Optimal. Leaf size=536 \[ -\frac{2 B g^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^3 i}-\frac{4 B^2 g^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac{B^2 g^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{d^3 i}+\frac{2 B^2 g^2 (b c-a d)^2 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 i}+\frac{b^2 g^2 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 d^3 i}-\frac{2 g^2 (a+b x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{d^2 i}-\frac{g^2 (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 i}-\frac{B g^2 (a+b x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^2 i}-\frac{4 B g^2 (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^3 i}+\frac{B g^2 (b c-a d)^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^3 i}+\frac{B^2 g^2 (b c-a d)^2 \log (c+d x)}{d^3 i} \]
[Out]
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Rubi [B] time = 4.82488, antiderivative size = 1666, normalized size of antiderivative = 3.11, number of steps used = 86, number of rules used = 27, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, 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 2528
Rule 2523
Rule 12
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 2525
Rule 2486
Rule 31
Rule 6688
Rule 6742
Rule 2499
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 2302
Rule 30
Rule 2500
Rule 2375
Rule 2317
Rule 2440
Rule 2434
Rubi steps
\begin{align*} \int \frac{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 c+85 d x} \, dx &=\int \left (-\frac{b (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{d^2 (85 c+85 d x)}+\frac{b g (a g+b g x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d}\right ) \, dx\\ &=\frac{(b g) \int (a g+b g x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{85 d}-\frac{\left (b (b c-a d) g^2\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{85 d^2}+\frac{\left ((b c-a d)^2 g^2\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 c+85 d x} \, dx}{d^2}\\ &=-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac{B \int \frac{(b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{85 d}+\frac{\left (2 b B (b c-a d) g^2\right ) \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}{85 d^2}-\frac{\left (2 B (b c-a d)^2 g^2\right ) \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 (85 c+85 d x)}{e (a+b x)} \, dx}{85 d^3}\\ &=-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac{\left (B (b c-a d) g^2\right ) \int \frac{(a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{85 d}+\frac{\left (2 b B (b c-a d)^2 g^2\right ) \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}{85 d^2}-\frac{\left (2 B (b c-a d)^2 g^2\right ) \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 (85 c+85 d x)}{a+b x} \, dx}{85 d^3 e}\\ &=-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac{\left (B (b c-a d) g^2\right ) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d}+\frac{(-b c+a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d (c+d x)}\right ) \, dx}{85 d}+\frac{\left (2 b B (b c-a d)^2 g^2\right ) \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}{85 d^2}-\frac{\left (2 B (b c-a d)^2 g^2\right ) \int \frac{(b c-a d) e \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{(a+b x) (c+d x)} \, dx}{85 d^3 e}\\ &=-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac{\left (b B (b c-a d) g^2\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{85 d^2}-\frac{\left (2 a b B (b c-a d) g^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{85 d^2}+\frac{\left (2 b B c (b c-a d) g^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{85 d^2}+\frac{\left (B (b c-a d)^2 g^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{85 d^2}-\frac{\left (2 B (b c-a d)^3 g^2\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{(a+b x) (c+d x)} \, dx}{85 d^3}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac{\left (2 b B^2 c (b c-a d) g^2\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}{85 d^3}+\frac{\left (2 a B^2 (b c-a d) g^2\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}{85 d^2}-\frac{\left (b B^2 (b c-a d) g^2\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{85 d^2}-\frac{\left (B^2 (b c-a d)^2 g^2\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}{85 d^3}-\frac{\left (2 B (b c-a d)^3 g^2\right ) \int \left (\frac{d \left (-A-B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{(b c-a d) (c+d x)}+\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{(b c-a d) (a+b x)}\right ) \, dx}{85 d^3}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac{\left (2 b B (b c-a d)^2 g^2\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{a+b x} \, dx}{85 d^3}-\frac{\left (2 B (b c-a d)^2 g^2\right ) \int \frac{\left (-A-B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (85 c+85 d x)}{c+d x} \, dx}{85 d^2}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \int \frac{1}{c+d x} \, dx}{85 d^2}-\frac{\left (2 b B^2 c (b c-a d) g^2\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}{85 d^3 e}+\frac{\left (2 a B^2 (b c-a d) g^2\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}{85 d^2 e}-\frac{\left (B^2 (b c-a d)^2 g^2\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}{85 d^3 e}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac{\left (2 b B (b c-a d)^2 g^2\right ) \int \left (\frac{A \log (85 c+85 d x)}{a+b x}+\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right ) \log (85 c+85 d x)}{a+b x}\right ) \, dx}{85 d^3}-\frac{\left (2 B (b c-a d)^2 g^2\right ) \int \left (\frac{A \log (85 c+85 d x)}{-c-d x}+\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right ) \log (85 c+85 d x)}{-c-d x}\right ) \, dx}{85 d^2}-\frac{\left (2 b B^2 c (b c-a d) g^2\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}{85 d^3 e}+\frac{\left (2 a