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

Optimal. Leaf size=645 \[ -\frac{x \text{PolyLog}(2,d x) (3 d (2 a d+b)+2 c)}{6 d^2}+\frac{\text{PolyLog}(3,1-d x) (3 d (2 a d+b)+2 c)}{3 d^3}-\frac{\log (1-d x) \text{PolyLog}(2,d x) (3 d (2 a d+b)+2 c)}{6 d^3}-\frac{\log (1-d x) \text{PolyLog}(2,1-d x) (3 d (2 a d+b)+2 c)}{3 d^3}+\frac{1}{6} \log (1-d x) \text{PolyLog}(2,d x) \left (6 a x+3 b x^2+2 c x^3\right )-\frac{x^2 (3 b d+2 c) \text{PolyLog}(2,d x)}{12 d}-\frac{1}{9} c x^3 \text{PolyLog}(2,d x)+\frac{x (3 d (2 a d+b)+2 c)}{6 d^2}-\frac{\log (d x) \log ^2(1-d x) (3 d (2 a d+b)+2 c)}{6 d^3}+\frac{(1-d x) \log (1-d x) (3 d (2 a d+b)+2 c)}{6 d^3}-\frac{a (1-d x) \log ^2(1-d x)}{d}+\frac{2 a (1-d x) \log (1-d x)}{d}+2 a x+\frac{x (3 b d+2 c)}{24 d^2}+\frac{(3 b d+2 c) \log (1-d x)}{24 d^3}+\frac{x^2 (3 b d+2 c)}{48 d}-\frac{x^2 (3 b d+2 c) \log (1-d x)}{24 d}+\frac{b (1-d x)^2}{8 d^2}+\frac{b (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac{b (1-d x) \log ^2(1-d x)}{2 d^2}-\frac{b (1-d x)^2 \log (1-d x)}{4 d^2}+\frac{b (1-d x) \log (1-d x)}{d^2}+\frac{b x}{d}+\frac{4 c x}{9 d^2}-\frac{c \log ^2(1-d x)}{9 d^3}+\frac{2 c (1-d x) \log (1-d x)}{9 d^3}+\frac{2 c \log (1-d x)}{9 d^3}+\frac{c x^2}{9 d}+\frac{1}{9} c x^3 \log ^2(1-d x)-\frac{1}{9} c x^3 \log (1-d x)-\frac{c x^2 \log (1-d x)}{9 d}+\frac{c x^3}{27} \]

[Out]

