Optimal. Leaf size=767 \[ -\frac{1}{12} d^2 \text{PolyLog}(3,d x) (d (3 a d+4 b)+6 c)-\frac{1}{6} d^2 \text{PolyLog}(3,1-d x) (d (3 a d+4 b)+6 c)+\frac{1}{12} d^2 \log (1-d x) \text{PolyLog}(2,d x) (d (3 a d+4 b)+6 c)+\frac{1}{6} d^2 \log (1-d x) \text{PolyLog}(2,1-d x) (d (3 a d+4 b)+6 c)-\frac{1}{12} \log (1-d x) \text{PolyLog}(2,d x) \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right )+\frac{d \text{PolyLog}(2,d x) (d (3 a d+4 b)+6 c)}{12 x}+\frac{d (3 a d+4 b) \text{PolyLog}(2,d x)}{24 x^2}-\frac{1}{8} a d^4 \text{PolyLog}(2,d x)+\frac{a d \text{PolyLog}(2,d x)}{12 x^3}-\frac{2}{9} b d^3 \text{PolyLog}(2,d x)-\frac{1}{2} c d^2 \text{PolyLog}(2,d x)+\frac{1}{12} d^2 \log (d x) \log ^2(1-d x) (d (3 a d+4 b)+6 c)-\frac{1}{12} d^2 \log (x) (d (3 a d+4 b)+6 c)+\frac{1}{12} d^2 \log (1-d x) (d (3 a d+4 b)+6 c)-\frac{d \log (1-d x) (d (3 a d+4 b)+6 c)}{12 x}+\frac{d^2 (3 a d+4 b)}{48 x}-\frac{1}{48} d^3 \log (x) (3 a d+4 b)+\frac{1}{48} d^3 (3 a d+4 b) \log (1-d x)-\frac{d (3 a d+4 b) \log (1-d x)}{48 x^2}+\frac{5 a d^2}{144 x^2}-\frac{a d^2 \log (1-d x)}{16 x^2}+\frac{19 a d^3}{144 x}-\frac{1}{16} a d^4 \log ^2(1-d x)-\frac{37}{144} a d^4 \log (x)+\frac{37}{144} a d^4 \log (1-d x)-\frac{a d^3 \log (1-d x)}{8 x}+\frac{a \log ^2(1-d x)}{16 x^4}-\frac{5 a d \log (1-d x)}{72 x^3}+\frac{b d^2}{9 x}-\frac{1}{9} b d^3 \log ^2(1-d x)-\frac{1}{3} b d^3 \log (x)+\frac{1}{3} b d^3 \log (1-d x)-\frac{2 b d^2 \log (1-d x)}{9 x}+\frac{b \log ^2(1-d x)}{9 x^3}-\frac{b d \log (1-d x)}{9 x^2}-\frac{1}{4} c d^2 \log ^2(1-d x)-\frac{1}{2} c d^2 \log (x)+\frac{1}{2} c d^2 \log (1-d x)+\frac{c \log ^2(1-d x)}{4 x^2}-\frac{c d \log (1-d x)}{2 x} \]
[Out]
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Rubi [A] time = 1.12465, antiderivative size = 767, normalized size of antiderivative = 1., number of steps used = 61, number of rules used = 19, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.731, Rules used = {6742, 6591, 2395, 44, 14, 6606, 2398, 2410, 36, 29, 31, 2391, 2390, 2301, 6589, 6596, 2396, 2433, 2374} \[ -\frac{1}{12} d^2 \text{PolyLog}(3,d x) (d (3 a d+4 b)+6 c)-\frac{1}{6} d^2 \text{PolyLog}(3,1-d x) (d (3 a d+4 b)+6 c)+\frac{1}{12} d^2 \log (1-d x) \text{PolyLog}(2,d x) (d (3 a d+4 b)+6 c)+\frac{1}{6} d^2 \log (1-d x) \text{PolyLog}(2,1-d x) (d (3 a d+4 b)+6 c)-\frac{1}{12} \log (1-d x) \text{PolyLog}(2,d x) \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right )+\frac{d \text{PolyLog}(2,d x) (d (3 a d+4 b)+6 c)}{12 x}+\frac{d (3 a d+4 b) \text{PolyLog}(2,d x)}{24 x^2}-\frac{1}{8} a d^4 \text{PolyLog}(2,d x)+\frac{a d \text{PolyLog}(2,d x)}{12 x^3}-\frac{2}{9} b d^3 \text{PolyLog}(2,d x)-\frac{1}{2} c d^2 \text{PolyLog}(2,d x)+\frac{1}{12} d^2 \log (d x) \log ^2(1-d x) (d (3 a d+4 b)+6 