Optimal. Leaf size=573 \[ \frac {b^2 g^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d^3 i}-\frac {2 B g^2 n (b c-a d)^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i}-\frac {g^2 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^3 i}-\frac {4 B g^2 n (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i}+\frac {B g^2 n (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i}-\frac {2 g^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^2 i}-\frac {B g^2 n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i}-\frac {4 B^2 g^2 n^2 (b c-a d)^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac {B^2 g^2 n^2 (b c-a d)^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{d^3 i}+\frac {2 B^2 g^2 n^2 (b c-a d)^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}+\frac {B^2 g^2 n^2 (b c-a d)^2 \log (c+d x)}{d^3 i} \]
[Out]
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Rubi [B] time = 4.75, antiderivative size = 1780, normalized size of antiderivative = 3.11, number of steps used = 82, number of rules used = 27, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 2486, 31, 6688, 6742, 2499, 2396, 2433, 2374, 6589, 2302, 30, 2500, 2375, 2317, 2440, 2434} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 31
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 2486
Rule 2499
Rule 2500
Rule 2523
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}{187 c+187 d x} \, dx &=\int \left (-\frac {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}{187 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}{d^2 (187 c+187 d x)}+\frac {b g (a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d}\right ) \, dx\\ &=\frac {(b g) \int (a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{187 d}-\frac {\left (b (b c-a d) g^2\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{187 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}{187 c+187 d x} \, dx}{d^2}\\ &=-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\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 \log (187 c+187 d x)}{187 d^3}-\frac {(B n) \int \frac {(b c-a d) g^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{187 d}+\frac {\left (2 b B (b c-a d) g^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}{187 d^2}-\frac {\left (2 B (b c-a d)^2 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 (187 c+187 d x)}{a+b x} \, dx}{187 d^3}\\ &=-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\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 \log (187 c+187 d x)}{187 d^3}-\frac {\left (B (b c-a d) g^2 n\right ) \int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{187 d}-\frac {\left (2 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 ) \log (187 c+187 d x)}{(a+b x) (c+d x)} \, dx}{187 d^3}+\frac {\left (2 b B (b c-a d)^2 g^2 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}{187 d^2}\\ &=-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\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 \log (187 c+187 d x)}{187 d^3}-\frac {\left (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 )}{d}+\frac {(-b c+a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x)}\right ) \, dx}{187 d}+\frac {\left (2 b B (b c-a d)^2 g^2 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}{187 d^2}-\frac {\left (2 B (b c-a d)^3 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 (187 c+187 d x)}{(a+b x) (c+d x)} \, dx}{187 d^3}\\ &=-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\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 \log (187 c+187 d x)}{187 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{187 d^2}-\frac {\left (2 a 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 )}{a+b x} \, dx}{187 d^2}+\frac {\left (2 b B c (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} \, dx}{187 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} \, dx}{187 d^2}-\frac {\left (2 B (b c-a d)^3 g^2 n\right ) \int \left (\frac {d \left (-A-B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (187 c+187 d x)}{(b c-a d) (c+d x)}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (187 c+187 d x)}{(b c-a d) (a+b x)}\right ) \, dx}{187 d^3}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 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 \log (187 c+187 d x)}{187 d^3}-\frac {\left (b B^2 (b c-a d) g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{187 d^2}-\frac {\left (2 b B (b c-a d)^2 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 (187 c+187 d x)}{a+b x} \, dx}{187 d^3}-\frac {\left (2 B (b c-a d)^2 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 (187 c+187 d x)}{c+d x} \, dx}{187 d^2}-\frac {\left (2 b B^2 c (b c-a d) 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}{187 d^3}+\frac {\left (2 a B^2 (b c-a d) 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}{187 d^2}-\frac {\left (B^2 (b c-a d)^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}{187 d^3}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}-\frac {B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 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 \log (187 c+187 d x)}{187 d^3}-\frac {\left (2 b B (b c-a d)^2 g^2 n\right ) \int \left (\frac {A \log (187 c+187 d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (187 c+187 d x)}{a+b x}\right ) \, dx}{187 d^3}-\frac {\left (2 B (b c-a d)^2 g^2 n\right ) \int \left (\frac {A \log (187 c+187 d x)}{-c-d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (187 