Optimal. Leaf size=900 \[ \frac {1}{16} c \log ^2(1-d x) x^4+\frac {3 c x^4}{256}-\frac {3}{64} c \log (1-d x) x^4-\frac {1}{16} c \text {Li}_2(d x) x^4+\frac {1}{9} b \log ^2(1-d x) x^3+\frac {2 b x^3}{81}+\frac {(3 c+4 b d) x^3}{324 d}-\frac {2}{27} b \log (1-d x) x^3-\frac {(3 c+4 b d) \log (1-d x) x^3}{108 d}-\frac {c \log (1-d x) x^3}{24 d}-\frac {(3 c+4 b d) \text {Li}_2(d x) x^3}{36 d}+\frac {17 c x^3}{576 d}+\frac {(3 c+4 b d) x^2}{216 d^2}+\frac {\left (6 a d^2+4 b d+3 c\right ) x^2}{96 d^2}-\frac {\left (6 a d^2+4 b d+3 c\right ) \log (1-d x) x^2}{48 d^2}-\frac {b \log (1-d x) x^2}{9 d}-\frac {c \log (1-d x) x^2}{16 d^2}-\frac {\left (6 a d^2+4 b d+3 c\right ) \text {Li}_2(d x) x^2}{24 d^2}+\frac {5 b x^2}{54 d}+\frac {29 c x^2}{384 d^2}+\frac {(3 c+4 b d) x}{108 d^3}+\frac {5 \left (6 a d^2+4 b d+3 c\right ) x}{48 d^3}-\frac {\left (6 a d^2+4 b d+3 c\right ) \text {Li}_2(d x) x}{12 d^3}+\frac {a x}{d}+\frac {11 b x}{27 d^2}+\frac {53 c x}{192 d^3}+\frac {a (1-d x)^2}{8 d^2}+\frac {a (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac {a (1-d x) \log ^2(1-d x)}{2 d^2}-\frac {\left (6 a d^2+4 b d+3 c\right ) \log (d x) \log ^2(1-d x)}{12 d^4}-\frac {b \log ^2(1-d x)}{9 d^3}-\frac {c \log ^2(1-d x)}{16 d^4}-\frac {a (1-d x)^2 \log (1-d x)}{4 d^2}+\frac {(3 c+4 b d) \log (1-d x)}{108 d^4}+\frac {\left (6 a d^2+4 b d+3 c\right ) \log (1-d x)}{48 d^4}+\frac {\left (6 a d^2+4 b d+3 c\right ) (1-d x) \log (1-d x)}{12 d^4}+\frac {a (1-d x) \log (1-d x)}{d^2}+\frac {2 b (1-d x) \log (1-d x)}{9 d^3}+\frac {c (1-d x) \log (1-d x)}{8 d^4}+\frac {5 b \log (1-d x)}{27 d^3}+\frac {29 c \log (1-d x)}{192 d^4}-\frac {\left (6 a d^2+4 b d+3 c\right ) \log (1-d x) \text {Li}_2(d x)}{12 d^4}+\frac {1}{12} \left (3 c x^4+4 b x^3+6 a x^2\right ) \log (1-d x) \text {Li}_2(d x)-\frac {\left (6 a d^2+4 b d+3 c\right ) \log (1-d x) \text {Li}_2(1-d x)}{6 d^4}+\frac {\left (6 a d^2+4 b d+3 c\right ) \text {Li}_3(1-d x)}{6 d^4} \]
[Out]
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Rubi [A] time = 1.18, antiderivative size = 900, normalized size of antiderivative = 1.00, number of steps used = 60, number of rules used = 22, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6742, 6591, 2395, 43, 14, 6604, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2398, 2410, 2301, 6586, 6596, 2396, 2433, 2374, 6589} \[ \frac {1}{16} c \log ^2(1-d x) x^4+\frac {3 c x^4}{256}-\frac {3}{64} c \log (1-d x) x^4-\frac {1}{16} c \text {PolyLog}(2,d x) x^4+\frac {1}{9} b \log ^2(1-d x) x^3+\frac {2 b x^3}{81}+\frac {(3 c+4 b d) x^3}{324 d}-\frac {2}{27} b \log (1-d