2*a*x + (4*c*x)/(9*d^2) + (b*x)/d + ((2*c + 3*b*d)*x)/(24*d^2) + ((2*c + 3*d*(b + 2*a*d))*x)/(6*d^2) + (c*x^2)
/(9*d) + ((2*c + 3*b*d)*x^2)/(48*d) + (c*x^3)/27 + (b*(1 - d*x)^2)/(8*d^2) + (2*c*Log[1 - d*x])/(9*d^3) + ((2*
c + 3*b*d)*Log[1 - d*x])/(24*d^3) - (c*x^2*Log[1 - d*x])/(9*d) - ((2*c + 3*b*d)*x^2*Log[1 - d*x])/(24*d) - (c*
x^3*Log[1 - d*x])/9 + (2*c*(1 - d*x)*Log[1 - d*x])/(9*d^3) + (b*(1 - d*x)*Log[1 - d*x])/d^2 + (2*a*(1 - d*x)*L
og[1 - d*x])/d + ((2*c + 3*d*(b + 2*a*d))*(1 - d*x)*Log[1 - d*x])/(6*d^3) - (b*(1 - d*x)^2*Log[1 - d*x])/(4*d^
2) - (c*Log[1 - d*x]^2)/(9*d^3) + (c*x^3*Log[1 - d*x]^2)/9 - (b*(1 - d*x)*Log[1 - d*x]^2)/(2*d^2) - (a*(1 - d*
x)*Log[1 - d*x]^2)/d + (b*(1 - d*x)^2*Log[1 - d*x]^2)/(4*d^2) - ((2*c + 3*d*(b + 2*a*d))*Log[d*x]*Log[1 - d*x]
^2)/(6*d^3) - ((2*c + 3*d*(b + 2*a*d))*x*PolyLog[2, d*x])/(6*d^2) - ((2*c + 3*b*d)*x^2*PolyLog[2, d*x])/(12*d)
- (c*x^3*PolyLog[2, d*x])/9 - ((2*c + 3*d*(b + 2*a*d))*Log[1 - d*x]*PolyLog[2, d*x])/(6*d^3) + ((6*a*x + 3*b*
x^2 + 2*c*x^3)*Log[1 - d*x]*PolyLog[2, d*x])/6 - ((2*c + 3*d*(b + 2*a*d))*Log[1 - d*x]*PolyLog[2, 1 - d*x])/(3
*d^3) + ((2*c + 3*d*(b + 2*a*d))*PolyLog[3, 1 - d*x])/(3*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.818251, antiderivative size = 645, normalized size of antiderivative = 1., number of steps used = 43, number of rules used = 21, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.913, Rules used = {6742, 6586, 2389, 2295, 6591, 2395, 43, 6604, 2296, 2401, 2390, 2305, 2304, 2398, 2410, 2301, 6596, 2396, 2433, 2374, 6589} \[ -\frac{x \text{PolyLog}(2,d x) (3 d (2 a d+b)+2 c)}{6 d^2}+\frac{\text{PolyLog}(3,1-d x) (3 d (2 a d+b)+2 c)}{3 d^3}-\frac{\log (1-d x) \text{PolyLog}(2,d x) (3 d (2 a d+b)+2 c)}{6 d^3}-\frac{\log (1-d x) \text{PolyLog}(2,1-d x) (3 d (2 a d+b)+2 c)}{3 d^3}+\frac{1}{6} \log (1-d x) \text{PolyLog}(2,d x) \left (6 a x+3 b x^2+2 c x^3\right )-\frac{x^2 (3 b d+2 c) \text{PolyLog}(2,d x)}{12 d}-\frac{1}{9} c x^3 \text{PolyLog}(2,d x)+\frac{x (3 d (2 a d+b)+2 c)}{6 d^2}-\frac{\log (d x) \log ^2(1-d x) (3 d (2 a d+b)+2 c)}{6 d^3}+\frac{(1-d x) \log (1-d x) (3 d (2 a d+b)+2 c)}{6 d^3}-\frac{a (1-d x) \log ^2(1-d x)}{d}+\frac{2 a (1-d x) \log (1-d x)}{d}+2 a x+\frac{x (3 b d+2 c)}{24 d^2}+\frac{(3 b d+2 c) \log (1-d x)}{24 d^3}+\frac{x^2 (3 b d+2 c)}{48 d}-\frac{x^2 (3 b d+2 c) \log (1-d x)}{24 d}+\frac{b (1-d x)^2}{8 d^2}+\frac{b (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac{b (1-d x) \log ^2(1-d x)}{2 d^2}-\frac{b (1-d x)^2 \log (1-d x)}{4 d^2}+\frac{b (1-d x) \log (1-d x)}{d^2}+\frac{b x}{d}+\frac{4 c x}{9 d^2}-\frac{c \log ^2(1-d x)}{9 d^3}+\frac{2 c (1-d x) \log (1-d x)}{9 d^3}+\frac{2 c \log (1-d x)}{9 d^3}+\frac{c x^2}{9 d}+\frac{1}{9} c x^3 \log ^2(1-d x)-\frac{1}{9} c x^3 \log (1-d x)-\frac{c x^2 \log (1-d x)}{9 d}+\frac{c x^3}{27} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*x + (4*c*x)/(9*d^2) + (b*x)/d + ((2*c + 3*b*d)*x)/(24*d^2) + ((2*c + 3*d*(b + 2*a*d))*x)/(6*d^2) + (c*x^2)
/(9*d) + ((2*c + 3*b*d)*x^2)/(48*d) + (c*x^3)/27 + (b*(1 - d*x)^2)/(8*d^2) + (2*c*Log[1 - d*x])/(9*d^3) + ((2*
c + 3*b*d)*Log[1 - d*x])/(24*d^3) - (c*x^2*Log[1 - d*x])/(9*d) - ((2*c + 3*b*d)*x^2*Log[1 - d*x])/(24*d) - (c*
x^3*Log[1 - d*x])/9 + (2*c*(1 - d*x)*Log[1 - d*x])/(9*d^3) + (b*(1 - d*x)*Log[1 - d*x])/d^2 + (2*a*(1 - d*x)*L
og[1 - d*x])/d + ((2*c + 3*d*(b + 2*a*d))*(1 - d*x)*Log[1 - d*x])/(6*d^3) - (b*(1 - d*x)^2*Log[1 - d*x])/(4*d^
2) - (c*Log[1 - d*x]^2)/(9*d^3) + (c*x^3*Log[1 - d*x]^2)/9 - (b*(1 - d*x)*Log[1 - d*x]^2)/(2*d^2) - (a*(1 - d*
x)*Log[1 - d*x]^2)/d + (b*(1 - d*x)^2*Log[1 - d*x]^2)/(4*d^2) - ((2*c + 3*d*(b + 2*a*d))*Log[d*x]*Log[1 - d*x]
^2)/(6*d^3) - ((2*c + 3*d*(b + 2*a*d))*x*PolyLog[2, d*x])/(6*d^2) - ((2*c + 3*b*d)*x^2*PolyLog[2, d*x])/(12*d)
- (c*x^3*PolyLog[2, d*x])/9 - ((2*c + 3*d*(b + 2*a*d))*Log[1 - d*x]*PolyLog[2, d*x])/(6*d^3) + ((6*a*x + 3*b*
x^2 + 2*c*x^3)*Log[1 - d*x]*PolyLog[2, d*x])/6 - ((2*c + 3*d*(b + 2*a*d))*Log[1 - d*x]*PolyLog[2, 1 - d*x])/(3
*d^3) + ((2*c + 3*d*(b + 2*a*d))*PolyLog[3, 1 - d*x])/(3*d^3)