c+187 d x)}{-c-d x}\right ) \, dx}{187 d^2}-\frac {\left (2 b B^2 c (b c-a d) 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}{187 d^3}+\frac {\left (2 a B^2 (b c-a d) 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}{187 d^2}-\frac {\left (B^2 (b c-a d)^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}{187 d^3}+\frac {\left (B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{187 d^2}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}-\frac {B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log (c+d x)}{187 d^3}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 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 \log (187 c+187 d x)}{187 d^3}-\frac {\left (2 A b B (b c-a d)^2 g^2 n\right ) \int \frac {\log (187 c+187 d x)}{a+b x} \, dx}{187 d^3}-\frac {\left (2 b B^2 (b c-a d)^2 g^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (187 c+187 d x)}{a+b x} \, dx}{187 d^3}-\frac {\left (2 A B (b c-a d)^2 g^2 n\right ) \int \frac {\log (187 c+187 d x)}{-c-d x} \, dx}{187 d^2}-\frac {\left (2 B^2 (b c-a d)^2 g^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (187 c+187 d x)}{-c-d x} \, dx}{187 d^2}-\frac {\left (2 b^2 B^2 c (b c-a d) g^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{187 d^3}+\frac {\left (2 a b B^2 (b c-a d) g^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{187 d^2}+\frac {\left (2 b B^2 c (b c-a d) g^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{187 d^2}-\frac {\left (2 a B^2 (b c-a d) g^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{187 d}-\frac {\left (b B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{187 d^3}+\frac {\left (B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{187 d^2}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}-\frac {B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log (c+d x)}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^2}-\frac {2 A B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (187 c+187 d x)}{187 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 \log (187 c+187 d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(187 c+187 d x)}{187 d^3}-\frac {\left (2 A B (b c-a d)^2 g^2 n\right ) \operatorname {Subst}\left (\int -\frac {187 \log (x)}{x} \, dx,x,187 c+187 d x\right )}{34969 d^3}-\frac {\left (2 b B^2 (b c-a d)^2 g^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (187 c+187 d x)}{a+b x} \, dx}{187 d^3}-\frac {\left (2 b B^2 (b c-a d)^2 g^2 n\right ) \int \frac {\log \left ((c+d x)^{-n}\right ) \log (187 c+187 d x)}{a+b x} \, dx}{187 d^3}+\frac {\left (2 A B (b c-a d)^2 g^2 n\right ) \int \frac {\log \left (\frac {187 d (a+b x)}{-187 b c+187 a d}\right )}{187 c+187 d x} \, dx}{d^2}+\frac {\left (2 b B^2 c (b c-a d) g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{187 d^3}+\frac {\left (2 a B^2 (b c-a d) g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{187 d^2}+\frac {\left (2 a b B^2 (b c-a d) g^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{187 d^2}+\frac {\left (2 b B^2 c (b c-a d) g^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{187 d^2}+\frac {\left (B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{187 d^3}-\frac {\left (b B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {\log ^2(187 c+187 d x)}{a+b x} \, dx}{187 d^3}+\frac {\left (B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{187 d^2}+\frac {\left (B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {\log ^2(187 c+187 d x)}{c+d x} \, dx}{187 d^2}-\frac {\left (2 b B^2 (b c-a d)^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 (187 c+187 d x)}{a+b x} \, dx}{187 d^3}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}+\frac {a B^2 (b c-a d) g^2 n^2 \log ^2(a+b x)}{187 d^2}-\frac {B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log (c+d x)}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 d^3}+\frac {b B^2 c (b c-a d) g^2 n^2 \log ^2(c+d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^2(c+d x)}{374 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^2}-\frac {2 A B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (187 c+187 d x)}{187 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 \log (187 c+187 d x)}{187 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 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 (187 c+187 d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(187 c+187 d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(187 c+187 d x)}{187 d^3}+\frac {\left (2 A B (b c-a d)^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,187 c+187 d x\right )}{187 d^3}+\frac {\left (2 A B (b c-a d)^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-187 b c+187 a d}\right )}{x} \, dx,x,187 c+187 d x\right )}{187 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {187 b c-187 a d}{b}+\frac {187 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{187 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right ) \log \left (-\frac {-187 b c+187 a d}{b}+\frac {187 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{187 d^3}+\frac {\left (2 b B^2 c (b c-a d) 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 )}{187 d^3}+\frac {\left (2 a B^2 (b c-a d) 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 )}{187 d^2}+\frac {\left (B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {187 \log ^2(x)}{x} \, dx,x,187 c+187 d x\right )}{34969 d^3}+\frac {\left (B^2 (b c-a d)^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 )}{187 d^3}+\frac {\left (2 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {\log \left (\frac {187 d (a+b x)}{-187 b c+187 a d}\right ) \log (187 c+187 d x)}{187 c+187 d x} \, dx}{d^2}+\frac {\left (2 B^2 (b c-a d)^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 {187 d (a+b x)}{-187 b c+187 a d}\right )}{187 c+187 d x} \, dx}{d^2}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}+\frac {a B^2 (b c-a d) g^2 n^2 \log ^2(a+b x)}{187 d^2}-\frac {B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log (c+d x)}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 d^3}+\frac {b B^2 c (b c-a d) g^2 n^2 \log ^2(c+d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^2(c+d x)}{374 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (187 (c+d x))}{187 d^3}+\frac {A B (b c-a d)^2 g^2 n \log ^2(187 (c+d x))}{187 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^2}-\frac {2 B^2 (b c-a d)^2 g^2 n \log (a+b x) \log (187 (c+d x)) \log \left ((c+d x)^{-n}\right )}{187 d^3}-\frac {2 A B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (187 c+187 d x)}{187 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 \log (187 c+187 d x)}{187 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 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 (187 c+187 d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(187 c+187 d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(187 c+187 d x)}{187 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{187 d^2}-\frac {2 A B (b c-a d)^2 g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}+\frac {\left (B^2 (b c-a d)^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {187 b c-187 a d}{b}+\frac {187 d x}{b}} \, dx,x,a+b x\right )}{b d^2}+\frac {\left (2 B^2 (b c-a d)^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 )}{187 b d^2}+\frac {\left (B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,187 c+187 d x\right )}{187 d^3}+\frac {\left (2 B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {187 d \left (\frac {-187 b c+187 a d}{187 d}+\frac {b x}{187 d}\right )}{-187 b c+187 a d}\right )}{x} \, dx,x,187 c+187 d x\right )}{187 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-187 b c+187 a d}{b}+\frac {187 d x}{b}\right )}{-\frac {-187 b c+187 a d}{b}+\frac {187 d x}{b}} \, dx,x,a+b x\right )}{b d^2}+\frac {\left (2 B^2 (b c-a d)^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}{-187 b c+187 a d}\right )}{x} \, dx,x,187 c+187 d x\right )}{187 d^3}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}+\frac {a B^2 (b c-a d) g^2 n^2 \log ^2(a+b x)}{187 d^2}-\frac {B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log (c+d x)}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 d^3}+\frac {b B^2 c (b c-a d) g^2 n^2 \log ^2(c+d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^2(c+d x)}{374 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (187 (c+d x))}{187 d^3}+\frac {A B (b c-a d)^2 g^2 n \log ^2(187 (c+d x))}{187 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^2}+\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n \log (a+b x) \log (187 (c+d x)) \log \left ((c+d x)^{-n}\right )}{187 d^3}-\frac {2 A B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (187 c+187 d x)}{187 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 \log (187 c+187 d x)}{187 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 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 (187 c+187 d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(187 c+187 d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(187 c+187 d x)}{187 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{187 d^2}-\frac {2 A B (b c-a d)^2 g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n^2 \log (187 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}+\frac {2 B^2 (b c-a d)^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 )}{187 d^3}+\frac {\left (2 B^2 (b c-a d)^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 )}{187 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {187 d x}{187 b c-187 a d}\right )}{x} \, dx,x,a+b x\right )}{187 d^3}+\frac {\left (B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (187 (c+d x))\right )}{187 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-187 b c+187 a d}{187 d}+\frac {b x}{187 d}\right )}{x} \, dx,x,187 (c+d x)\right )}{187 d^3}+\frac {\left (2 B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-187 b c+187 a d}\right )}{x} \, dx,x,187 c+187 d x\right )}{187 d^3}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}+\frac {a B^2 (b c-a d) g^2 n^2 \log ^2(a+b x)}{187 d^2}-\frac {B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log (c+d x)}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 d^3}+\frac {b B^2 c (b c-a d) g^2 n^2 \log ^2(c+d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^2(c+d x)}{374 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (187 (c+d x))}{187 d^3}+\frac {A B (b c-a d)^2 g^2 n \log ^2(187 (c+d x))}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log (a+b x) \log ^2(187 (c+d x))}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^3(187 (c+d x))}{561 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^2}+\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n \log (a+b x) \log (187 (c+d x)) \log \left ((c+d x)^{-n}\right )}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{187 d^3}-\frac {2 A B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (187 c+187 d