c)-\frac{1}{12} d^2 \log (x) (d (3 a d+4 b)+6 c)+\frac{1}{12} d^2 \log (1-d x) (d (3 a d+4 b)+6 c)-\frac{d \log (1-d x) (d (3 a d+4 b)+6 c)}{12 x}+\frac{d^2 (3 a d+4 b)}{48 x}-\frac{1}{48} d^3 \log (x) (3 a d+4 b)+\frac{1}{48} d^3 (3 a d+4 b) \log (1-d x)-\frac{d (3 a d+4 b) \log (1-d x)}{48 x^2}+\frac{5 a d^2}{144 x^2}-\frac{a d^2 \log (1-d x)}{16 x^2}+\frac{19 a d^3}{144 x}-\frac{1}{16} a d^4 \log ^2(1-d x)-\frac{37}{144} a d^4 \log (x)+\frac{37}{144} a d^4 \log (1-d x)-\frac{a d^3 \log (1-d x)}{8 x}+\frac{a \log ^2(1-d x)}{16 x^4}-\frac{5 a d \log (1-d x)}{72 x^3}+\frac{b d^2}{9 x}-\frac{1}{9} b d^3 \log ^2(1-d x)-\frac{1}{3} b d^3 \log (x)+\frac{1}{3} b d^3 \log (1-d x)-\frac{2 b d^2 \log (1-d x)}{9 x}+\frac{b \log ^2(1-d x)}{9 x^3}-\frac{b d \log (1-d x)}{9 x^2}-\frac{1}{4} c d^2 \log ^2(1-d x)-\frac{1}{2} c d^2 \log (x)+\frac{1}{2} c d^2 \log (1-d x)+\frac{c \log ^2(1-d x)}{4 x^2}-\frac{c d \log (1-d x)}{2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 6591
Rule 2395
Rule 44
Rule 14
Rule 6606
Rule 2398
Rule 2410
Rule 36
Rule 29
Rule 31
Rule 2391
Rule 2390
Rule 2301
Rule 6589
Rule 6596
Rule 2396
Rule 2433
Rule 2374
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right ) \log (1-d x) \text{Li}_2(d x)}{x^5} \, dx &=-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)+d \int \left (-\frac{a \text{Li}_2(d x)}{4 x^4}+\frac{(-4 b-3 a d) \text{Li}_2(d x)}{12 x^3}+\frac{\left (-6 c-4 b d-3 a d^2\right ) \text{Li}_2(d x)}{12 x^2}+\frac{d (-6 c-d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{d^2 (-6 c-d (4 b+3 a d)) \text{Li}_2(d x)}{12 (1-d x)}\right ) \, dx+\int \left (-\frac{a \log ^2(1-d x)}{4 x^5}-\frac{b \log ^2(1-d x)}{3 x^4}-\frac{c \log ^2(1-d x)}{2 x^3}\right ) \, dx\\ &=-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{4} a \int \frac{\log ^2(1-d x)}{x^5} \, dx-\frac{1}{3} b \int \frac{\log ^2(1-d x)}{x^4} \, dx-\frac{1}{2} c \int \frac{\log ^2(1-d x)}{x^3} \, dx-\frac{1}{4} (a d) \int \frac{\text{Li}_2(d x)}{x^4} \, dx-\frac{1}{12} (d (4 b+3 a d)) \int \frac{\text{Li}_2(d x)}{x^3} \, dx-\frac{1}{12} (d (6 c+d (4 b+3 a d))) \int \frac{\text{Li}_2(d x)}{x^2} \, dx-\frac{1}{12} \left (d^2 (6 c+d (4 b+3 a d))\right ) \int \frac{\text{Li}_2(d x)}{x} \, dx-\frac{1}{12} \left (d^3 (6 c+d (4 b+3 a d))\right ) \int \frac{\text{Li}_2(d x)}{1-d x} \, dx\\ &=\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)+\frac{1}{12} (a d) \int \frac{\log (1-d x)}{x^4} \, dx+\frac{1}{8} (a d) \int \frac{\log (1-d x)}{x^4 (1-d x)} \, dx+\frac{1}{9} (2 b d) \int \frac{\log (1-d x)}{x^3 (1-d x)} \, dx+\frac{1}{2} (c d) \int \frac{\log (1-d x)}{x^2 (1-d x)} \, dx+\frac{1}{24} (d (4 b+3 a d)) \int \frac{\log (1-d x)}{x^3} \, dx+\frac{1}{12} (d (6 c+d (4 b+3 a d))) \int \frac{\log (1-d x)}{x^2} \, dx+\frac{1}{12} \left (d^2 (6 c+d (4 b+3 a d))\right ) \int \frac{\log ^2(1-d x)}{x} \, dx\\ &=-\frac{a d \log (1-d x)}{36 x^3}-\frac{d (4 b+3 a d) \log (1-d x)}{48 x^2}-\frac{d (6 c+d (4 b+3 a d)) \log (1-d x)}{12 x}+\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (d x) \log ^2(1-d x)+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)+\frac{1}{8} (a d) \int \left (\frac{\log (1-d x)}{x^4}+\frac{d \log (1-d x)}{x^3}+\frac{d^2 \log (1-d x)}{x^2}+\frac{d^3 \log (1-d x)}{x}-\frac{d^4 \log (1-d x)}{-1+d x}\right ) \, dx+\frac{1}{9} (2 b d) \int \left (\frac{\log (1-d x)}{x^3}+\frac{d \log (1-d x)}{x^2}+\frac{d^2 \log (1-d x)}{x}-\frac{d^3 \log (1-d x)}{-1+d x}\right ) \, dx+\frac{1}{2} (c d) \int \left (\frac{\log (1-d x)}{x^2}+\frac{d \log (1-d x)}{x}-\frac{d^2 \log (1-d x)}{-1+d x}\right ) \, dx-\frac{1}{36} \left (a d^2\right ) \int \frac{1}{x^3 (1-d x)} \, dx-\frac{1}{48} \left (d^2 (4 b+3 a d)\right ) \int \frac{1}{x^2 (1-d x)} \, dx-\frac{1}{12} \left (d^2 (6 c+d (4 b+3 a d))\right ) \int \frac{1}{x (1-d x)} \, dx+\frac{1}{6} \left (d^3 (6 c+d (4 b+3 a d))\right ) \int \frac{\log (d x) \log (1-d x)}{1-d x} \, dx\\ &=-\frac{a d \log (1-d x)}{36 x^3}-\frac{d (4 b+3 a d) \log (1-d x)}{48 x^2}-\frac{d (6 c+d (4 b+3 a d)) \log (1-d x)}{12 x}+\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (d x) \log ^2(1-d x)+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)+\frac{1}{8} (a d) \int \frac{\log (1-d x)}{x^4} \, dx+\frac{1}{9} (2 b d) \int \frac{\log (1-d x)}{x^3} \, dx+\frac{1}{2} (c d) \int \frac{\log (1-d x)}{x^2} \, dx-\frac{1}{36} \left (a d^2\right ) \int \left (\frac{1}{x^3}+\frac{d}{x^2}+\frac{d^2}{x}-\frac{d^3}{-1+d x}\right ) \, dx+\frac{1}{8} \left (a d^2\right ) \int \frac{\log (1-d x)}{x^3} \, dx+\frac{1}{9} \left (2 b d^2\right ) \int \frac{\log (1-d x)}{x^2} \, dx+\frac{1}{2} \left (c d^2\right ) \int \frac{\log (1-d x)}{x} \, dx+\frac{1}{8} \left (a d^3\right ) \int \frac{\log (1-d x)}{x^2} \, dx+\frac{1}{9} \left (2 b d^3\right ) \int \frac{\log (1-d x)}{x} \, dx-\frac{1}{2} \left (c d^3\right ) \int \frac{\log (1-d x)}{-1+d x} \, dx+\frac{1}{8} \left (a d^4\right ) \int \frac{\log (1-d x)}{x} \, dx-\frac{1}{9} \left (2 b d^4\right ) \int \frac{\log (1-d x)}{-1+d x} \, dx-\frac{1}{8} \left (a d^5\right ) \int \frac{\log (1-d x)}{-1+d x} \, dx-\frac{1}{48} \left (d^2 (4 b+3 a d)\right ) \int \left (\frac{1}{x^2}+\frac{d}{x}-\frac{d^2}{-1+d x}\right ) \, dx-\frac{1}{12} \left (d^2 (6 c+d (4 b+3 a d))\right ) \int \frac{1}{x} \, dx-\frac{1}{6} \left (d^2 (6 c+d (4 b+3 a d))\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (d \left (\frac{1}{d}-\frac{x}{d}\right )\right )}{x} \, dx,x,1-d x\right )-\frac{1}{12} \left (d^3 (6 c+d (4 b+3 a d))\right ) \int \frac{1}{1-d x} \, dx\\ &=\frac{a