B^2 (b c-a d) g^2\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}{85 d^2 e}-\frac{\left (B^2 (b c-a d)^2 g^2\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}{85 d^3 e}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}-\frac{\left (2 b^2 B^2 c (b c-a d) g^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{85 d^3}+\frac{\left (2 a b B^2 (b c-a d) g^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{85 d^2}+\frac{\left (2 b B^2 c (b c-a d) g^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{85 d^2}-\frac{\left (2 a B^2 (b c-a d) g^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{85 d}-\frac{\left (2 A b B (b c-a d)^2 g^2\right ) \int \frac{\log (85 c+85 d x)}{a+b x} \, dx}{85 d^3}-\frac{\left (b B^2 (b c-a d)^2 g^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{85 d^3}-\frac{\left (2 b B^2 (b c-a d)^2 g^2\right ) \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \log (85 c+85 d x)}{a+b x} \, dx}{85 d^3}-\frac{\left (2 A B (b c-a d)^2 g^2\right ) \int \frac{\log (85 c+85 d x)}{-c-d x} \, dx}{85 d^2}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{85 d^2}-\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \log (85 c+85 d x)}{-c-d x} \, dx}{85 d^2}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 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 (c+d x)}{85 d^3}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}-\frac{2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^2}-\frac{2 A B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac{\left (2 b B^2 c (b c-a d) g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{85 d^3}+\frac{\left (2 a B^2 (b c-a d) g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{85 d^2}+\frac{\left (2 a b B^2 (b c-a d) g^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{85 d^2}+\frac{\left (2 b B^2 c (b c-a d) g^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{85 d^2}-\frac{\left (2 A B (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int -\frac{85 \log (x)}{x} \, dx,x,85 c+85 d x\right )}{7225 d^3}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{85 d^3}-\frac{\left (b B^2 (b c-a d)^2 g^2\right ) \int \frac{\log ^2(85 c+85 d x)}{a+b x} \, dx}{85 d^3}-\frac{\left (2 b B^2 (b c-a d)^2 g^2\right ) \int \frac{\log (a+b x) \log (85 c+85 d x)}{a+b x} \, dx}{85 d^3}-\frac{\left (2 b B^2 (b c-a d)^2 g^2\right ) \int \frac{\log \left (\frac{1}{c+d x}\right ) \log (85 c+85 d x)}{a+b x} \, dx}{85 d^3}+\frac{\left (2 A B (b c-a d)^2 g^2\right ) \int \frac{\log \left (\frac{85 d (a+b x)}{-85 b c+85 a d}\right )}{85 c+85 d x} \, dx}{d^2}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{85 d^2}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \int \frac{\log ^2(85 c+85 d x)}{c+d x} \, dx}{85 d^2}-\frac{\left (2 b B^2 (b c-a d)^2 g^2 \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 (85 c+85 d x)}{a+b x} \, dx}{85 d^3}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}+\frac{a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 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 (c+d x)}{85 d^3}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac{2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^2}-\frac{2 A B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 (85 c+85 d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 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(85 c+85 d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}+\frac{\left (2 b B^2 c (b c-a d) g^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 )}{85 d^3}+\frac{\left (2 a B^2 (b c-a d) g^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 )}{85 d^2}+\frac{\left (2 A B (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}+\frac{\left (2 A B (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-85 b c+85 a d}\right )}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{85 \log ^2(x)}{x} \, dx,x,85 c+85 d x\right )}{7225 d^3}+\frac{\left (B^2 (b c-a d)^2 g^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 )}{85 d^3}-\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{85 b c-85 a d}{b}+\frac{85 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{85 d^3}-\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{-\frac{-b c+a d}{b}+\frac{d x}{b}}\right ) \log \left (-\frac{-85 b c+85 a d}{b}+\frac{85 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{85 d^3}+\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \int \frac{\log \left (\frac{85 d (a+b x)}{-85 b c+85 a d}\right ) \log (85 c+85 d x)}{85 c+85 d x} \, dx}{d^2}+\frac{\left (2 B^2 (b c-a d)^2 g^2 \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{85 d (a+b x)}{-85 b c+85 a d}\right )}{85 c+85 d x} \, dx}{d^2}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}+\frac{a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 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 (c+d x)}{85 d^3}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac{A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac{2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^2}-\frac{2 A B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 (85 c+85 d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 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(85 c+85 d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac{2 a B^2 (b c-a d) g^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^2}-\frac{2 b B^2 c (b c-a d) g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{B^2 (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}+\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{85 d \left (\frac{-85 b c+85 a d}{85 d}+\frac{b x}{85 d}\right )}{-85 b c+85 a d}\right )}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}+\frac{\left (2 B^2 (b c-a d)^2 g^2\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 )}{85 b d^2}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{85 b c-85 a d}{b}+\frac{85 d x}{b}} \, dx,x,a+b x\right )}{b d^2}-\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{-85 b c+85 a d}{b}+\frac{85 d x}{b}\right )}{-\frac{-85 b c+85 a d}{b}+\frac{85 d x}{b}} \, dx,x,a+b x\right )}{b d^2}+\frac{\left (2 B^2 (b c-a d)^2 g^2 \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}{-85 b c+85 a d}\right )}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}+\frac{a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 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 (c+d x)}{85 d^3}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac{A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac{2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^2}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 (85 c+85 d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 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(85 c+85 d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac{2 a B^2 (b c-a d) g^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^2}-\frac{2 b B^2 c (b c-a d) g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{B^2 (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 )}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log (85 (c+d x)) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac{\left (B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (85 (c+d x))\right )}{85 d^3}-\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{-85 b c+85 a d}{85 d}+\frac{b x}{85 d}\right )}{x} \, dx,x,85 (c+d x)\right )}{85 d^3}+\frac{\left (2 B^2 (b c-a d)^2 g^2\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 )}{85 d^3}-\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1+\frac{85 d x}{85 b c-85 a d}\right )}{x} \, dx,x,a+b x\right )}{85 d^3}+\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-85 b c+85 a d}\right )}{x} \, dx,x,85 c+85 d x\right )}{85 d^3}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}+\frac{a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac{B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{85 d^3}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 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 (c+d x)}{85 d^3}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac{A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac{B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2(85 (c+d x))}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^3(85 (c+d x))}{255 d^3}-\frac{2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^2}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 (85 c+85 d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 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(85 c+85 d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac{2 a B^2 (b c-a d) g^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^2}+\frac{2 B^2 (b c-a d)^2 g^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{B^2 (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 )}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log (85 (c+d x)) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac{\left (b B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{-85 b c+85 a d}{85 d}+\frac{b x}{85 d}} \, dx,x,85 (c+d x)\right )}{7225 d^4}+\frac{\left (b B^2 (b c-a d)^2 g^2\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 )}{85 d^4}-\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{85 d x}{85 b c-85 a d}\right )}{x} \, dx,x,a+b x\right )}{85 d^3}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}+\frac{a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac{B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{85 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 (\frac{1}{c+d x}\right )}{85 d^3}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 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 (c+d x)}{85 d^3}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac{A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac{B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2(85 (c+d x))}{85 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(85 (c+d x))}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^3(85 (c+d x))}{255 d^3}-\frac{2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^2}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 (85 c+85 d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 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(85 c+85 d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac{2 a B^2 (b c-a d) g^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^2}+\frac{2 B^2 (b c-a d)^2 g^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{B^2 (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 )}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log (85 (c+d x)) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac{\left (2 B^2 (b c-a d)^2 g^2\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 )}{85 d^3}-\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1+\frac{b x}{-85 b c+85 a d}\right )}{x} \, dx,x,85 (c+d x)\right )}{85 d^3}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}+\frac{a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac{B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{85 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 (\frac{1}{c+d x}\right )}{85 d^3}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 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 (c+d x)}{85 d^3}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac{A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac{B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2(85 (c+d x))}{85 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(85 (c+d