Rule 6742

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

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 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 2295

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

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 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 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 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 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 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 \left (a+b x+c x^2\right ) \log (1-d x) \text{Li}_2(d x) \, dx &=\frac{1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text{Li}_2(d x)+d \int \left (\frac{(-2 c-3 d (b+2 a d)) \text{Li}_2(d x)}{6 d^3}-\frac{(2 c+3 b d) x \text{Li}_2(d x)}{6 d^2}-\frac{c x^2 \text{Li}_2(d x)}{3 d}+\frac{(2 c+3 d (b+2 a d)) \text{Li}_2(d x)}{6 d^3 (1-d x)}\right ) \, dx+\int \left (a \log ^2(1-d x)+\frac{1}{2} b x \log ^2(1-d x)+\frac{1}{3} c x^2 \log ^2(1-d x)\right ) \, dx\\ &=\frac{1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text{Li}_2(d x)+a \int \log ^2(1-d x) \, dx+\frac{1}{2} b \int x \log ^2(1-d x) \, dx+\frac{1}{3} c \int x^2 \log ^2(1-d x) \, dx-\frac{1}{3} c \int x^2 \text{Li}_2(d x) \, dx-\frac{(2 c+3 b d) \int x \text{Li}_2(d x) \, dx}{6 d}-\frac{(2 c+3 d (b+2 a d)) \int \text{Li}_2(d x) \, dx}{6 d^2}+\frac{(2 c+3 d (b+2 a d)) \int \frac{\text{Li}_2(d x)}{1-d x} \, dx}{6 d^2}\\ &=\frac{1}{9} c x^3 \log ^2(1-d x)-\frac{(2 c+3 d (b+2 a d)) x \text{Li}_2(d x)}{6 d^2}-\frac{(2 c+3 b d) x^2 \text{Li}_2(d x)}{12 d}-\frac{1}{9} c x^3 \text{Li}_2(d x)-\frac{(2 c+3 d (b+2 a d)) \log (1-d x) \text{Li}_2(d x)}{6 d^3}+\frac{1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text{Li}_2(d x)+\frac{1}{2} b \int \left (\frac{\log ^2(1-d x)}{d}-\frac{(1-d x) \log ^2(1-d x)}{d}\right ) \, dx-\frac{1}{9} c \int x^2 \log (1-d x) \, dx-\frac{a \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1-d x\right )}{d}+\frac{1}{9} (2 c d) \int \frac{x^3 \log (1-d x)}{1-d x} \, dx-\frac{(2 c+3 b d) \int x \log (1-d x) \, dx}{12 d}-\frac{(2 c+3 d (b+2 a d)) \int \frac{\log ^2(1-d x)}{x} \, dx}{6 d^3}-\frac{(2 c+3 d (b+2 a d)) \int \log (1-d x) \, dx}{6 d^2}\\ &=-\frac{(2 c+3 b d) x^2 \log (1-d x)}{24 d}-\frac{1}{27} c x^3 \log (1-d x)+\frac{1}{9} c x^3 \log ^2(1-d x)-\frac{a (1-d x) \log ^2(1-d x)}{d}-\frac{(2 c+3 d (b+2 a d)) \log (d x) \log ^2(1-d x)}{6 d^3}-\frac{(2 c+3 d (b+2 a d)) x \text{Li}_2(d x)}{6 d^2}-\frac{(2 c+3 b d) x^2 \text{Li}_2(d x)}{12 d}-\frac{1}{9} c x^3 \text{Li}_2(d x)-\frac{(2 c+3 d (b+2 a d)) \log (1-d x) \text{Li}_2(d x)}{6 d^3}+\frac{1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text{Li}_2(d x)+\frac{(2 a) \operatorname{Subst}(\int \log (x) \, dx,x,1-d x)}{d}+\frac{b \int \log ^2(1-d x) \, dx}{2 d}-\frac{b \int (1-d