d^2}{72 x^2}+\frac{a d^3}{36 x}+\frac{d^2 (4 b+3 a d)}{48 x}-\frac{1}{36} a d^4 \log (x)-\frac{1}{48} d^3 (4 b+3 a d) \log (x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (x)+\frac{1}{36} a d^4 \log (1-d x)+\frac{1}{48} d^3 (4 b+3 a d) \log (1-d x)+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x)-\frac{5 a d \log (1-d x)}{72 x^3}-\frac{b d \log (1-d x)}{9 x^2}-\frac{a d^2 \log (1-d x)}{16 x^2}-\frac{d (4 b+3 a d) \log (1-d x)}{48 x^2}-\frac{c d \log (1-d x)}{2 x}-\frac{2 b d^2 \log (1-d x)}{9 x}-\frac{a d^3 \log (1-d x)}{8 x}-\frac{d (6 c+d (4 b+3 a d)) \log (1-d x)}{12 x}+\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (d x) \log ^2(1-d x)-\frac{1}{2} c d^2 \text{Li}_2(d x)-\frac{2}{9} b d^3 \text{Li}_2(d x)-\frac{1}{8} a d^4 \text{Li}_2(d x)+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)+\frac{1}{6} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(1-d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)-\frac{1}{24} \left (a d^2\right ) \int \frac{1}{x^3 (1-d x)} \, dx-\frac{1}{9} \left (b d^2\right ) \int \frac{1}{x^2 (1-d x)} \, dx-\frac{1}{2} \left (c d^2\right ) \int \frac{1}{x (1-d x)} \, dx-\frac{1}{2} \left (c d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-d x\right )-\frac{1}{16} \left (a d^3\right ) \int \frac{1}{x^2 (1-d x)} \, dx-\frac{1}{9} \left (2 b d^3\right ) \int \frac{1}{x (1-d x)} \, dx-\frac{1}{9} \left (2 b d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-d x\right )-\frac{1}{8} \left (a d^4\right ) \int \frac{1}{x (1-d x)} \, dx-\frac{1}{8} \left (a d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-d x\right )-\frac{1}{6} \left (d^2 (6 c+d (4 b+3 a d))\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-d x\right )\\ &=\frac{a d^2}{72 x^2}+\frac{a d^3}{36 x}+\frac{d^2 (4 b+3 a d)}{48 x}-\frac{1}{36} a d^4 \log (x)-\frac{1}{48} d^3 (4 b+3 a d) \log (x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (x)+\frac{1}{36} a d^4 \log (1-d x)+\frac{1}{48} d^3 (4 b+3 a d) \log (1-d x)+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x)-\frac{5 a d \log (1-d x)}{72 x^3}-\frac{b d \log (1-d x)}{9 x^2}-\frac{a d^2 \log (1-d x)}{16 x^2}-\frac{d (4 b+3 a d) \log (1-d x)}{48 x^2}-\frac{c d \log (1-d x)}{2 x}-\frac{2 b d^2 \log (1-d x)}{9 x}-\frac{a d^3 \log (1-d x)}{8 x}-\frac{d (6 c+d (4 b+3 a d)) \log (1-d x)}{12 x}-\frac{1}{4} c d^2 \log ^2(1-d x)-\frac{1}{9} b d^3 \log ^2(1-d x)-\frac{1}{16} a d^4 \log ^2(1-d x)+\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (d x) \log ^2(1-d x)-\frac{1}{2} c d^2 \text{Li}_2(d x)-\frac{2}{9} b d^3 \text{Li}_2(d x)-\frac{1}{8} a d^4 \text{Li}_2(d x)+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)+\frac{1}{6} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(1-d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)-\frac{1}{6} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(1-d