3.192 \(\int x (a+b x+c x^2) \log (1-d x) \text{PolyLog}(2,d x) \, dx\)

Optimal. Leaf size=900 \[ \frac{1}{16} c \log ^2(1-d x) x^4+\frac{3 c x^4}{256}-\frac{3}{64} c \log (1-d x) x^4-\frac{1}{16} c \text{PolyLog}(2,d x) x^4+\frac{1}{9} b \log ^2(1-d x) x^3+\frac{2 b x^3}{81}+\frac{(3 c+4 b d) x^3}{324 d}-\frac{2}{27} b \log (1-d x) x^3-\frac{(3 c+4 b d) \log (1-d x) x^3}{108 d}-\frac{c \log (1-d x) x^3}{24 d}-\frac{(3 c+4 b d) \text{PolyLog}(2,d x) x^3}{36 d}+\frac{17 c x^3}{576 d}+\frac{(3 c+4 b d) x^2}{216 d^2}+\frac{\left (6 a d^2+4 b d+3 c\right ) x^2}{96 d^2}-\frac{\left (6 a d^2+4 b d+3 c\right ) \log (1-d x) x^2}{48 d^2}-\frac{b \log (1-d x) x^2}{9 d}-\frac{c \log (1-d x) x^2}{16 d^2}-\frac{\left (6 a d^2+4 b d+3 c\right ) \text{PolyLog}(2,d x) x^2}{24 d^2}+\frac{5 b x^2}{54 d}+\frac{29 c x^2}{384 d^2}+\frac{(3 c+4 b d) x}{108 d^3}+\frac{5 \left (6 a d^2+4 b d+3 c\right ) x}{48 d^3}-\frac{\left (6 a d^2+4 b d+3 c\right ) \text{PolyLog}(2,d x) x}{12 d^3}+\frac{a x}{d}+\frac{11 b x}{27 d^2}+\frac{53 c x}{192 d^3}+\frac{a (1-d x)^2}{8 d^2}+\frac{a (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac{a (1-d x) \log ^2(1-d x)}{2 d^2}-\frac{\left (6 a d^2+4 b d+3 c\right ) \log (d x) \log ^2(1-d x)}{12 d^4}-\frac{b \log ^2(1-d x)}{9 d^3}-\frac{c \log ^2(1-d x)}{16 d^4}-\frac{a (1-d x)^2 \log (1-d x)}{4 d^2}+\frac{(3 c+4 b d) \log (1-d x)}{108 d^4}+\frac{\left (6 a d^2+4 b d+3 c\right ) \log (1-d x)}{48 d^4}+\frac{\left (6 a d^2+4 b d+3 c\right ) (1-d x) \log (1-d x)}{12 d^4}+\frac{a (1-d x) \log (1-d x)}{d^2}+\frac{2 b (1-d x) \log (1-d x)}{9 d^3}+\frac{c (1-d x) \log (1-d x)}{8 d^4}+\frac{5 b \log (1-d x)}{27 d^3}+\frac{29 c \log (1-d x)}{192 d^4}-\frac{\left (6 a d^2+4 b d+3 c\right ) \log (1-d x) \text{PolyLog}(2,d x)}{12 d^4}+\frac{1}{12} \left (3 c x^4+4 b x^3+6 a x^2\right ) \log (1-d x) \text{PolyLog}(2,d x)-\frac{\left (6 a d^2+4 b d+3 c\right ) \log (1-d x) \text{PolyLog}(2,1-d x)}{6 d^4}+\frac{\left (6 a d^2+4 b d+3 c\right ) \text{PolyLog}(3,1-d x)}{6 d^4} \]

[Out]