x) \log ^2(1-d x) \, dx}{2 d}-\frac{1}{27} (c d) \int \frac{x^3}{1-d x} \, dx+\frac{1}{9} (2 c 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}{24} (2 c+3 b d) \int \frac{x^2}{1-d x} \, dx+\frac{(2 c+3 d (b+2 a d)) \operatorname{Subst}(\int \log (x) \, dx,x,1-d x)}{6 d^3}-\frac{(2 c+3 d (b+2 a d)) \int \frac{\log (d x) \log (1-d x)}{1-d x} \, dx}{3 d^2}\\ &=2 a x+\frac{(2 c+3 d (b+2 a d)) x}{6 d^2}-\frac{(2 c+3 b d) x^2 \log (1-d x)}{24 d}-\frac{1}{27} c x^3 \log (1-d x)+\frac{2 a (1-d x) \log (1-d x)}{d}+\frac{(2 c+3 d (b+2 a d)) (1-d x) \log (1-d x)}{6 d^3}+\frac{1}{9} c x^3 \log ^2(1-d x)-\frac{a (1-d x) \log ^2(1-d x)}{d}-\frac{(2 c+3 d (b+2 a d)) \log (d x) \log ^2(1-d x)}{6 d^3}-\frac{(2 c+3 d (b+2 a d)) x \text{Li}_2(d x)}{6 d^2}-\frac{(2 c+3 b d) x^2 \text{Li}_2(d x)}{12 d}-\frac{1}{9} c x^3 \text{Li}_2(d x)-\frac{(2 c+3 d (b+2 a d)) \log (1-d x) \text{Li}_2(d x)}{6 d^3}+\frac{1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{9} (2 c) \int x^2 \log (1-d x) \, dx-\frac{b \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1-d x\right )}{2 d^2}+\frac{b \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1-d x\right )}{2 d^2}-\frac{(2 c) \int \log (1-d x) \, dx}{9 d^2}-\frac{(2 c) \int \frac{\log (1-d x)}{-1+d x} \, dx}{9 d^2}-\frac{(2 c) \int x \log (1-d x) \, dx}{9 d}-\frac{1}{27} (c 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}{24} (2 c+3 b d) \int \left (-\frac{1}{d^2}-\frac{x}{d}-\frac{1}{d^2 (-1+d x)}\right ) \, dx+\frac{(2 c+3 d (b+2 a d)) \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 )}{3 d^3}\\ &=2 a x+\frac{c x}{27 d^2}+\frac{(2 c+3 b d) x}{24 d^2}+\frac{(2 c+3 d (b+2 a d)) x}{6 d^2}+\frac{c x^2}{54 d}+\frac{(2 c+3 b d) x^2}{48 d}+\frac{c x^3}{81}+\frac{c \log (1-d x)}{27 d^3}+\frac{(2 c+3 b d) \log (1-d x)}{24 d^3}-\frac{c x^2 \log (1-d x)}{9 d}-\frac{(2 c+3 b d) x^2 \log (1-d x)}{24 d}-\frac{1}{9} c x^3 \log (1-d x)+\frac{2 a (1-d x) \log (1-d x)}{d}+\frac{(2 c+3 d (b+2 a d)) (1-d x) \log (1-d x)}{6 d^3}+\frac{1}{9} c x^3 \log ^2(1-d x)-\frac{b (1-d x) \log ^2(1-d x)}{2 d^2}-\frac{a (1-d x) \log ^2(1-d x)}{d}+\frac{b (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac{(2 c+3 d (b+2 a d)) \log (d x) \log ^2(1-d x)}{6 d^3}-\frac{(2 c+3 d (b+2 a d)) x \text{Li}_2(d x)}{6 d^2}-\frac{(2 c+3 b d) x^2 \text{Li}_2(d x)}{12 d}-\frac{1}{9} c x^3 \text{Li}_2(d x)-\frac{(2 c+3 d (b+2 a d)) \log (1-d x) \text{Li}_2(d x)}{6 d^3}+\frac{1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text{Li}_2(d x)-\frac{(2 c+3 d (b+2 a d)) \log (1-d x) \text{Li}_2(1-d x)}{3 d^3}-\frac{1}{9} c \int \frac{x^2}{1-d x} \, dx+\frac{(2 c) \operatorname{Subst}(\int \log (x) \, dx,x,1-d x)}{9 d^3}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-d x\right )}{9 d^3}-\frac{b \operatorname{Subst}(\int x \log (x) \, dx,x,1-d x)}{2 d^2}+\frac{b \operatorname{Subst}(\int \log (x) \, dx,x,1-d x)}{d^2}-\frac{1}{27} (2 c d) \int \frac{x^3}{1-d x} \, dx+\frac{(2 c+3 d (b+2 a d)) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-d x\right )}{3 d^3}\\ &=2 a x+\frac{7 c x}{27 d^2}+\frac{b x}{d}+\frac{(2 c+3 b d) x}{24 d^2}+\frac{(2 c+3 d (b+2 a d)) x}{6 d^2}+\frac{c x^2}{54 d}+\frac{(2 c+3 b d) x^2}{48 d}+\frac{c x^3}{81}+\frac{b (1-d x)^2}{8 d^2}+\frac{c \log (1-d x)}{27 d^3}+\frac{(2 c+3 b d) \log (1-d x)}{24 d^3}-\frac{c x^2 \log (1-d x)}{9 d}-\frac{(2 c+3 b d) x^2 \log (1-d x)}{24 d}-\frac{1}{9} c x^3 \log (1-d x)+\frac{2 c (1-d x) \log (1-d x)}{9 d^3}+\frac{b (1-d x) \log (1-d x)}{d^2}+\frac{2 a (1-d x) \log (1-d x)}{d}+\frac{(2 c+3 d (b+2 a d)) (1-d x) \log (1-d x)}{6 d^3}-\frac{b (1-d x)^2 \log (1-d x)}{4 d^2}-\frac{c \log ^2(1-d x)}{9 d^3}+\frac{1}{9} c x^3 \log ^2(1-d x)-\frac{b (1-d x) \log ^2(1-d x)}{2 d^2}-\frac{a (1-d x) \log ^2(1-d x)}{d}+\frac{b (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac{(2 c+3 d (b+2 a d)) \log (d x) \log ^2(1-d x)}{6 d^3}-\frac{(2 c+3 d (b+2 a d)) x \text{Li}_2(d x)}{6 d^2}-\frac{(2 c+3 b d) x^2 \text{Li}_2(d x)}{12 d}-\frac{1}{9} c x^3 \text{Li}_2(d x)-\frac{(2 c+3 d (b+2 a d)) \log (1-d x) \text{Li}_2(d x)}{6 d^3}+\frac{1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text{Li}_2(d x)-\frac{(2 c+3 d (b+2 a d)) \log (1-d x) \text{Li}_2(1-d x)}{3 d^3}+\frac{(2 c+3 d (b+2 a d)) \text{Li}_3(1-d x)}{3 d^3}-\frac{1}{9} c \int \left (-\frac{1}{d^2}-\frac{x}{d}-\frac{1}{d^2 (-1+d x)}\right ) \, dx-\frac{1}{27} (2 c d) \int \left (-\frac{1}{d^3}-\frac{x}{d^2}-\frac{x^2}{d}-\frac{1}{d^3 (-1+d x)}\right ) \, dx\\ &=2 a x+\frac{4 c x}{9 d^2}+\frac{b x}{d}+\frac{(2 c+3 b d) x}{24 d^2}+\frac{(2 c+3 d (b+2 a d)) x}{6 d^2}+\frac{c x^2}{9 d}+\frac{(2 c+3 b d) x^2}{48 d}+\frac{c x^3}{27}+\frac{b (1-d x)^2}{8 d^2}+\frac{2 c \log (1-d x)}{9 d^3}+\frac{(2 c+3 b d) \log (1-d x)}{24 d^3}-\frac{c x^2 \log (1-d x)}{9 d}-\frac{(2 c+3 b d) x^2 \log (1-d x)}{24 d}-\frac{1}{9} c x^3 \log (1-d x)+\frac{2 c (1-d x) \log (1-d x)}{9 d^3}+\frac{b (1-d x) \log (1-d x)}{d^2}+\frac{2 a (1-d x) \log (1-d x)}{d}+\frac{(2 c+3 d (b+2 a d)) (1-d x) \log (1-d x)}{6 d^3}-\frac{b (1-d x)^2 \log (1-d x)}{4 d^2}-\frac{c \log ^2(1-d x)}{9 d^3}+\frac{1}{9} c x^3 \log ^2(1-d x)-\frac{b (1-d x) \log ^2(1-d x)}{2 d^2}-\frac{a (1-d x) \log ^2(1-d x)}{d}+\frac{b (1-d x)^2 \log ^2(1-d x)}{4 d^2}-\frac{(2 c+3 d (b+2 a d)) \log (d x) \log ^2(1-d x)}{6 d^3}-\frac{(2 c+3 d (b+2 a d)) x \text{Li}_2(d x)}{6 d^2}-\frac{(2 c+3 b d) x^2 \text{Li}_2(d x)}{12 d}-\frac{1}{9} c x^3 \text{Li}_2(d x)-\frac{(2 c+3 d (b+2 a d)) \log (1-d x) \text{Li}_2(d x)}{6 d^3}+\frac{1}{6} \left (6 a x+3 b x^2+2 c x^3\right ) \log (1-d x) \text{Li}_2(d x)-\frac{(2 c+3 d (b+2 a d)) \log (1-d x) \text{Li}_2(1-d x)}{3 d^3}+\frac{(2 c+3 d (b+2 a d)) \text{Li}_3(1-d x)}{3 d^3}\\ \end{align*}