x) x^3-\frac {(3 c+4 b d) \log (1-d x) x^3}{108 d}-\frac {c \log (1-d x) x^3}{24 d}-\frac {(3 c+4 b d) \text {PolyLog}(2,d x) x^3}{36 d}+\frac {17 c x^3}{576 d}+\frac {(3 c+4 b d) x^2}{216 d^2}+\frac {\left (6 a d^2+4 b d+3 c\right ) x^2}{96 d^2}-\frac {\left (6 a d^2+4 b d+3 c\right ) \log (1-d x) x^2}{48 d^2}-\frac {b \log (1-d x) x^2}{9 d}-\frac {c \log (1-d x) x^2}{16 d^2}-\frac {\left (6 a d^2+4 b d+3 c\right ) \text {PolyLog}(2,d x) x^2}{24 d^2}+\frac {5 b x^2}{54 d}+\frac {29 c x^2}{384 d^2}+\frac {(3 c+4 b d) x}{108 d^3}+\frac {5 \left (6 a d^2+4 b d+3 c\right ) x}{48 d^3}-\frac {\left (6 a d^2+4 b d+3 c\right ) \text {PolyLog}(2,d x) x}{12 d^3}+\frac {a x}{d}+\frac {11 b x}{27 d^2}+\frac {53 c x}{192 d^3}+\frac {a (1-d x)^2}{8 d^2}+\frac {a (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac {a (1-d x) \log ^2(1-d x)}{2 d^2}-\frac {\left (6 a d^2+4 b d+3 c\right ) \log (d x) \log ^2(1-d x)}{12 d^4}-\frac {b \log ^2(1-d x)}{9 d^3}-\frac {c \log ^2(1-d x)}{16 d^4}-\frac {a (1-d x)^2 \log (1-d x)}{4 d^2}+\frac {(3 c+4 b d) \log (1-d x)}{108 d^4}+\frac {\left (6 a d^2+4 b d+3 c\right ) \log (1-d x)}{48 d^4}+\frac {\left (6 a d^2+4 b d+3 c\right ) (1-d x) \log (1-d x)}{12 d^4}+\frac {a (1-d x) \log (1-d x)}{d^2}+\frac {2 b (1-d x) \log (1-d x)}{9 d^3}+\frac {c (1-d x) \log (1-d x)}{8 d^4}+\frac {5 b \log (1-d x)}{27 d^3}+\frac {29 c \log (1-d x)}{192 d^4}-\frac {\left (6 a d^2+4 b d+3 c\right ) \log (1-d x) \text {PolyLog}(2,d x)}{12 d^4}+\frac {1}{12} \left (3 c x^4+4 b x^3+6 a x^2\right ) \log (1-d x) \text {PolyLog}(2,d x)-\frac {\left (6 a d^2+4 b d+3 c\right ) \log (1-d x) \text {PolyLog}(2,1-d x)}{6 d^4}+\frac {\left (6 a d^2+4 b d+3 c\right ) \text {PolyLog}(3,1-d x)}{6 d^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 43
Rule 2295
Rule 2296
Rule 2301
Rule 2304
Rule 2305
Rule 2374
Rule 2389
Rule 2390
Rule 2395
Rule 2396
Rule 2398
Rule 2401
Rule 2410
Rule 2433
Rule 6586
Rule 6589
Rule 6591
Rule 6596
Rule 6604
Rule 6742
Rubi steps
\begin {align*} \int x \left (a+b x+c x^2\right ) \log (1-d x) \text {Li}_2(d x) \, dx &=\frac {1}{12} \left (6 a x^2+4 b x^3+3 c x^4\right ) \log (1-d x) \text {Li}_2(d x)+d \int \left (\frac {\left (-3 c-4 b d-6 a d^2\right ) \text {Li}_2(d x)}{12 d^4}-\frac {\left (3 c+4 b d+6 a d^2\right ) x \text {Li}_2(d x)}{12 d^3}-\frac {(3 c+4 b d) x^2 \text {Li}_2(d x)}{12 d^2}-\frac {c x^3 \text {Li}_2(d x)}{4 d}+\frac {\left (-3 c-4 b d-6 a d^2\right ) \text {Li}_2(d x)}{12 d^4 (-1+d x)}\right ) \, dx+\int \left (\frac {1}{2} a x \log ^2(1-d x)+\frac {1}{3} b x^2 \log ^2(1-d x)+\frac {1}{4} c x^3 \log ^2(1-d