x)}{187 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 \log (187 c+187 d x)}{187 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 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 (187 c+187 d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(187 c+187 d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(187 c+187 d x)}{187 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{187 d^2}+\frac {2 B^2 (b c-a d)^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 )}{187 d^3}-\frac {2 A B (b c-a d)^2 g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n^2 \log (187 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}+\frac {2 B^2 (b c-a d)^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 )}{187 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}+\frac {\left (b B^2 (b c-a d)^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 )}{187 d^4}+\frac {\left (b B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-187 b c+187 a d}{187 d}+\frac {b x}{187 d}} \, dx,x,187 (c+d x)\right )}{34969 d^4}-\frac {\left (2 B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {187 d x}{187 b c-187 a d}\right )}{x} \, dx,x,a+b x\right )}{187 d^3}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}+\frac {a B^2 (b c-a d) g^2 n^2 \log ^2(a+b x)}{187 d^2}-\frac {B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log (c+d x)}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 d^3}+\frac {b B^2 c (b c-a d) g^2 n^2 \log ^2(c+d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^2(c+d x)}{374 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (187 (c+d x))}{187 d^3}+\frac {A B (b c-a d)^2 g^2 n \log ^2(187 (c+d x))}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log (a+b x) \log ^2(187 (c+d x))}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(187 (c+d x))}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^3(187 (c+d x))}{561 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^2}+\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n \log (a+b x) \log (187 (c+d x)) \log \left ((c+d x)^{-n}\right )}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{187 d^3}-\frac {2 A B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (187 c+187 d x)}{187 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 \log (187 c+187 d x)}{187 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 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 (187 c+187 d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(187 c+187 d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(187 c+187 d x)}{187 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{187 d^2}+\frac {2 B^2 (b c-a d)^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 )}{187 d^3}-\frac {2 A B (b c-a d)^2 g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n^2 \log (187 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}+\frac {2 B^2 (b c-a d)^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 )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{187 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}+\frac {\left (2 B^2 (b c-a d)^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 )}{187 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-187 b c+187 a d}\right )}{x} \, dx,x,187 (c+d x)\right )}{187 d^3}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}+\frac {a B^2 (b c-a d) g^2 n^2 \log ^2(a+b x)}{187 d^2}-\frac {B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log (c+d x)}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 d^3}+\frac {b B^2 c (b c-a d) g^2 n^2 \log ^2(c+d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^2(c+d x)}{374 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (187 (c+d x))}{187 d^3}+\frac {A B (b c-a d)^2 g^2 n \log ^2(187 (c+d x))}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log (a+b x) \log ^2(187 (c+d x))}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(187 (c+d x))}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^3(187 (c+d x))}{561 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^2}+\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n \log (a+b x) \log (187 (c+d x)) \log \left ((c+d x)^{-n}\right )}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{187 d^3}-\frac {2 A B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (187 c+187 d x)}{187 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 \log (187 c+187 d x)}{187 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 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 (187 c+187 d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(187 c+187 d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(187 c+187 d x)}{187 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{187 d^2}+\frac {2 B^2 (b c-a d)^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 )}{187 d^3}-\frac {2 A B (b c-a d)^2 g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^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 )}{187 d^3}+\frac {2 B^2 (b c-a d)^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 )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{187 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {\left (2 B^2 (b c-a d)^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 )}{187 d^3}-\frac {\left (2 B^2 (b c-a d)^2 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-187 b c+187 a d}\right )}{x} \, dx,x,187 (c+d x)\right )}{187 d^3}\\ &=-\frac {A b B (b c-a d) g^2 n x}{187 d^2}+\frac {a B^2 (b c-a