(53*c*x)/(192*d^3) + (11*b*x)/(27*d^2) + (a*x)/d + ((3*c + 4*b*d)*x)/(108*d^3) + (5*(3*c + 4*b*d + 6*a*d^2)*x)
/(48*d^3) + (29*c*x^2)/(384*d^2) + (5*b*x^2)/(54*d) + ((3*c + 4*b*d)*x^2)/(216*d^2) + ((3*c + 4*b*d + 6*a*d^2)
*x^2)/(96*d^2) + (2*b*x^3)/81 + (17*c*x^3)/(576*d) + ((3*c + 4*b*d)*x^3)/(324*d) + (3*c*x^4)/256 + (a*(1 - d*x
)^2)/(8*d^2) + (29*c*Log[1 - d*x])/(192*d^4) + (5*b*Log[1 - d*x])/(27*d^3) + ((3*c + 4*b*d)*Log[1 - d*x])/(108
*d^4) + ((3*c + 4*b*d + 6*a*d^2)*Log[1 - d*x])/(48*d^4) - (c*x^2*Log[1 - d*x])/(16*d^2) - (b*x^2*Log[1 - d*x])
/(9*d) - ((3*c + 4*b*d + 6*a*d^2)*x^2*Log[1 - d*x])/(48*d^2) - (2*b*x^3*Log[1 - d*x])/27 - (c*x^3*Log[1 - d*x]
)/(24*d) - ((3*c + 4*b*d)*x^3*Log[1 - d*x])/(108*d) - (3*c*x^4*Log[1 - d*x])/64 + (c*(1 - d*x)*Log[1 - d*x])/(
8*d^4) + (2*b*(1 - d*x)*Log[1 - d*x])/(9*d^3) + (a*(1 - d*x)*Log[1 - d*x])/d^2 + ((3*c + 4*b*d + 6*a*d^2)*(1 -
d*x)*Log[1 - d*x])/(12*d^4) - (a*(1 - d*x)^2*Log[1 - d*x])/(4*d^2) - (c*Log[1 - d*x]^2)/(16*d^4) - (b*Log[1 -
d*x]^2)/(9*d^3) + (b*x^3*Log[1 - d*x]^2)/9 + (c*x^4*Log[1 - d*x]^2)/16 - (a*(1 - d*x)*Log[1 - d*x]^2)/(2*d^2)
+ (a*(1 - d*x)^2*Log[1 - d*x]^2)/(4*d^2) - ((3*c + 4*b*d + 6*a*d^2)*Log[d*x]*Log[1 - d*x]^2)/(12*d^4) - ((3*c
+ 4*b*d + 6*a*d^2)*x*PolyLog[2, d*x])/(12*d^3) - ((3*c + 4*b*d + 6*a*d^2)*x^2*PolyLog[2, d*x])/(24*d^2) - ((3
*c + 4*b*d)*x^3*PolyLog[2, d*x])/(36*d) - (c*x^4*PolyLog[2, d*x])/16 - ((3*c + 4*b*d + 6*a*d^2)*Log[1 - d*x]*P
olyLog[2, d*x])/(12*d^4) + ((6*a*x^2 + 4*b*x^3 + 3*c*x^4)*Log[1 - d*x]*PolyLog[2, d*x])/12 - ((3*c + 4*b*d + 6
*a*d^2)*Log[1 - d*x]*PolyLog[2, 1 - d*x])/(6*d^4) + ((3*c + 4*b*d + 6*a*d^2)*PolyLog[3, 1 - d*x])/(6*d^4)

________________________________________________________________________________________

Rubi [A]  time = 1.18491, antiderivative size = 900, normalized size of antiderivative = 1., 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]

Int[x*(a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x],x]

[Out]