x)\right ) \, dx\\ &=\frac {1}{12} \left (6 a x^2+4 b x^3+3 c x^4\right ) \log (1-d x) \text {Li}_2(d x)+\frac {1}{2} a \int x \log ^2(1-d x) \, dx+\frac {1}{3} b \int x^2 \log ^2(1-d x) \, dx+\frac {1}{4} c \int x^3 \log ^2(1-d x) \, dx-\frac {1}{4} c \int x^3 \text {Li}_2(d x) \, dx-\frac {(3 c+4 b d) \int x^2 \text {Li}_2(d x) \, dx}{12 d}-\frac {\left (3 c+4 b d+6 a d^2\right ) \int \text {Li}_2(d x) \, dx}{12 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) \int \frac {\text {Li}_2(d x)}{-1+d x} \, dx}{12 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) \int x \text {Li}_2(d x) \, dx}{12 d^2}\\ &=\frac {1}{9} b x^3 \log ^2(1-d x)+\frac {1}{16} c x^4 \log ^2(1-d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) x \text {Li}_2(d x)}{12 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \text {Li}_2(d x)}{24 d^2}-\frac {(3 c+4 b d) x^3 \text {Li}_2(d x)}{36 d}-\frac {1}{16} c x^4 \text {Li}_2(d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x) \text {Li}_2(d x)}{12 d^4}+\frac {1}{12} \left (6 a x^2+4 b x^3+3 c x^4\right ) \log (1-d x) \text {Li}_2(d x)+\frac {1}{2} a \int \left (\frac {\log ^2(1-d x)}{d}-\frac {(1-d x) \log ^2(1-d x)}{d}\right ) \, dx-\frac {1}{16} c \int x^3 \log (1-d x) \, dx+\frac {1}{9} (2 b d) \int \frac {x^3 \log (1-d x)}{1-d x} \, dx+\frac {1}{8} (c d) \int \frac {x^4 \log (1-d x)}{1-d x} \, dx-\frac {(3 c+4 b d) \int x^2 \log (1-d x) \, dx}{36 d}-\frac {\left (3 c+4 b d+6 a d^2\right ) \int \frac {\log ^2(1-d x)}{x} \, dx}{12 d^4}-\frac {\left (3 c+4 b d+6 a d^2\right ) \int \log (1-d x) \, dx}{12 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) \int x \log (1-d x) \, dx}{24 d^2}\\ &=-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \log (1-d x)}{48 d^2}-\frac {(3 c+4 b d) x^3 \log (1-d x)}{108 d}-\frac {1}{64} c x^4 \log (1-d x)+\frac {1}{9} b x^3 \log ^2(1-d x)+\frac {1}{16} c x^4 \log ^2(1-d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (d x) \log ^2(1-d x)}{12 d^4}-\frac {\left (3 c+4 b d+6 a d^2\right ) x \text {Li}_2(d x)}{12 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \text {Li}_2(d x)}{24 d^2}-\frac {(3 c+4 b d) x^3 \text {Li}_2(d x)}{36 d}-\frac {1}{16} c x^4 \text {Li}_2(d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x) \text {Li}_2(d x)}{12 d^4}+\frac {1}{12} \left (6 a x^2+4 b x^3+3 c x^4\right ) \log (1-d x) \text {Li}_2(d x)+\frac {a \int \log ^2(1-d x) \, dx}{2 d}-\frac {a \int (1-d x) \log ^2(1-d x) \, dx}{2 d}+\frac {1}{9} (2 b d) \int \left (-\frac {\log (1-d x)}{d^3}-\frac {x \log (1-d x)}{d^2}-\frac {x^2 \log (1-d x)}{d}-\frac {\log (1-d x)}{d^3 (-1+d