Optimal. Leaf size=283 \[ \frac {2 B g (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i}+\frac {2 B g (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2 i}+\frac {g (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^2 i}+\frac {g (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d i}+\frac {2 B^2 g (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i}-\frac {2 B^2 g (b c-a d) \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]
[Out]
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Rubi [B] time = 4.13, antiderivative size = 1072, normalized size of antiderivative = 3.79, number of steps used = 68, number of rules used = 24, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2500, 2433, 2375, 2317, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ -\frac {B^2 (b c-a d) g \log ^3(c+d x)}{3 d^2 i}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2(c+d x)}{d^2 i}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{d^2 i}-\frac {A B (b c-a d) g \log ^2(c+d x)}{d^2 i}-\frac {b B^2 c g \log ^2(c+d x)}{d^2 i}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{d^2 i}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{d^2 i}+\frac {2 A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 i}+\frac {2 b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 i}+\frac {2 B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{d^2 i}-\frac {2 B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{d^2 i}-\frac {2 b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{d^2 i}-\frac {a B^2 g \log ^2(a+b x)}{d i}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{d^2 i}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{d^2 i}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d i}+\frac {2 a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac {2 a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d i}-\frac {2 B^2 (b c-a d) g \log (a+b x) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i}+\frac {2 a B^2 g \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d i}+\frac {2 B^2 (b c-a d) g \log \left (\frac {1}{c+d x}\right ) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}-\frac {2 B^2 (b c-a d) g \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac {2 A B (b c-a d) g \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac {2 b B^2 c g \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}+\frac {2 B^2 (b c-a d) g \text {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i}+\frac {2 B^2 (b c-a d) g \text {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
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 2523
Rule 2524
Rule 2528
Rule 6589
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 c+86 d x} \, dx &=\int \left (\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {(-b c+a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d (c+d x)}\right ) \, dx\\ &=\frac {(b g) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{86 d}-\frac {((b c-a d) g) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{86 d}\\ &=\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {(b B g) \int \frac {(b c-a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{43 d}+\frac {(B (b c-a d) g) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{e (a+b x)} \, dx}{43 d^2}\\ &=\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {(b B (b c-a d) g) \int \frac {x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{43 d}+\frac {(B (b c-a d) g) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{43 d^2 e}\\ &=\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {(b B (b c-a d) g) \int \left (-\frac {a \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)}+\frac {c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{43 d}+\frac {(B (b c-a d) g) \int \frac {(b c-a d) e \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{43 d^2 e}\\ &=\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}+\frac {(a b B g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{43 d}-\frac {(b B c g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{43 d}+\frac {\left (B (b c-a d)^2 g\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{43 d^2}\\ &=\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}+\frac {\left (b B^2 c g\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{43 d^2}-\frac {\left (a B^2 g\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{43 d}+\frac {\left (B (b c-a d)^2 g\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{43 d^2}\\ &=\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}+\frac {(b B (b c-a d) g) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{43 d^2}-\frac {(B (b c-a d) g) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{c+d x} \, dx}{43 d}+\frac {\left (b B^2 c g\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{43 d^2 e}-\frac {\left (a B^2 g\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{43 d e}\\ &=\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}+\frac {(b B (b c-a d) g) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x}\right ) \, dx}{43 d^2}-\frac {(B (b c-a d) g) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x}\right ) \, dx}{43 d}+\frac {\left (b B^2 c g\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{43 d^2 e}-\frac {\left (a B^2 g\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{43 d e}\\ &=\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}+\frac {1}{43} \left (a B^2 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx+\frac {\left (b^2 B^2 c g\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{43 d^2}-\frac {\left (a b B^2 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{43 d}-\frac {\left (b B^2 c g\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{43 d}+\frac {(A b B (b c-a d) g) \int \frac {\log (c+d x)}{a+b x} \, dx}{43 d^2}+\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{43 d^2}-\frac {(A B (b c-a d) g) \int \frac {\log (c+d x)}{c+d x} \, dx}{43 d}-\frac {\left (B^2 (b c-a d) g\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{43 d}\\ &=\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {\left (b B^2 c g\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{43 d^2}-\frac {\left (a B^2 g\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{43 d}-\frac {\left (a b B^2 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{43 d}-\frac {\left (b B^2 c g\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{43 d}-\frac {(A B (b c-a d) g) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{43 d^2}+\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{86 d^2}+\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{43 d^2}+\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{43 d^2}-\frac {(A B (b c-a d) g) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{43 d}-\frac {\left (B^2 (b c-a d) g\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{86 d}+\frac {\left (b B^2 (b c-a d) g \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {\left (b B^2 c g\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{43 d^2}-\frac {\left (a B^2 g\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{43 d}-\frac {(A B (b c-a d) g) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{86 d^2}+\frac {\left (B^2 (b c-a d) g\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{43 d^2}+\frac {\left (B^2 (b c-a d) g\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right ) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{43 d}-\frac {\left (B^2 (b c-a d) g \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{43 d}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{86 d^2}-\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{86 b d}-\frac {\left (B^2 (b c-a d) g\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {1}{-\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 )}{43 b d}+\frac {\left (B^2 (b c-a d) g\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 )}{43 b d}-\frac {\left (B^2 (b c-a d) g \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\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 )}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log ^3(c+d x)}{258 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{86 d^2}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{43 d^2}+\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}+\frac {\left (B^2 (b c-a d) g\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log ^3(c+d x)}{258 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{86 d^2}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {\left (b B^2 (b c-a d) g\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (\frac {1}{x}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{86 d^3}-\frac {\left (b B^2 (b c-a d) g\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 )}{86 d^3}+\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log ^3(c+d x)}{258 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{86 d^2}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {\left (B^2 (b c-a d) g\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{43 d^2}+\frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log ^3(c+d x)}{258 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{86 d^2}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+2 \frac {\left (B^2 (b c-a d) g\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 )}{43 d^2}\\ &=-\frac {a B^2 g \log ^2(a+b x)}{86 d}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{86 d^2}+\frac {a B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{43 d}+\frac {b g x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{86 d}+\frac {B^2 (b c-a d) g \log ^2(a+b x) \log (c+d x)}{86 d^2}+\frac {b B^2 c g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {A B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{43 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{43 d^2}-\frac {B^2 (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {b B c g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{43 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{86 d^2}-\frac {b B^2 c g \log ^2(c+d x)}{86 d^2}-\frac {A B (b c-a d) g \log ^2(c+d x)}{86 d^2}+\frac {B^2 (b c-a d) g \log (a+b x) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{86 d^2}-\frac {B^2 (b c-a d) g \log ^3(c+d x)}{258 d^2}+\frac {a B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{86 d^2}+\frac {a B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d}-\frac {B^2 (b c-a d) g \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}+\frac {b B^2 c g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {A B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}-\frac {B^2 (b c-a d) g \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{43 d^2}+\frac {B^2 (b c-a d) g \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{43 d^2}\\ \end {align*}
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Mathematica [B] time = 0.72, size = 646, normalized size = 2.28 \[ -\frac {g \left (A^2 (b c-a d) \log (c+d x)+a A B d \left (2 \log (c+d x) \left (-\log \left (\frac {e (a+b x)}{c+d x}\right )+\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )\right )-2 \left (\text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )+\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+\log ^2\left (\frac {c}{d}+x\right )\right )+A B \left (2 b (d x-c \log (c+d x)) \left (-\log \left (\frac {e (a+b x)}{c+d x}\right )+\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )\right )+2 b c \left (\text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )+\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 d (a+b x) \left (\log \left (\frac {a}{b}+x\right )-1\right )-b c \log ^2\left (\frac {c}{d}+x\right )+2 b (c+d x) \left (\log \left (\frac {c}{d}+x\right )-1\right )\right )-B^2 \left (-(b c-a d) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (-2 \log \left (\frac {e (a+b x)}{c+d x}\right )+2 \log \left (\frac {d (a+b x)}{a d-b c}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 b c \left (\text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-\text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )+d (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+b c \log \left (\frac {b c-a d}{b c+b d x}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )\right )+a B^2 d \left (2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )-2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )+A^2 (-b) d x\right )}{d^2 i} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} b g x + A^{2} a g + {\left (B^{2} b g x + B^{2} a g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left (A B b g x + A B a g\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{d i x + c i}, x\right ) \]
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 = 1.74, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right ) \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}}{d i x +c i}\, dx \]
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 \[ A^{2} b g {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {A^{2} a g \log \left (d i x + c i\right )}{d i} + \frac {3 \, B^{2} b d g x \log \left (d x + c\right )^{2} - {\left (b c g - a d g\right )} B^{2} \log \left (d x + c\right )^{3}}{3 \, d^{2} i} - \int -\frac {B^{2} a g \log \relax (e)^{2} + 2 \, A B a g \log \relax (e) + {\left (B^{2} b g x + B^{2} a g\right )} \log \left (b x + a\right )^{2} + {\left (B^{2} b g \log \relax (e)^{2} + 2 \, A B b g \log \relax (e)\right )} x + 2 \, {\left (B^{2} a g \log \relax (e) + A B a g + {\left (B^{2} b g \log \relax (e) + A B b g\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (B^{2} a g \log \relax (e) + A B a g + {\left ({\left (g \log \relax (e) + g\right )} B^{2} b + A B b g\right )} x + {\left (B^{2} b g x + B^{2} a g\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{d i x + c i}\,{d x} \]
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 \[ \int \frac {\left (a\,g+b\,g\,x\right )\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{c\,i+d\,i\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {g \left (\int \frac {A^{2} a}{c + d x}\, dx + \int \frac {A^{2} b x}{c + d x}\, dx + \int \frac {B^{2} a \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B a \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {B^{2} b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx\right )}{i} \]
Verification of antiderivative is not currently implemented for this CAS.
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