(53*c*x)/(192*d^3) + (11*b*x)/(27*d^2) + (a*x)/d + ((3*c + 4*b*d)*x)/(108*d^3) + (5*(3*c + 4*b*d + 6*a*d^2)*x)
/(48*d^3) + (29*c*x^2)/(384*d^2) + (5*b*x^2)/(54*d) + ((3*c + 4*b*d)*x^2)/(216*d^2) + ((3*c + 4*b*d + 6*a*d^2)
*x^2)/(96*d^2) + (2*b*x^3)/81 + (17*c*x^3)/(576*d) + ((3*c + 4*b*d)*x^3)/(324*d) + (3*c*x^4)/256 + (a*(1 - d*x
)^2)/(8*d^2) + (29*c*Log[1 - d*x])/(192*d^4) + (5*b*Log[1 - d*x])/(27*d^3) + ((3*c + 4*b*d)*Log[1 - d*x])/(108
*d^4) + ((3*c + 4*b*d + 6*a*d^2)*Log[1 - d*x])/(48*d^4) - (c*x^2*Log[1 - d*x])/(16*d^2) - (b*x^2*Log[1 - d*x])
/(9*d) - ((3*c + 4*b*d + 6*a*d^2)*x^2*Log[1 - d*x])/(48*d^2) - (2*b*x^3*Log[1 - d*x])/27 - (c*x^3*Log[1 - d*x]
)/(24*d) - ((3*c + 4*b*d)*x^3*Log[1 - d*x])/(108*d) - (3*c*x^4*Log[1 - d*x])/64 + (c*(1 - d*x)*Log[1 - d*x])/(
8*d^4) + (2*b*(1 - d*x)*Log[1 - d*x])/(9*d^3) + (a*(1 - d*x)*Log[1 - d*x])/d^2 + ((3*c + 4*b*d + 6*a*d^2)*(1 -
d*x)*Log[1 - d*x])/(12*d^4) - (a*(1 - d*x)^2*Log[1 - d*x])/(4*d^2) - (c*Log[1 - d*x]^2)/(16*d^4) - (b*Log[1 -
d*x]^2)/(9*d^3) + (b*x^3*Log[1 - d*x]^2)/9 + (c*x^4*Log[1 - d*x]^2)/16 - (a*(1 - d*x)*Log[1 - d*x]^2)/(2*d^2)
+ (a*(1 - d*x)^2*Log[1 - d*x]^2)/(4*d^2) - ((3*c + 4*b*d + 6*a*d^2)*Log[d*x]*Log[1 - d*x]^2)/(12*d^4) - ((3*c
+ 4*b*d + 6*a*d^2)*x*PolyLog[2, d*x])/(12*d^3) - ((3*c + 4*b*d + 6*a*d^2)*x^2*PolyLog[2, d*x])/(24*d^2) - ((3
*c + 4*b*d)*x^3*PolyLog[2, d*x])/(36*d) - (c*x^4*PolyLog[2, d*x])/16 - ((3*c + 4*b*d + 6*a*d^2)*Log[1 - d*x]*P
olyLog[2, d*x])/(12*d^4) + ((6*a*x^2 + 4*b*x^3 + 3*c*x^4)*Log[1 - d*x]*PolyLog[2, d*x])/12 - ((3*c + 4*b*d + 6
*a*d^2)*Log[1 - d*x]*PolyLog[2, 1 - d*x])/(6*d^4) + ((3*c + 4*b*d + 6*a*d^2)*PolyLog[3, 1 - d*x])/(6*d^4)

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 6591

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[((d*x)^(m + 1)*PolyLog[n
, a*(b*x^p)^q])/(d*(m + 1)), x] - Dist[(p*q)/(m + 1), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ
[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 6604

Int[((g_.) + Log[(f_.)*((d_.) + (e_.)*(x_))^(n_.)]*(h_.))*(Px_)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x_Symb
ol] :> With[{u = IntHide[Px, x]}, Simp[u*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x] + (Dist[b, Int
[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], u/(a + b*x), x], x], x] - Dist[e*h*n, Int[Ex
pandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x], x])] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && P
olyQ[Px, x]

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2410

Int[(Log[(c_.)*((d_) + (e_.)*(x_))]*(x_)^(m_.))/((f_) + (g_.)*(x_)), x_Symbol] :> Int[ExpandIntegrand[Log[c*(d
+ e*x)], x^m/(f + g*x), x], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[e*f - d*g, 0] && EqQ[c*d, 1] && IntegerQ[m
]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 6586

Int[PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[x*PolyLog[n, a*(b*x^p)^q], x] - Dist[p*q, I
nt[PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, p, q}, x] && GtQ[n, 0]