x)-\frac{1}{24} \left (a d^2\right ) \int \left (\frac{1}{x^3}+\frac{d}{x^2}+\frac{d^2}{x}-\frac{d^3}{-1+d x}\right ) \, dx-\frac{1}{9} \left (b d^2\right ) \int \left (\frac{1}{x^2}+\frac{d}{x}-\frac{d^2}{-1+d x}\right ) \, dx-\frac{1}{2} \left (c d^2\right ) \int \frac{1}{x} \, dx-\frac{1}{16} \left (a d^3\right ) \int \left (\frac{1}{x^2}+\frac{d}{x}-\frac{d^2}{-1+d x}\right ) \, dx-\frac{1}{9} \left (2 b d^3\right ) \int \frac{1}{x} \, dx-\frac{1}{2} \left (c d^3\right ) \int \frac{1}{1-d x} \, dx-\frac{1}{8} \left (a d^4\right ) \int \frac{1}{x} \, dx-\frac{1}{9} \left (2 b d^4\right ) \int \frac{1}{1-d x} \, dx-\frac{1}{8} \left (a d^5\right ) \int \frac{1}{1-d x} \, dx\\ &=\frac{5 a d^2}{144 x^2}+\frac{b d^2}{9 x}+\frac{19 a d^3}{144 x}+\frac{d^2 (4 b+3 a d)}{48 x}-\frac{1}{2} c d^2 \log (x)-\frac{1}{3} b d^3 \log (x)-\frac{37}{144} a d^4 \log (x)-\frac{1}{48} d^3 (4 b+3 a d) \log (x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (x)+\frac{1}{2} c d^2 \log (1-d x)+\frac{1}{3} b d^3 \log (1-d x)+\frac{37}{144} a d^4 \log (1-d x)+\frac{1}{48} d^3 (4 b+3 a d) \log (1-d x)+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x)-\frac{5 a d \log (1-d x)}{72 x^3}-\frac{b d \log (1-d x)}{9 x^2}-\frac{a d^2 \log (1-d x)}{16 x^2}-\frac{d (4 b+3 a d) \log (1-d x)}{48 x^2}-\frac{c d \log (1-d x)}{2 x}-\frac{2 b d^2 \log (1-d x)}{9 x}-\frac{a d^3 \log (1-d x)}{8 x}-\frac{d (6 c+d (4 b+3 a d)) \log (1-d x)}{12 x}-\frac{1}{4} c d^2 \log ^2(1-d x)-\frac{1}{9} b d^3 \log ^2(1-d x)-\frac{1}{16} a d^4 \log ^2(1-d x)+\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (d x) \log ^2(1-d x)-\frac{1}{2} c d^2 \text{Li}_2(d x)-\frac{2}{9} b d^3 \text{Li}_2(d x)-\frac{1}{8} a d^4 \text{Li}_2(d x)+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)+\frac{1}{6} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(1-d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)-\frac{1}{6} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(1-d x)\\ \end{align*}
Mathematica [A] time = 1.95317, size = 621, normalized size = 0.81 \[ \frac{1}{144} \left (\frac{6 \text{PolyLog}(2,d x) \left (d x \left (a \left (6 d^2 x^2+3 d x+2\right )+4 x (2 b d x+b+3 c x)\right )+2 \log (1-d x) \left (3 a \left (d^4 x^4-1\right )+4 b d^3 x^4-4 b x+6 c d^2 x^4-6 c x^2\right )\right )}{x^4}+2 d^2 \text{PolyLog}(2,1-d x) \left (12 \log (1-d x) \left (3 a d^2+4 b d+6 c\right )+9 a d^2+16 b d+36 c\right )-36 a d^4 \text{PolyLog}(3,d x)-72 a d^4 \text{PolyLog}(3,1-d x)-48 b d^3 \text{PolyLog}(3,d x)-96 b d^3 \text{PolyLog}(3,1-d x)-72 c d^2 \text{PolyLog}(3,d x)-144 c d^2 \text{PolyLog}(3,1-d x)+\frac{5 a d^2}{x^2}-\frac{18 a d^2 \log (1-d x)}{x^2}+\frac{28 