x)}\right ) \, dx-\frac {1}{64} (c d) \int \frac {x^4}{1-d x} \, dx+\frac {1}{8} (c d) \int \left (-\frac {\log (1-d x)}{d^4}-\frac {x \log (1-d x)}{d^3}-\frac {x^2 \log (1-d x)}{d^2}-\frac {x^3 \log (1-d x)}{d}-\frac {\log (1-d x)}{d^4 (-1+d x)}\right ) \, dx-\frac {1}{108} (3 c+4 b d) \int \frac {x^3}{1-d x} \, dx+\frac {\left (3 c+4 b d+6 a d^2\right ) \operatorname {Subst}(\int \log (x) \, dx,x,1-d x)}{12 d^4}-\frac {\left (3 c+4 b d+6 a d^2\right ) \int \frac {\log (d x) \log (1-d x)}{1-d x} \, dx}{6 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) \int \frac {x^2}{1-d x} \, dx}{48 d}\\ &=\frac {\left (3 c+4 b d+6 a d^2\right ) x}{12 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \log (1-d x)}{48 d^2}-\frac {(3 c+4 b d) x^3 \log (1-d x)}{108 d}-\frac {1}{64} c x^4 \log (1-d x)+\frac {\left (3 c+4 b d+6 a d^2\right ) (1-d x) \log (1-d x)}{12 d^4}+\frac {1}{9} b x^3 \log ^2(1-d x)+\frac {1}{16} c x^4 \log ^2(1-d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (d x) \log ^2(1-d x)}{12 d^4}-\frac {\left (3 c+4 b d+6 a d^2\right ) x \text {Li}_2(d x)}{12 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \text {Li}_2(d x)}{24 d^2}-\frac {(3 c+4 b d) x^3 \text {Li}_2(d x)}{36 d}-\frac {1}{16} c x^4 \text {Li}_2(d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x) \text {Li}_2(d x)}{12 d^4}+\frac {1}{12} \left (6 a x^2+4 b x^3+3 c x^4\right ) \log (1-d x) \text {Li}_2(d x)-\frac {1}{9} (2 b) \int x^2 \log (1-d x) \, dx-\frac {1}{8} c \int x^3 \log (1-d x) \, dx-\frac {c \int \log (1-d x) \, dx}{8 d^3}-\frac {c \int \frac {\log (1-d x)}{-1+d x} \, dx}{8 d^3}-\frac {a \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1-d x\right )}{2 d^2}+\frac {a \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1-d x\right )}{2 d^2}-\frac {(2 b) \int \log (1-d x) \, dx}{9 d^2}-\frac {(2 b) \int \frac {\log (1-d x)}{-1+d x} \, dx}{9 d^2}-\frac {c \int x \log (1-d x) \, dx}{8 d^2}-\frac {(2 b) \int x \log (1-d x) \, dx}{9 d}-\frac {c \int x^2 \log (1-d x) \, dx}{8 d}-\frac {1}{64} (c d) \int \left (-\frac {1}{d^4}-\frac {x}{d^3}-\frac {x^2}{d^2}-\frac {x^3}{d}-\frac {1}{d^4 (-1+d x)}\right ) \, dx-\frac {1}{108} (3 c+4 b d) \int \left (-\frac {1}{d^3}-\frac {x}{d^2}-\frac {x^2}{d}-\frac {1}{d^3 (-1+d x)}\right ) \, dx+\frac {\left (3 c+4 b d+6 a d^2\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 )}{6 d^4}-\frac {\left (3 c+4 b d+6 a d^2\right ) \int \left (-\frac {1}{d^2}-\frac {x}{d}-\frac {1}{d^2 (-1+d x)}\right ) \, dx}{48 d}\\ &=\frac {c x}{64 d^3}+\frac {(3 c+4 b d) x}{108 d^3}+\frac {5 \left (3 c+4 b d+6 a d^2\right ) x}{48 