Rule 6596

Int[PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 - a*c - b*c*x]*PolyL
og[2, c*(a + b*x)])/e, x] + Dist[b/e, Int[Log[1 - a*c - b*c*x]^2/(a + b*x), x], x] /; FreeQ[{a, b, c, d, e}, x
] && EqQ[c*(b*d - a*e) + e, 0]

Rule 2396

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*
(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^p)/g, x] - Dist[(b*e*n*p)/g, Int[(Log[(e*(f + g*x))/(e*f -
d*g)]*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*
f - d*g, 0] && IGtQ[p, 1]

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

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*}

Mathematica [A]  time = 1.27973, size = 583, normalized size = 0.65 \[ \frac{\text{PolyLog}(2,d x) \left (12 \log (1-d x) \left (6 a d^4 x^2-6 a d^2+4 b d^4 x^3-4 b d+3 c \left (d^4 x^4-1\right )\right )-d x \left (4 d \left (9 a d (d x+2)+2 b \left (2 d^2 x^2+3 d x+6\right )\right )+3 c \left (3 d^3 x^3+4 d^2 x^2+6 d x+12\right )\right )\right )+24 \text{PolyLog}(3,1-d x) \left (6 a d^2+4 b d+3 c\right )-24 \log (1-d x) \text{PolyLog}(2,1-d x) \left (6 a d^2+4 b d+3 c\right )+27 a d^4 x^2+36 a d^4 x^2 \log ^2(1-d x)-54 a d^4 x^2 \log (1-d x)+198 a d^3 x-36 a d^2 \log ^2(1-d x)-72 a d^2 \log (d x) \log ^2(1-d x)-144 a d^3 x \log (1-d x)+198 a d^2 \log (1-d x)+\frac{16}{3} b d^4 x^3+22 b d^3 x^2+16 b d^4 x^3 \log ^2(1-d x)-16 b d^4 x^3 \log (1-d x)-28 b d^3 x^2 \log (1-d x)+124 b d^2 x-80 b d^2 x \log (1-d x)-16 b d \log ^2(1-d x)-48 b d \log (d x) \log ^2(1-d x)+124 b d \log (1-d x)+\frac{27}{16} c d^4 x^4+\frac{67}{12} c d^3 x^3+\frac{139}{8} c d^2 x^2+9 c d^4 x^4 \log ^2(1-d x)-\frac{27}{4} c d^4 x^4 \log (1-d x)-10 c d^3 x^3 \log (1-d x)-18 c d^2 x^2 \log (1-d x)+\frac{355 c d x}{4}-9 c \log ^2(1-d x)-36 c \log (d x) \log ^2(1-d x)-54 c d x \log (1-d x)+\frac{355}{4} c \log (1-d x)}{144 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x],x]

[Out]

((355*c*d*x)/4 + 124*b*d^2*x + 198*a*d^3*x + (139*c*d^2*x^2)/8 + 22*b*d^3*x^2 + 27*a*d^4*x^2 + (67*c*d^3*x^3)/
12 + (16*b*d^4*x^3)/3 + (27*c*d^4*x^4)/16 + (355*c*Log[1 - d*x])/4 + 124*b*d*Log[1 - d*x] + 198*a*d^2*Log[1 -
d*x] - 54*c*d*x*Log[1 - d*x] - 80*b*d^2*x*Log[1 - d*x] - 144*a*d^3*x*Log[1 - d*x] - 18*c*d^2*x^2*Log[1 - d*x]
- 28*b*d^3*x^2*Log[1 - d*x] - 54*a*d^4*x^2*Log[1 - d*x] - 10*c*d^3*x^3*Log[1 - d*x] - 16*b*d^4*x^3*Log[1 - d*x
] - (27*c*d^4*x^4*Log[1 - d*x])/4 - 9*c*Log[1 - d*x]^2 - 16*b*d*Log[1 - d*x]^2 - 36*a*d^2*Log[1 - d*x]^2 + 36*
a*d^4*x^2*Log[1 - d*x]^2 + 16*b*d^4*x^3*Log[1 - d*x]^2 + 9*c*d^4*x^4*Log[1 - d*x]^2 - 36*c*Log[d*x]*Log[1 - d*
x]^2 - 48*b*d*Log[d*x]*Log[1 - d*x]^2 - 72*a*d^2*Log[d*x]*Log[1 - d*x]^2 + (-(d*x*(3*c*(12 + 6*d*x + 4*d^2*x^2
+ 3*d^3*x^3) + 4*d*(9*a*d*(2 + d*x) + 2*b*(6 + 3*d*x + 2*d^2*x^2)))) + 12*(-4*b*d - 6*a*d^2 + 6*a*d^4*x^2 + 4
*b*d^4*x^3 + 3*c*(-1 + d^4*x^4))*Log[1 - d*x])*PolyLog[2, d*x] - 24*(3*c + 4*b*d + 6*a*d^2)*Log[1 - d*x]*PolyL
og[2, 1 - d*x] + 24*(3*c + 4*b*d + 6*a*d^2)*PolyLog[3, 1 - d*x])/(144*d^4)

