Optimal. Leaf size=767 \[ -\frac {1}{12} d^2 \text {Li}_3(d x) (d (3 a d+4 b)+6 c)-\frac {1}{6} d^2 \text {Li}_3(1-d x) (d (3 a d+4 b)+6 c)+\frac {1}{12} d^2 \text {Li}_2(d x) \log (1-d x) (d (3 a d+4 b)+6 c)+\frac {1}{6} d^2 \text {Li}_2(1-d x) \log (1-d x) (d (3 a d+4 b)+6 c)+\frac {1}{12} d^2 \log (d x) \log ^2(1-d x) (d (3 a d+4 b)+6 c)-\frac {1}{12} d^2 \log (x) (d (3 a d+4 b)+6 c)+\frac {1}{12} d^2 \log (1-d x) (d (3 a d+4 b)+6 c)-\frac {1}{12} \text {Li}_2(d x) \log (1-d x) \left (\frac {3 a}{x^4}+\frac {4 b}{x^3}+\frac {6 c}{x^2}\right )+\frac {d \text {Li}_2(d x) (d (3 a d+4 b)+6 c)}{12 x}-\frac {d \log (1-d x) (d (3 a d+4 b)+6 c)}{12 x}-\frac {1}{48} d^3 \log (x) (3 a d+4 b)+\frac {1}{48} d^3 (3 a d+4 b) \log (1-d x)+\frac {d^2 (3 a d+4 b)}{48 x}+\frac {d (3 a d+4 b) \text {Li}_2(d x)}{24 x^2}-\frac {d (3 a d+4 b) \log (1-d x)}{48 x^2}-\frac {1}{8} a d^4 \text {Li}_2(d x)-\frac {1}{16} a d^4 \log ^2(1-d x)-\frac {37}{144} a d^4 \log (x)+\frac {37}{144} a d^4 \log (1-d x)+\frac {19 a d^3}{144 x}-\frac {a d^3 \log (1-d x)}{8 x}+\frac {5 a d^2}{144 x^2}-\frac {a d^2 \log (1-d x)}{16 x^2}+\frac {a d \text {Li}_2(d x)}{12 x^3}+\frac {a \log ^2(1-d x)}{16 x^4}-\frac {5 a d \log (1-d x)}{72 x^3}-\frac {2}{9} b d^3 \text {Li}_2(d x)-\frac {1}{9} b d^3 \log ^2(1-d x)-\frac {1}{3} b d^3 \log (x)+\frac {1}{3} b d^3 \log (1-d x)+\frac {b d^2}{9 x}-\frac {2 b d^2 \log (1-d x)}{9 x}+\frac {b \log ^2(1-d x)}{9 x^3}-\frac {b d \log (1-d x)}{9 x^2}-\frac {1}{2} c d^2 \text {Li}_2(d x)-\frac {1}{4} c d^2 \log ^2(1-d x)-\frac {1}{2} c d^2 \log (x)+\frac {1}{2} c d^2 \log (1-d x)+\frac {c \log ^2(1-d x)}{4 x^2}-\frac {c d \log (1-d x)}{2 x} \]
[Out]
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Rubi [A] time = 1.12, antiderivative size = 767, normalized size of antiderivative = 1.00, 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 14
Rule 29
Rule 31
Rule 36
Rule 44
Rule 2301
Rule 2374
Rule 2390
Rule 2391
Rule 2395
Rule 2396
Rule 2398
Rule 2410
Rule 2433
Rule 6589
Rule 6591
Rule 6596
Rule 6606
Rule 6742
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*}
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Mathematica [A] time = 1.91, size = 621, normalized size = 0.81 \[ \frac {1}{144} \left (2 d^2 \text {Li}_2(1-d x) \left (12 \log (1-d x) \left (3 a d^2+4 b d+6 c\right )+9 a d^2+16 b d+36 c\right )+\frac {6 \text {Li}_2(d x) \left (d x \left (a \left (6 d^2 x^2+3 d x+2\right )+4 x (2 b d x+b+3 c x)\right )+2 \log (1-d x) \left (3 a \left (d^4 x^4-1\right )+4 b d^3 x^4-4 b x+6 c d^2 x^4-6 c x^2\right )\right )}{x^4}-36 a d^4 \text {Li}_3(d x)-72 a d^4 \text {Li}_3(1-d x)-9 a d^4 \log ^2(1-d x)+36 a d^4 \log (d x) \log ^2(1-d x)-82 a d^4 \log (d x)+82 a d^4 \log (1-d x)+18 a d^4 \log (d x) \log (1-d x)-33 a d^4+\frac {28 a d^3}{x}-\frac {54 a d^3 \log (1-d x)}{x}+\frac {5 a d^2}{x^2}-\frac {18 a d^2 \log (1-d x)}{x^2}+\frac {9 a \log ^2(1-d x)}{x^4}-\frac {10 a d \log (1-d x)}{x^3}-48 b d^3 \text {Li}_3(d x)-96 b d^3 \text {Li}_3(1-d x)-16 b d^3 \log ^2(1-d x)+48 b d^3 \log (d x) \log ^2(1-d x)-108 b d^3 \log (d x)+108 b d^3 \log (1-d x)+32 b d^3 \log (d x) \log (1-d x)-28 b d^3+\frac {28 b d^2}{x}-\frac {80 b d^2 \log (1-d x)}{x}+\frac {16 b \log ^2(1-d x)}{x^3}-\frac {28 b d \log (1-d x)}{x^2}-72 c d^2 \text {Li}_3(d x)-144 c d^2 \text {Li}_3(1-d x)-36 c d^2 \log ^2(1-d x)+72 c d^2 \log (d x) \log ^2(1-d x)-144 c d^2 \log (d x)+144 c d^2 \log (1-d x)+72 c d^2 \log (d x) \log (1-d x)+\frac {36 c \log ^2(1-d x)}{x^2}-\frac {144 c d \log (1-d x)}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 2.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )} {\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )} {\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \,x^{2}+b x +a \right ) \ln \left (-d x +1\right ) \polylog \left (2, d x \right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 403, normalized size = 0.53 \[ \frac {1}{12} \, {\left (3 \, a d^{4} + 4 \, b d^{3} + 6 \, c d^{2}\right )} {\left (\log \left (d x\right ) \log \left (-d x + 1\right )^{2} + 2 \, {\rm Li}_2\left (-d x + 1\right ) \log \left (-d x + 1\right ) - 2 \, {\rm Li}_{3}(-d x + 1)\right )} + \frac {1}{72} \, {\left (9 \, a d^{4} + 16 \, b d^{3} + 36 \, c d^{2}\right )} {\left (\log \left (d x\right ) \log \left (-d x + 1\right ) + {\rm Li}_2\left (-d x + 1\right )\right )} - \frac {1}{72} \, {\left (41 \, a d^{4} + 54 \, b d^{3} + 72 \, c d^{2}\right )} \log \relax (x) - \frac {1}{12} \, {\left (3 \, a d^{4} + 4 \, b d^{3} + 6 \, c d^{2}\right )} {\rm Li}_{3}(d x) + \frac {5 \, a d^{2} x^{2} + 28 \, {\left (a d^{3} + b d^{2}\right )} x^{3} - {\left ({\left (9 \, a d^{4} + 16 \, b d^{3} + 36 \, c d^{2}\right )} x^{4} - 36 \, c x^{2} - 16 \, b x - 9 \, a\right )} \log \left (-d x + 1\right )^{2} + 6 \, {\left (2 \, {\left (3 \, a d^{3} + 4 \, b d^{2} + 6 \, c d\right )} x^{3} + 2 \, a d x + {\left (3 \, a d^{2} + 4 \, b d\right )} x^{2} + 2 \, {\left ({\left (3 \, a d^{4} + 4 \, b d^{3} + 6 \, c d^{2}\right )} x^{4} - 6 \, c x^{2} - 4 \, b x - 3 \, a\right )} \log \left (-d x + 1\right )\right )} {\rm Li}_2\left (d x\right ) + 2 \, {\left ({\left (41 \, a d^{4} + 54 \, b d^{3} + 72 \, c d^{2}\right )} x^{4} - {\left (27 \, a d^{3} + 40 \, b d^{2} + 72 \, c d\right )} x^{3} - 5 \, a d x - {\left (9 \, a d^{2} + 14 \, b d\right )} x^{2}\right )} \log \left (-d x + 1\right )}{144 \, x^{4}} \]
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 \[ \int \frac {\ln \left (1-d\,x\right )\,\mathrm {polylog}\left (2,d\,x\right )\,\left (c\,x^2+b\,x+a\right )}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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