d^3}+\frac {c x^2}{128 d^2}+\frac {(3 c+4 b d) x^2}{216 d^2}+\frac {\left (3 c+4 b d+6 a d^2\right ) x^2}{96 d^2}+\frac {c x^3}{192 d}+\frac {(3 c+4 b d) x^3}{324 d}+\frac {c x^4}{256}+\frac {c \log (1-d x)}{64 d^4}+\frac {(3 c+4 b d) \log (1-d x)}{108 d^4}+\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x)}{48 d^4}-\frac {c x^2 \log (1-d x)}{16 d^2}-\frac {b x^2 \log (1-d x)}{9 d}-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \log (1-d x)}{48 d^2}-\frac {2}{27} b x^3 \log (1-d x)-\frac {c x^3 \log (1-d x)}{24 d}-\frac {(3 c+4 b d) x^3 \log (1-d x)}{108 d}-\frac {3}{64} c x^4 \log (1-d x)+\frac {\left (3 c+4 b d+6 a d^2\right ) (1-d x) \log (1-d x)}{12 d^4}+\frac {1}{9} b x^3 \log ^2(1-d x)+\frac {1}{16} c x^4 \log ^2(1-d x)-\frac {a (1-d x) \log ^2(1-d x)}{2 d^2}+\frac {a (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (d x) \log ^2(1-d x)}{12 d^4}-\frac {\left (3 c+4 b d+6 a d^2\right ) x \text {Li}_2(d x)}{12 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \text {Li}_2(d x)}{24 d^2}-\frac {(3 c+4 b d) x^3 \text {Li}_2(d x)}{36 d}-\frac {1}{16} c x^4 \text {Li}_2(d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x) \text {Li}_2(d x)}{12 d^4}+\frac {1}{12} \left (6 a x^2+4 b x^3+3 c x^4\right ) \log (1-d x) \text {Li}_2(d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x) \text {Li}_2(1-d x)}{6 d^4}-\frac {1}{9} b \int \frac {x^2}{1-d x} \, dx-\frac {1}{24} c \int \frac {x^3}{1-d x} \, dx+\frac {c \operatorname {Subst}(\int \log (x) \, dx,x,1-d x)}{8 d^4}-\frac {c \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-d x\right )}{8 d^4}+\frac {(2 b) \operatorname {Subst}(\int \log (x) \, dx,x,1-d x)}{9 d^3}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-d x\right )}{9 d^3}-\frac {a \operatorname {Subst}(\int x \log (x) \, dx,x,1-d x)}{2 d^2}+\frac {a \operatorname {Subst}(\int \log (x) \, dx,x,1-d x)}{d^2}-\frac {c \int \frac {x^2}{1-d x} \, dx}{16 d}-\frac {1}{27} (2 b d) \int \frac {x^3}{1-d x} \, dx-\frac {1}{32} (c d) \int \frac {x^4}{1-d x} \, dx+\frac {\left (3 c+4 b d+6 a d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-d x\right )}{6 d^4}\\ &=\frac {9 c x}{64 d^3}+\frac {2 b x}{9 d^2}+\frac {a x}{d}+\frac {(3 c+4 b d) x}{108 d^3}+\frac {5 \left (3 c+4 b d+6 a d^2\right ) x}{48 d^3}+\frac {c x^2}{128 d^2}+\frac {(3 c+4 b d) x^2}{216 d^2}+\frac {\left (3 c+4 b d+6 a d^2\right ) x^2}{96 d^2}+\frac {c x^3}{192 d}+\frac {(3 c+4 b d) x^3}{324 d}+\frac {c x^4}{256}+\frac {a (1-d x)^2}{8 d^2}+\frac {c \log (1-d x)}{64 d^4}+\frac {(3 c+4 b d) \log (1-d x)}{108 d^4}+\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x)}{48 