Mathematica [A]  time = 1.03267, size = 472, normalized size = 0.73 \[ \frac{\text{PolyLog}(2,d x) \left (6 (d x-1) \log (1-d x) \left (3 d (2 a d+b d x+b)+2 c \left (d^2 x^2+d x+1\right )\right )-d x \left (9 d (4 a d+b d x+2 b)+2 c \left (2 d^2 x^2+3 d x+6\right )\right )\right )+12 \text{PolyLog}(3,1-d x) (3 d (2 a d+b)+2 c)-12 \log (1-d x) \text{PolyLog}(2,1-d x) (3 d (2 a d+b)+2 c)+108 a d^3 x+36 a d^3 x \log ^2(1-d x)-36 a d^2 \log ^2(1-d x)-36 a d^2 \log (d x) \log ^2(1-d x)-108 a d^3 x \log (1-d x)+108 a d^2 \log (1-d x)+\frac{27}{4} b d^3 x^2+9 b d^3 x^2 \log ^2(1-d x)-\frac{27}{2} b d^3 x^2 \log (1-d x)+\frac{99}{2} b d^2 x-36 b d^2 x \log (1-d x)-9 b d \log ^2(1-d x)-18 b d \log (d x) \log ^2(1-d x)+\frac{99}{2} b d \log (1-d x)+\frac{4}{3} c d^3 x^3+\frac{11}{2} c d^2 x^2+4 c d^3 x^3 \log ^2(1-d x)-4 c d^3 x^3 \log (1-d x)-7 c d^2 x^2 \log (1-d x)+31 c d x-4 c \log ^2(1-d x)-12 c \log (d x) \log ^2(1-d x)-20 c d x \log (1-d x)+31 c \log (1-d x)}{36 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(31*c*d*x + (99*b*d^2*x)/2 + 108*a*d^3*x + (11*c*d^2*x^2)/2 + (27*b*d^3*x^2)/4 + (4*c*d^3*x^3)/3 + 31*c*Log[1
- d*x] + (99*b*d*Log[1 - d*x])/2 + 108*a*d^2*Log[1 - d*x] - 20*c*d*x*Log[1 - d*x] - 36*b*d^2*x*Log[1 - d*x] -
108*a*d^3*x*Log[1 - d*x] - 7*c*d^2*x^2*Log[1 - d*x] - (27*b*d^3*x^2*Log[1 - d*x])/2 - 4*c*d^3*x^3*Log[1 - d*x]
- 4*c*Log[1 - d*x]^2 - 9*b*d*Log[1 - d*x]^2 - 36*a*d^2*Log[1 - d*x]^2 + 36*a*d^3*x*Log[1 - d*x]^2 + 9*b*d^3*x
^2*Log[1 - d*x]^2 + 4*c*d^3*x^3*Log[1 - d*x]^2 - 12*c*Log[d*x]*Log[1 - d*x]^2 - 18*b*d*Log[d*x]*Log[1 - d*x]^2
- 36*a*d^2*Log[d*x]*Log[1 - d*x]^2 + (-(d*x*(9*d*(2*b + 4*a*d + b*d*x) + 2*c*(6 + 3*d*x + 2*d^2*x^2))) + 6*(-
1 + d*x)*(3*d*(b + 2*a*d + b*d*x) + 2*c*(1 + d*x + d^2*x^2))*Log[1 - d*x])*PolyLog[2, d*x] - 12*(2*c + 3*d*(b
+ 2*a*d))*Log[1 - d*x]*PolyLog[2, 1 - d*x] + 12*(2*c + 3*d*(b + 2*a*d))*PolyLog[3, 1 - d*x])/(36*d^3)