x))}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^3(85 (c+d x))}{255 d^3}-\frac{2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^2}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 (85 c+85 d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 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(85 c+85 d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac{2 a B^2 (b c-a d) g^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^2}+\frac{2 B^2 (b c-a d)^2 g^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{B^2 (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log \left (\frac{1}{c+d x}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 )}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{\left (2 B^2 (b c-a d)^2 g^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 )}{85 d^3}-\frac{\left (2 B^2 (b c-a d)^2 g^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-85 b c+85 a d}\right )}{x} \, dx,x,85 (c+d x)\right )}{85 d^3}\\ &=-\frac{A b B (b c-a d) g^2 x}{85 d^2}+\frac{a B^2 (b c-a d) g^2 \log ^2(a+b x)}{85 d^2}-\frac{B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{85 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 (\frac{1}{c+d x}\right )}{85 d^3}-\frac{B^2 (b c-a d) g^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{85 d^2}-\frac{2 a B (b c-a d) g^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{85 d^2}-\frac{b (b c-a d) g^2 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{85 d^2}+\frac{g^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{170 d}+\frac{B^2 (b c-a d)^2 g^2 \log (c+d x)}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{85 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 (c+d x)}{85 d^3}+\frac{2 b B c (b c-a d) g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{B (b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{85 d^3}+\frac{b B^2 c (b c-a d) g^2 \log ^2(c+d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(c+d x)}{170 d^3}-\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log (85 (c+d x))}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (85 (c+d x))}{85 d^3}+\frac{A B (b c-a d)^2 g^2 \log ^2(85 (c+d x))}{85 d^3}-\frac{B^2 (b c-a d)^2 g^2 \log (a+b x) \log ^2(85 (c+d x))}{85 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(85 (c+d x))}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log ^3(85 (c+d x))}{255 d^3}-\frac{2 a B^2 (b c-a d) g^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^2}+\frac{B^2 (b c-a d)^2 g^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (85 c+85 d x)}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 (85 c+85 d x)}{85 d^3}+\frac{(b c-a d)^2 g^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (85 c+85 d x)}{85 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(85 c+85 d x)}{85 d^3}+\frac{B^2 (b c-a d)^2 g^2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(85 c+85 d x)}{85 d^3}-\frac{2 a B^2 (b c-a d) g^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^2}+\frac{2 B^2 (b c-a d)^2 g^2 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^3}-\frac{2 b B^2 c (b c-a d) g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 A B (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{B^2 (b c-a d)^2 g^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \log \left (\frac{1}{c+d x}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}+\frac{2 B^2 (b c-a d)^2 g^2 \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 )}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{85 d^3}-\frac{2 B^2 (b c-a d)^2 g^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{85 d^3}\\ \end{align*}
Mathematica [B] time = 1.67134, size = 1514, normalized size = 2.82 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.291, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bgx+ag \right ) ^{2}}{dix+ci} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} 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{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 )^{3} + 3 \,{\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\right )} \log \left (d x + c\right )^{2}}{6 \, d^{3} i} - \int -\frac{B^{2} a^{2} d g^{2} \log \left (e\right )^{2} + 2 \, A B a^{2} d g^{2} \log \left (e\right ) +{\left (B^{2} b^{2} d g^{2} \log \left (e\right )^{2} + 2 \, A B b^{2} d g^{2} \log \left (e\right )\right )} x^{2} +{\left (B^{2} b^{2} d g^{2} x^{2} + 2 \, B^{2} a b d g^{2} x + B^{2} a^{2} d g^{2}\right )} \log \left (b x + a\right )^{2} + 2 \,{\left (B^{2} a b d g^{2} \log \left (e\right )^{2} + 2 \, A B a b d g^{2} \log \left (e\right )\right )} x + 2 \,{\left (B^{2} a^{2} d g^{2} \log \left (e\right ) + A B a^{2} d g^{2} +{\left (B^{2} b^{2} d g^{2} \log \left (e\right ) + A B b^{2} d g^{2}\right )} x^{2} + 2 \,{\left (B^{2} a b d g^{2} \log \left (e\right ) + A B a b d g^{2}\right )} x\right )} \log \left (b x + a\right ) -{\left (2 \, B^{2} a^{2} d g^{2} \log \left (e\right ) + 2 \, A B a^{2} d g^{2} +{\left (2 \, A B b^{2} d g^{2} +{\left (2 \, g^{2} \log \left (e\right ) + g^{2}\right )} B^{2} b^{2} d\right )} x^{2} + 2 \,{\left (2 \, A B a b d g^{2} -{\left (b^{2} c g^{2} - 2 \,{\left (g^{2} \log \left (e\right ) + g^{2}\right )} a b d\right )} B^{2}\right )} x + 2 \,{\left (B^{2} b^{2} d g^{2} x^{2} + 2 \, B^{2} a b d g^{2} x + B^{2} a^{2} d g^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{d^{2} i x + c d i}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 (\frac{b e x + a e}{d x + c}\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 (\frac{b e x + a e}{d x + c}\right )}{d i x + c i}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b g x + a g\right )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{d i x + c i}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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