d^4}-\frac {c x^2 \log (1-d x)}{16 d^2}-\frac {b x^2 \log (1-d x)}{9 d}-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \log (1-d x)}{48 d^2}-\frac {2}{27} b x^3 \log (1-d x)-\frac {c x^3 \log (1-d x)}{24 d}-\frac {(3 c+4 b d) x^3 \log (1-d x)}{108 d}-\frac {3}{64} c x^4 \log (1-d x)+\frac {c (1-d x) \log (1-d x)}{8 d^4}+\frac {2 b (1-d x) \log (1-d x)}{9 d^3}+\frac {a (1-d x) \log (1-d x)}{d^2}+\frac {\left (3 c+4 b d+6 a d^2\right ) (1-d x) \log (1-d x)}{12 d^4}-\frac {a (1-d x)^2 \log (1-d x)}{4 d^2}-\frac {c \log ^2(1-d x)}{16 d^4}-\frac {b \log ^2(1-d x)}{9 d^3}+\frac {1}{9} b x^3 \log ^2(1-d x)+\frac {1}{16} c x^4 \log ^2(1-d x)-\frac {a (1-d x) \log ^2(1-d x)}{2 d^2}+\frac {a (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (d x) \log ^2(1-d x)}{12 d^4}-\frac {\left (3 c+4 b d+6 a d^2\right ) x \text {Li}_2(d x)}{12 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \text {Li}_2(d x)}{24 d^2}-\frac {(3 c+4 b d) x^3 \text {Li}_2(d x)}{36 d}-\frac {1}{16} c x^4 \text {Li}_2(d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x) \text {Li}_2(d x)}{12 d^4}+\frac {1}{12} \left (6 a x^2+4 b x^3+3 c x^4\right ) \log (1-d x) \text {Li}_2(d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x) \text {Li}_2(1-d x)}{6 d^4}+\frac {\left (3 c+4 b d+6 a d^2\right ) \text {Li}_3(1-d x)}{6 d^4}-\frac {1}{9} b \int \left (-\frac {1}{d^2}-\frac {x}{d}-\frac {1}{d^2 (-1+d x)}\right ) \, dx-\frac {1}{24} c \int \left (-\frac {1}{d^3}-\frac {x}{d^2}-\frac {x^2}{d}-\frac {1}{d^3 (-1+d x)}\right ) \, dx-\frac {c \int \left (-\frac {1}{d^2}-\frac {x}{d}-\frac {1}{d^2 (-1+d x)}\right ) \, dx}{16 d}-\frac {1}{27} (2 b d) \int \left (-\frac {1}{d^3}-\frac {x}{d^2}-\frac {x^2}{d}-\frac {1}{d^3 (-1+d x)}\right ) \, dx-\frac {1}{32} (c d) \int \left (-\frac {1}{d^4}-\frac {x}{d^3}-\frac {x^2}{d^2}-\frac {x^3}{d}-\frac {1}{d^4 (-1+d x)}\right ) \, dx\\ &=\frac {53 c x}{192 d^3}+\frac {11 b x}{27 d^2}+\frac {a x}{d}+\frac {(3 c+4 b d) x}{108 d^3}+\frac {5 \left (3 c+4 b d+6 a d^2\right ) x}{48 d^3}+\frac {29 c x^2}{384 d^2}+\frac {5 b x^2}{54 d}+\frac {(3 c+4 b d) x^2}{216 d^2}+\frac {\left (3 c+4 b d+6 a d^2\right ) x^2}{96 d^2}+\frac {2 b x^3}{81}+\frac {17 c x^3}{576 d}+\frac {(3 c+4 b d) x^3}{324 d}+\frac {3 c x^4}{256}+\frac {a (1-d x)^2}{8 d^2}+\frac {29 c \log (1-d x)}{192 d^4}+\frac {5 b \log (1-d x)}{27 d^3}+\frac {(3 c+4 b d) \log (1-d x)}{108 d^4}+\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x)}{48 d^4}-\frac {c x^2 \log (1-d x)}{16 d^2}-\frac {b x^2 \log (1-d x)}{9 d}-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \log (1-d x)}{48 d^2}-\frac {2}{27} b x^3 \log (1-d x)-\frac {c x^3 \log (1-d x)}{24 d}-\frac {(3 c+4 b d) x^3 \log (1-d x)}{108 d}-\frac {3}{64} c x^4 \log (1-d x)+\frac {c (1-d x) \log (1-d x)}{8 d^4}+\frac {2 b (1-d x) \log (1-d x)}{9 d^3}+\frac {a (1-d x) \log (1-d x)}{d^2}+\frac {\left (3 c+4 b d+6 a d^2\right ) (1-d x) \log (1-d x)}{12 d^4}-\frac {a (1-d x)^2 \log (1-d x)}{4 d^2}-\frac {c \log ^2(1-d x)}{16 d^4}-\frac {b \log ^2(1-d x)}{9 d^3}+\frac {1}{9} b x^3 \log ^2(1-d x)+\frac {1}{16} c x^4 \log ^2(1-d x)-\frac {a (1-d x) \log ^2(1-d x)}{2 d^2}+\frac {a (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (d x) \log ^2(1-d x)}{12 d^4}-\frac {\left (3 c+4 b d+6 a d^2\right ) x \text {Li}_2(d x)}{12 d^3}-\frac {\left (3 c+4 b d+6 a d^2\right ) x^2 \text {Li}_2(d x)}{24 d^2}-\frac {(3 c+4 b d) x^3 \text {Li}_2(d x)}{36 d}-\frac {1}{16} c x^4 \text {Li}_2(d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x) \text {Li}_2(d x)}{12 d^4}+\frac {1}{12} \left (6 a x^2+4 b x^3+3 c x^4\right ) \log (1-d x) \text {Li}_2(d x)-\frac {\left (3 c+4 b d+6 a d^2\right ) \log (1-d x) \text {Li}_2(1-d x)}{6 d^4}+\frac {\left (3 c+4 b d+6 a d^2\right ) \text {Li}_3(1-d x)}{6 d^4}\\ \end {align*}
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Mathematica [A] time = 1.21, size = 583, normalized size = 0.65 \[ \frac {24 \text {Li}_3(1-d x) \left (6 a d^2+4 b d+3 c\right )-24 \text {Li}_2(1-d x) \log (1-d x) \left (6 a d^2+4 b d+3 c\right )+\text {Li}_2(d x) \left (12 \log (1-d x) \left (6 a d^4 x^2-6 a d^2+4 b d^4 x^3-4 b d+3 c \left (d^4 x^4-1\right )\right )-d x \left (4 d \left (9 a d (d x+2)+2 b \left (2 d^2 x^2+3 d x+6\right )\right )+3 c \left (3 d^3 x^3+4 d^2 x^2+6 d x+12\right )\right )\right )+27 a d^4 x^2+36 a d^4 x^2 \log ^2(1-d x)-54 a d^4 x^2 \log (1-d x)+198 a d^3 x-144 a d^3 x \log (1-d x)-36 a d^2 \log ^2(1-d x)-72 a d^2 \log (d x) \log ^2(1-d x)+198 a d^2 \log (1-d x)+\frac {16}{3} b d^4 x^3+16 b d^4 x^3 \log ^2(1-d x)-16 b d^4 x^3 \log (1-d x)+22 b d^3 x^2-28 b d^3 x^2 \log (1-d x)+124 b d^2 x-80 b d^2 x \log (1-d x)-16 b d \log ^2(1-d x)-48 b d \log (d x) \log ^2(1-d x)+124 b d \log (1-d x)+\frac {27}{16} c d^4 x^4+9 c d^4 x^4 \log ^2(1-d x)-\frac {27}{4} c d^4 x^4 \log (1-d x)+\frac {67}{12} c d^3 x^3-10 c d^3 x^3 \log (1-d x)+\frac {139}{8} c d^2 x^2-18 c d^2 x^2 \log (1-d x)+\frac {355 c d x}{4}-9 c \log ^2(1-d x)-36 c \log (d x) \log ^2(1-d x)-54 c d x \log (1-d x)+\frac {355}{4} c \log (1-d x)}{144 d^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{3} + b x^{2} + a x\right )} {\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right ), x\right ) \]
