac {B^2 d (b c-a d) n^2 \log ^2(c+d x) i^2}{b^3 g^2}-\frac {2 B^2 d (b c-a d) n^2 \log (a+b x) i^2}{b^3 g^2}+\frac {2 a B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) i^2}{b^3 g^2}-\frac {2 B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) i^2}{b^3 g^2}-\frac {2 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) i^2}{b^3 g^2 (a+b x)}+\frac {2 B^2 d (b c-a d) n^2 \log (c+d x) i^2}{b^3 g^2}+\frac {2 B^2 c d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) i^2}{b^2 g^2}-\frac {2 B^2 d (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) i^2}{b^3 g^2}-\frac {2 B c d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) i^2}{b^2 g^2}+\frac {2 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) i^2}{b^3 g^2}+\frac {2 a B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}-\frac {2 B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac {4 A B d (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac {2 a B^2 d^2 n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^2}-\frac {2 B^2 d (b c-a d) n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac {4 A B d (b c-a d) n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac {2 B^2 c d n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) i^2}{b^2 g^2}-\frac {2 B^2 d (b c-a d) n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) i^2}{b^3 g^2}+\frac {4 B^2 d (b c-a d) n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right ) i^2}{b^3 g^2}+\frac {4 B^2 d (b c-a d) n^2 \text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right ) i^2}{b^3 g^2}-\frac {2 B^2 (b c-a d)^2 n^2 i^2}{b^3 g^2 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 44
Rule 2301
Rule 2317
Rule 2344
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2411
Rule 2418
Rule 2488
Rule 2506
Rule 2507
Rule 2523
Rule 2524
Rule 2525
Rule 2528
Rule 6610
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int \frac {(173 c+173 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2} \, dx &=\int \left (\frac {29929 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}+\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)^2}+\frac {59858 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2 (a+b x)}\right ) \, dx\\ &=\frac {\left (29929 d^2\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2 g^2}+\frac {(59858 d (b c-a d)) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x} \, dx}{b^2 g^2}+\frac {\left (29929 (b c-a d)^2\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2} \, dx}{b^2 g^2}\\ &=\frac {29929 d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {59858 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}-\frac {\left (59858 B d^2 n\right ) \int \frac {(b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac {(119716 B d (b c-a d) n) \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) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{b^3 g^2}+\frac {\left (59858 B (b c-a d)^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)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=\frac {29929 d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {59858 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}-\frac {(119716 B d (b c-a d) n) \int \frac {(b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^2}-\frac {\left (59858 B d^2 (b c-a d) n\right ) \int \frac {x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}+\frac {\left (59858 B (b c-a d)^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=\frac {29929 d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {59858 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}-\frac {\left (59858 B d^2 (b c-a d) n\right ) \int \left (-\frac {a \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)}+\frac {c \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)}\right ) \, dx}{b^2 g^2}-\frac {\left (119716 B d (b c-a d)^2 n\right ) \int \frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^2}+\frac {\left (59858 B (b c-a d)^3 n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b 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 (a+b x)}+\frac {d^2 \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}{b^3 g^2}\\ &=\frac {29929 d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {59858 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac {\left (59858 a B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g^2}-\frac {\left (59858 B c d^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}{b^2 g^2}-\frac {(59858 B d (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g^2}+\frac {\left (59858 B d^2 (b c-a d) n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 g^2}+\frac {\left (59858 B (b c-a d)^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 g^2}-\frac {\left (119716 B d (b c-a d)^2 n\right ) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{b^3 g^2}\\ &=-\frac {59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {59858 a B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}-\frac {59858 B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac {29929 d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {59858 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}-\frac {59858 B c d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {59858 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}-\frac {\left (119716 A B d (b c-a d)^2 n\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^3 g^2}-\frac {\left (119716 B^2 d (b c-a d)^2 n\right ) \int \frac {\log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^2}+\frac {\left (59858 B^2 c d 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}{b^2 g^2}-\frac {\left (59858 a B^2 d^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 g^2}+\frac {\left (59858 B^2 d (b c-a d) n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 g^2}-\frac {\left (59858 B^2 d (b c-a d) 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}{b^3 g^2}+\frac {\left (59858 B^2 (b c-a d)^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=-\frac {59858 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {59858 a B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}-\frac {59858 B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac {29929 d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {59858 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}-\frac {59858 B c d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {59858 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {\left (59858 B^2 d (b c-a d)\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g^2}-\frac {\left (119716 A B d (b c-a d)^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^4 g^2}+\frac {\left (59858 B^2 c d 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}{b^2 g^2}-\frac {\left (59858 a B^2 d^2 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^3 g^2}+\frac {\left (59858 B^2 d (b c-a d) n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^3 g^2}-\frac {\left (59858 B^2 d (b c-a d) 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}{b^3 g^2}+\frac {\left (59858 B^2 (b c-a d)^3 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=-\frac {59858 B^2 d (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {59858 a B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}-\frac {59858 B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac {29929 d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {59858 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}-\frac {59858 B c d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {59858 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}-\frac {(119716 A B d (b c-a d) n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac {\left (119716 A B d^2 (b c-a d) n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b^4 g^2}+\frac {\left (119716 B^2 d (b c-a d)^2 n\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^2}+\frac {\left (59858 B^2 c d n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b g^2}-\frac {\left (59858 a B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^2}-\frac {\left (59858 B^2 c d^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 g^2}+\frac {\left (59858 a B^2 d^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^2}+\frac {\left (59858 B^2 d (b c-a d) n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^2}-\frac {\left (59858 B^2 d (b c-a d) n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 g^2}-\frac {\left (59858 B^2 d^2 (b c-a d) n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^2}+\frac {\left (59858 B^2 d^2 (b c-a d) n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 g^2}+\frac {\left (59858 B^2 (b c-a d)^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^2}\\ &=-\frac {59858 B^2 (b c-a d)^2 n^2}{b^3 g^2 (a+b x)}-\frac {59858 B^2 d (b c-a d) n^2 \log (a+b x)}{b^3 g^2}-\frac {59858 A B d (b c-a d) n \log ^2(a+b x)}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {59858 a B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}-\frac {59858 B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac {29929 d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {59858 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac {59858 B^2 d (b c-a d) n^2 \log (c+d x)}{b^3 g^2}+\frac {59858 B^2 c d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac {59858 B^2 d (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac {59858 B c d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {59858 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {119716 A B d (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {59858 a B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {119716 B^2 d (b c-a d) n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 g^2}-\frac {(119716 A B d (b c-a d) n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^2}-\frac {\left (59858 B^2 c d n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 g^2}-\frac {\left (59858 a B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}-\frac {\left (59858 a B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^2}-\frac {\left (59858 B^2 c d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 g^2}+\frac {\left (59858 B^2 d (b c-a d) n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac {\left (59858 B^2 d (b c-a d) n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 g^2}+\frac {\left (59858 B^2 d (b c-a d) n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^2}+\frac {\left (59858 B^2 d^2 (b c-a d) n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 g^2}-\frac {\left (119716 B^2 d (b c-a d)^2 n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^2}\\ &=-\frac {59858 B^2 (b c-a d)^2 n^2}{b^3 g^2 (a+b x)}-\frac {59858 B^2 d (b c-a d) n^2 \log (a+b x)}{b^3 g^2}-\frac {59858 A B d (b c-a d) n \log ^2(a+b x)}{b^3 g^2}-\frac {29929 a B^2 d^2 n^2 \log ^2(a+b x)}{b^3 g^2}+\frac {29929 B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {59858 a B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}-\frac {59858 B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac {29929 d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {59858 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac {59858 B^2 d (b c-a d) n^2 \log (c+d x)}{b^3 g^2}+\frac {59858 B^2 c d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac {59858 B^2 d (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac {59858 B c d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {59858 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}-\frac {29929 B^2 c d n^2 \log ^2(c+d x)}{b^2 g^2}+\frac {29929 B^2 d (b c-a d) n^2 \log ^2(c+d x)}{b^3 g^2}+\frac {119716 A B d (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {59858 a B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {119716 A B d (b c-a d) n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac {119716 B^2 d (b c-a d) n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 g^2}+\frac {119716 B^2 d (b c-a d) n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 g^2}-\frac {\left (59858 B^2 c d 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 )}{b^2 g^2}-\frac {\left (59858 a B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac {\left (59858 B^2 d (b c-a d) n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^2}+\frac {\left (59858 B^2 d (b c-a d) 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 )}{b^3 g^2}\\ &=-\frac {59858 B^2 (b c-a d)^2 n^2}{b^3 g^2 (a+b x)}-\frac {59858 B^2 d (b c-a d) n^2 \log (a+b x)}{b^3 g^2}-\frac {59858 A B d (b c-a d) n \log ^2(a+b x)}{b^3 g^2}-\frac {29929 a B^2 d^2 n^2 \log ^2(a+b x)}{b^3 g^2}+\frac {29929 B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac {59858 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}+\frac {59858 a B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}-\frac {59858 B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac {29929 d^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^2}-\frac {29929 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {59858 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^2}+\frac {59858 B^2 d (b c-a d) n^2 \log (c+d x)}{b^3 g^2}+\frac {59858 B^2 c d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}-\frac {59858 B^2 d (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac {59858 B c d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g^2}+\frac {59858 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^3 g^2}-\frac {29929 B^2 c d n^2 \log ^2(c+d x)}{b^2 g^2}+\frac {29929 B^2 d (b c-a d) n^2 \log ^2(c+d x)}{b^3 g^2}+\frac {119716 A B d (b c-a d) n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {59858 a B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {119716 A B d (b c-a d) n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac {59858 a B^2 d^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}-\frac {59858 B^2 d (b c-a d) n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac {59858 B^2 c d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {59858 B^2 d (b c-a d) n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {119716 B^2 d (b c-a d) n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 g^2}+\frac {119716 B^2 d (b c-a d) n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^3 g^2}\\ \end {align*}
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Mathematica [B] time = 12.40, size = 2834, normalized size = 6.00 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} d^{2} i^{2} x^{2} + 2 \, A^{2} c d i^{2} x + A^{2} c^{2} i^{2} + {\left (B^{2} d^{2} i^{2} x^{2} + 2 \, B^{2} c d i^{2} x + B^{2} c^{2} i^{2}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B d^{2} i^{2} x^{2} + 2 \, A B c d i^{2} x + A B c^{2} i^{2}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, x\right ) \]
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 = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right )^{2}}\, dx \]
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 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (c\,i+d\,i\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \]
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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