a d^3}{x}-9 a d^4 \log ^2(1-d x)+36 a d^4 \log (d x) \log ^2(1-d x)-82 a d^4 \log (d x)+82 a d^4 \log (1-d x)+18 a d^4 \log (d x) \log (1-d x)-\frac{54 a d^3 \log (1-d x)}{x}-33 a d^4+\frac{9 a \log ^2(1-d x)}{x^4}-\frac{10 a d \log (1-d x)}{x^3}+\frac{28 b d^2}{x}-16 b d^3 \log ^2(1-d x)+48 b d^3 \log (d x) \log ^2(1-d x)-108 b d^3 \log (d x)+108 b d^3 \log (1-d x)+32 b d^3 \log (d x) \log (1-d x)-\frac{80 b d^2 \log (1-d x)}{x}-28 b d^3+\frac{16 b \log ^2(1-d x)}{x^3}-\frac{28 b d \log (1-d x)}{x^2}-36 c d^2 \log ^2(1-d x)+72 c d^2 \log (d x) \log ^2(1-d x)-144 c d^2 \log (d x)+144 c d^2 \log (1-d x)+72 c d^2 \log (d x) \log (1-d x)+\frac{36 c \log ^2(1-d x)}{x^2}-\frac{144 c d \log (1-d x)}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c{x}^{2}+bx+a \right ) \ln \left ( -dx+1 \right ){\it polylog} \left ( 2,dx \right ) }{{x}^{5}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23365, size = 544, normalized size = 0.71 \begin{align*} \frac{1}{12} \,{\left (3 \, a d^{4} + 4 \, b d^{3} + 6 \, c d^{2}\right )}{\left (\log \left (d x\right ) \log \left (-d x + 1\right )^{2} + 2 \,{\rm Li}_2\left (-d x + 1\right ) \log \left (-d x + 1\right ) - 2 \,{\rm Li}_{3}(-d x + 1)\right )} + \frac{1}{72} \,{\left (9 \, a d^{4} + 16 \, b d^{3} + 36 \, c d^{2}\right )}{\left (\log \left (d x\right ) \log \left (-d x + 1\right ) +{\rm Li}_2\left (-d x + 1\right )\right )} - \frac{1}{72} \,{\left (41 \, a d^{4} + 54 \, b d^{3} + 72 \, c d^{2}\right )} \log \left (x\right ) - \frac{1}{12} \,{\left (3 \, a d^{4} + 4 \, b d^{3} + 6 \, c d^{2}\right )}{\rm Li}_{3}(d x) + \frac{5 \, a d^{2} x^{2} + 28 \,{\left (a d^{3} + b d^{2}\right )} x^{3} -{\left ({\left (9 \, a d^{4} + 16 \, b d^{3} + 36 \, c d^{2}\right )} x^{4} - 36 \, c x^{2} - 16 \, b x - 9 \, a\right )} \log \left (-d x + 1\right )^{2} + 6 \,{\left (2 \,{\left (3 \, a d^{3} + 4 \, b d^{2} + 6 \, c d\right )} x^{3} + 2 \, a d x +{\left (3 \, a d^{2} + 4 \, b d\right )} x^{2} + 2 \,{\left ({\left (3 \, a d^{4} + 4 \, b d^{3} + 6 \, c d^{2}\right )} x^{4} - 6 \, c x^{2} - 4 \, b x - 3 \, a\right )} \log \left (-d x + 1\right )\right )}{\rm Li}_2\left (d x\right ) + 2 \,{\left ({\left (41 \, a d^{4} + 54 \, b d^{3} + 72 \, c d^{2}\right )} x^{4} -{\left (27 \, a d^{3} + 40 \, b d^{2} + 72 \, c d\right )} x^{3} - 5 \, a d x -{\left (9 \, a d^{2} + 14 \, b d\right )} x^{2}\right )} \log \left (-d x + 1\right )}{144 \, x^{4}} \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{{\left (c x^{2} + b x + a\right )}{\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{5}}, 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 (c x^{2} + b x + a\right )}{\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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