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 \[ \int {\left (c x^{2} + b x + a\right )} x {\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )\,{d x} \]
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 \[ \int x \left (c \,x^{2}+b x +a \right ) \ln \left (-d x +1\right ) \polylog \left (2, d x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 518, normalized size = 0.58 \[ -\frac {1}{6912} \, d {\left (\frac {576 \, {\left (6 \, a d^{2} + 4 \, b d + 3 \, c\right )} {\left (\log \left (d x\right ) \log \left (-d x + 1\right )^{2} + 2 \, {\rm Li}_2\left (-d x + 1\right ) \log \left (-d x + 1\right ) - 2 \, {\rm Li}_{3}(-d x + 1)\right )}}{d^{5}} - \frac {81 \, c d^{4} x^{4} + 4 \, {\left (64 \, b d^{4} + 67 \, c d^{3}\right )} x^{3} + 6 \, {\left (216 \, a d^{4} + 176 \, b d^{3} + 139 \, c d^{2}\right )} x^{2} + 12 \, {\left (792 \, a d^{3} + 496 \, b d^{2} + 355 \, c d\right )} x - 48 \, {\left (9 \, c d^{4} x^{4} + 4 \, {\left (4 \, b d^{4} + 3 \, c d^{3}\right )} x^{3} + 6 \, {\left (6 \, a d^{4} + 4 \, b d^{3} + 3 \, c d^{2}\right )} x^{2} + 12 \, {\left (6 \, a d^{3} + 4 \, b d^{2} + 3 \, c d\right )} x + 12 \, {\left (6 \, a d^{2} + 4 \, b d + 3 \, c\right )} \log \left (-d x + 1\right )\right )} {\rm Li}_2\left (d x\right ) - 4 \, {\left (54 \, c d^{4} x^{4} + 4 \, {\left (32 \, b d^{4} + 21 \, c d^{3}\right )} x^{3} - 2376 \, a d^{2} + 6 \, {\left (72 \, a d^{4} + 40 \, b d^{3} + 27 \, c d^{2}\right )} x^{2} - 1488 \, b d + 12 \, {\left (108 \, a d^{3} + 64 \, b d^{2} + 45 \, c d\right )} x - 1065 \, c\right )} \log \left (-d x + 1\right )}{d^{5}}\right )} + \frac {1}{1728} \, {\left (\frac {216 \, {\left (4 \, d^{2} x^{2} {\rm Li}_2\left (d x\right ) - d^{2} x^{2} - 2 \, d x + 2 \, {\left (d^{2} x^{2} - 1\right )} \log \left (-d x + 1\right )\right )} a}{d^{2}} + \frac {32 \, {\left (18 \, d^{3} x^{3} {\rm Li}_2\left (d x\right ) - 2 \, d^{3} x^{3} - 3 \, d^{2} x^{2} - 6 \, d x + 6 \, {\left (d^{3} x^{3} - 1\right )} \log \left (-d x + 1\right )\right )} b}{d^{3}} + \frac {9 \, {\left (48 \, d^{4} x^{4} {\rm Li}_2\left (d x\right ) - 3 \, d^{4} x^{4} - 4 \, d^{3} x^{3} - 6 \, d^{2} x^{2} - 12 \, d x + 12 \, {\left (d^{4} x^{4} - 1\right )} \log \left (-d x + 1\right )\right )} c}{d^{4}}\right )} \log \left (-d x + 1\right ) \]
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 \[ \int x\,\ln \left (1-d\,x\right )\,\mathrm {polylog}\left (2,d\,x\right )\,\left (c\,x^2+b\,x+a\right ) \,d x \]
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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