d) g^2 n^2 \log ^2(a+b x)}{187 d^2}-\frac {B^2 (b c-a d) g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{187 d^2}-\frac {2 a B (b c-a d) 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 )}{187 d^2}-\frac {b (b c-a d) g^2 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{187 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{374 d}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log (c+d x)}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{187 d^3}+\frac {2 b B c (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 ) \log (c+d x)}{187 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 ) \log (c+d x)}{187 d^3}+\frac {b B^2 c (b c-a d) g^2 n^2 \log ^2(c+d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^2(c+d x)}{374 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log (187 (c+d x))}{187 d^3}+\frac {A B (b c-a d)^2 g^2 n \log ^2(187 (c+d x))}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log (a+b x) \log ^2(187 (c+d x))}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(187 (c+d x))}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log ^3(187 (c+d x))}{561 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^2}+\frac {B^2 (b c-a d)^2 g^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n \log (a+b x) \log (187 (c+d x)) \log \left ((c+d x)^{-n}\right )}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{187 d^3}-\frac {2 A B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (187 c+187 d x)}{187 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 \log (187 c+187 d x)}{187 d^3}+\frac {2 B^2 (b c-a d)^2 g^2 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 (187 c+187 d x)}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(187 c+187 d x)}{187 d^3}+\frac {B^2 (b c-a d)^2 g^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(187 c+187 d x)}{187 d^3}-\frac {2 a B^2 (b c-a d) g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{187 d^2}+\frac {2 B^2 (b c-a d)^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 )}{187 d^3}-\frac {2 A B (b c-a d)^2 g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 b B^2 c (b c-a d) g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {B^2 (b c-a d)^2 g^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^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 )}{187 d^3}+\frac {2 B^2 (b c-a d)^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 )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{187 d^3}-\frac {2 B^2 (b c-a d)^2 g^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{187 d^3}\\ \end {align*}
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Mathematica [B] time = 1.65, size = 1741, normalized size = 3.04 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} b^{2} g^{2} x^{2} + 2 \, A^{2} a b g^{2} x + A^{2} a^{2} g^{2} + {\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b^{2} g^{2} x^{2} + 2 \, A B a b g^{2} x + A B a^{2} g^{2}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{d i x + c i}, x\right ) \]
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 \[ \int \frac {\left (b g x +a g \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{d i x +c i}\, dx \]
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^{2} a b g^{2} {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {1}{2} \, A^{2} b^{2} g^{2} {\left (\frac {2 \, c^{2} \log \left (d x + c\right )}{d^{3} i} + \frac {d x^{2} - 2 \, c x}{d^{2} i}\right )} + \frac {A^{2} a^{2} g^{2} \log \left (d i x + c i\right )}{d i} + \frac {{\left (B^{2} b^{2} d^{2} g^{2} x^{2} - 2 \, {\left (b^{2} c d g^{2} - 2 \, a b d^{2} g^{2}\right )} B^{2} x + 2 \, {\left (b^{2} c^{2} g^{2} - 2 \, a b c d g^{2} + a^{2} d^{2} g^{2}\right )} B^{2} \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{2 \, d^{3} i} - \int -\frac {B^{2} a^{2} d^{2} g^{2} \log \relax (e)^{2} + 2 \, A B a^{2} d^{2} g^{2} \log \relax (e) + {\left (B^{2} b^{2} d^{2} g^{2} \log \relax (e)^{2} + 2 \, A B b^{2} d^{2} g^{2} \log \relax (e)\right )} x^{2} + {\left (B^{2} b^{2} d^{2} g^{2} x^{2} + 2 \, B^{2} a b d^{2} g^{2} x + B^{2} a^{2} d^{2} g^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + 2 \, {\left (B^{2} a b d^{2} g^{2} \log \relax (e)^{2} + 2 \, A B a b d^{2} g^{2} \log \relax (e)\right )} x + 2 \, {\left (B^{2} a^{2} d^{2} g^{2} \log \relax (e) + A B a^{2} d^{2} g^{2} + {\left (B^{2} b^{2} d^{2} g^{2} \log \relax (e) + A B b^{2} d^{2} g^{2}\right )} x^{2} + 2 \, {\left (B^{2} a b d^{2} g^{2} \log \relax (e) + A B a b d^{2} g^{2}\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (2 \, B^{2} a^{2} d^{2} g^{2} \log \relax (e) + 2 \, A B a^{2} d^{2} g^{2} + 2 \, {\left (b^{2} c^{2} g^{2} n - 2 \, a b c d g^{2} n + a^{2} d^{2} g^{2} n\right )} B^{2} \log \left (d x + c\right ) + {\left (2 \, A B b^{2} d^{2} g^{2} + {\left (g^{2} n + 2 \, g^{2} \log \relax (e)\right )} B^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (2 \, A B a b d^{2} g^{2} - {\left (b^{2} c d g^{2} n - 2 \, {\left (g^{2} n + g^{2} \log \relax (e)\right )} a b d^{2}\right )} B^{2}\right )} x + 2 \, {\left (B^{2} b^{2} d^{2} g^{2} x^{2} + 2 \, B^{2} a b d^{2} g^{2} x + B^{2} a^{2} d^{2} g^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{d^{3} i x + c d^{2} i}\,{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 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{c\,i+d\,i\,x} \,d x \]
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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