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int \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((c*x^2+b*x+a)*ln(-d*x+1)*polylog(2,d*x),x)

[Out]

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

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Maxima [A]  time = 1.05019, size = 556, normalized size = 0.86 \begin{align*} -\frac{1}{432} \, d{\left (\frac{72 \,{\left (6 \, a d^{2} + 3 \, b d + 2 \, 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^{4}} - \frac{16 \, c d^{3} x^{3} + 3 \,{\left (27 \, b d^{3} + 22 \, c d^{2}\right )} x^{2} + 6 \,{\left (216 \, a d^{3} + 99 \, b d^{2} + 62 \, c d\right )} x - 12 \,{\left (4 \, c d^{3} x^{3} + 3 \,{\left (3 \, b d^{3} + 2 \, c d^{2}\right )} x^{2} + 6 \,{\left (6 \, a d^{3} + 3 \, b d^{2} + 2 \, c d\right )} x + 6 \,{\left (6 \, a d^{2} + 3 \, b d + 2 \, c\right )} \log \left (-d x + 1\right )\right )}{\rm Li}_2\left (d x\right ) - 2 \,{\left (16 \, c d^{3} x^{3} - 648 \, a d^{2} + 6 \,{\left (9 \, b d^{3} + 5 \, c d^{2}\right )} x^{2} - 297 \, b d + 6 \,{\left (72 \, a d^{3} + 27 \, b d^{2} + 16 \, c d\right )} x - 186 \, c\right )} \log \left (-d x + 1\right )}{d^{4}}\right )} + \frac{1}{216} \,{\left (\frac{216 \,{\left (d x{\rm Li}_2\left (d x\right ) - d x +{\left (d x - 1\right )} \log \left (-d x + 1\right )\right )} a}{d} + \frac{27 \,{\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 )} b}{d^{2}} + \frac{4 \,{\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 )} c}{d^{3}}\right )} \log \left (-d x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/432*d*(72*(6*a*d^2 + 3*b*d + 2*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^4 - (16*c*d^3*x^3 + 3*(27*b*d^3 + 22*c*d^2)*x^2 + 6*(216*a*d^3 + 99*b*d^2 + 62*c*d)*x - 12*(4*c
*d^3*x^3 + 3*(3*b*d^3 + 2*c*d^2)*x^2 + 6*(6*a*d^3 + 3*b*d^2 + 2*c*d)*x + 6*(6*a*d^2 + 3*b*d + 2*c)*log(-d*x +
1))*dilog(d*x) - 2*(16*c*d^3*x^3 - 648*a*d^2 + 6*(9*b*d^3 + 5*c*d^2)*x^2 - 297*b*d + 6*(72*a*d^3 + 27*b*d^2 +
16*c*d)*x - 186*c)*log(-d*x + 1))/d^4) + 1/216*(216*(d*x*dilog(d*x) - d*x + (d*x - 1)*log(-d*x + 1))*a/d + 27*
(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))*b/d^2 + 4*(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))*c/d^3)*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^{2} + b x + a\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((c*x^2+b*x+a)*log(-d*x+1)*polylog(2,d*x),x, algorithm="fricas")

[Out]

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