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int x \left ( c{x}^{2}+bx+a \right ) \ln \left ( -dx+1 \right ){\it polylog} \left ( 2,dx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*x^2+b*x+a)*ln(-d*x+1)*polylog(2,d*x),x)

[Out]

int(x*(c*x^2+b*x+a)*ln(-d*x+1)*polylog(2,d*x),x)

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Maxima [A]  time = 1.05354, size = 699, normalized size = 0.78 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)*log(-d*x+1)*polylog(2,d*x),x, algorithm="maxima")

[Out]

-1/6912*d*(576*(6*a*d^2 + 4*b*d + 3*c)*(log(d*x)*log(-d*x + 1)^2 + 2*dilog(-d*x + 1)*log(-d*x + 1) - 2*polylog
(3, -d*x + 1))/d^5 - (81*c*d^4*x^4 + 4*(64*b*d^4 + 67*c*d^3)*x^3 + 6*(216*a*d^4 + 176*b*d^3 + 139*c*d^2)*x^2 +
12*(792*a*d^3 + 496*b*d^2 + 355*c*d)*x - 48*(9*c*d^4*x^4 + 4*(4*b*d^4 + 3*c*d^3)*x^3 + 6*(6*a*d^4 + 4*b*d^3 +
3*c*d^2)*x^2 + 12*(6*a*d^3 + 4*b*d^2 + 3*c*d)*x + 12*(6*a*d^2 + 4*b*d + 3*c)*log(-d*x + 1))*dilog(d*x) - 4*(5
4*c*d^4*x^4 + 4*(32*b*d^4 + 21*c*d^3)*x^3 - 2376*a*d^2 + 6*(72*a*d^4 + 40*b*d^3 + 27*c*d^2)*x^2 - 1488*b*d + 1
2*(108*a*d^3 + 64*b*d^2 + 45*c*d)*x - 1065*c)*log(-d*x + 1))/d^5) + 1/1728*(216*(4*d^2*x^2*dilog(d*x) - d^2*x^
2 - 2*d*x + 2*(d^2*x^2 - 1)*log(-d*x + 1))*a/d^2 + 32*(18*d^3*x^3*dilog(d*x) - 2*d^3*x^3 - 3*d^2*x^2 - 6*d*x +
6*(d^3*x^3 - 1)*log(-d*x + 1))*b/d^3 + 9*(48*d^4*x^4*dilog(d*x) - 3*d^4*x^4 - 4*d^3*x^3 - 6*d^2*x^2 - 12*d*x
+ 12*(d^4*x^4 - 1)*log(-d*x + 1))*c/d^4)*log(-d*x + 1)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)*log(-d*x+1)*polylog(2,d*x),x, algorithm="fricas")

[Out]

integral((c*x^3 + b*x^2 + a*x)*dilog(d*x)*log(-d*x + 1), x)

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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.

[In]

integrate(x*(c*x**2+b*x+a)*ln(-d*x+1)*polylog(2,d*x),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)*log(-d*x+1)*polylog(2,d*x),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)*x*dilog(d*x)*log(-d*x + 1), x)