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

Optimal. Leaf size=661 \[ -\frac{x^2 (4 a c+3 b) \text{PolyLog}(2,c x)}{24 c^2}-\frac{x (4 a c+3 b) \text{PolyLog}(2,c x)}{12 c^3}+\frac{(4 a c+3 b) \text{PolyLog}(3,1-c x)}{6 c^4}-\frac{(4 a c+3 b) \log (1-c x) \text{PolyLog}(2,c x)}{12 c^4}-\frac{(4 a c+3 b) \log (1-c x) \text{PolyLog}(2,1-c x)}{6 c^4}-\frac{x^3 (4 a c+3 b) \text{PolyLog}(2,c x)}{36 c}+\frac{1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \text{PolyLog}(2,c x)-\frac{1}{16} b x^4 \text{PolyLog}(2,c x)+\frac{13 x^2 (4 a c+3 b)}{864 c^2}-\frac{x^2 (4 a c+3 b) \log (1-c x)}{48 c^2}+\frac{49 x (4 a c+3 b)}{432 c^3}-\frac{(4 a c+3 b) \log (c x) \log ^2(1-c x)}{12 c^4}+\frac{13 (4 a c+3 b) \log (1-c x)}{432 c^4}+\frac{(1-c x) (4 a c+3 b) \log (1-c x)}{12 c^4}+\frac{x^3 (4 a c+3 b)}{324 c}-\frac{x^3 (4 a c+3 b) \log (1-c x)}{108 c}+\frac{11 a x}{27 c^2}-\frac{a \log ^2(1-c x)}{9 c^3}+\frac{2 a (1-c x) \log (1-c x)}{9 c^3}+\frac{5 a \log (1-c x)}{27 c^3}+\frac{5 a x^2}{54 c}+\frac{1}{9} a x^3 \log ^2(1-c x)-\frac{2}{27} a x^3 \log (1-c x)-\frac{a x^2 \log (1-c x)}{9 c}+\frac{2 a x^3}{81}+\frac{29 b x^2}{384 c^2}-\frac{b x^2 \log (1-c x)}{16 c^2}+\frac{53 b x}{192 c^3}-\frac{b \log ^2(1-c x)}{16 c^4}+\frac{b (1-c x) \log (1-c x)}{8 c^4}+\frac{29 b \log (1-c x)}{192 c^4}+\frac{17 b x^3}{576 c}+\frac{1}{16} b x^4 \log ^2(1-c x)-\frac{3}{64} b x^4 \log (1-c x)-\frac{b x^3 \log (1-c x)}{24 c}+\frac{3 b x^4}{256} \]

[Out]

(53*b*x)/(192*c^3) + (11*a*x)/(27*c^2) + (49*(3*b + 4*a*c)*x)/(432*c^3) + (29*b*x^2)/(384*c^2) + (5*a*x^2)/(54
*c) + (13*(3*b + 4*a*c)*x^2)/(864*c^2) + (2*a*x^3)/81 + (17*b*x^3)/(576*c) + ((3*b + 4*a*c)*x^3)/(324*c) + (3*
b*x^4)/256 + (29*b*Log[1 - c*x])/(192*c^4) + (5*a*Log[1 - c*x])/(27*c^3) + (13*(3*b + 4*a*c)*Log[1 - c*x])/(43
2*c^4) - (b*x^2*Log[1 - c*x])/(16*c^2) - (a*x^2*Log[1 - c*x])/(9*c) - ((3*b + 4*a*c)*x^2*Log[1 - c*x])/(48*c^2
) - (2*a*x^3*Log[1 - c*x])/27 - (b*x^3*Log[1 - c*x])/(24*c) - ((3*b + 4*a*c)*x^3*Log[1 - c*x])/(108*c) - (3*b*
x^4*Log[1 - c*x])/64 + (b*(1 - c*x)*Log[1 - c*x])/(8*c^4) + (2*a*(1 - c*x)*Log[1 - c*x])/(9*c^3) + ((3*b + 4*a
*c)*(1 - c*x)*Log[1 - c*x])/(12*c^4) - (b*Log[1 - c*x]^2)/(16*c^4) - (a*Log[1 - c*x]^2)/(9*c^3) + (a*x^3*Log[1
- c*x]^2)/9 + (b*x^4*Log[1 - c*x]^2)/16 - ((3*b + 4*a*c)*Log[c*x]*Log[1 - c*x]^2)/(12*c^4) - ((3*b + 4*a*c)*x
*PolyLog[2, c*x])/(12*c^3) - ((3*b + 4*a*c)*x^2*PolyLog[2, c*x])/(24*c^2) - ((3*b + 4*a*c)*x^3*PolyLog[2, c*x]
)/(36*c) - (b*x^4*PolyLog[2, c*x])/16 - ((3*b + 4*a*c)*Log[1 - c*x]*PolyLog[2, c*x])/(12*c^4) + ((4*a*x^3 + 3*
b*x^4)*Log[1 - c*x]*PolyLog[2, c*x])/12 - ((3*b + 4*a*c)*Log[1 - c*x]*PolyLog[2, 1 - c*x])/(6*c^4) + ((3*b + 4
*a*c)*PolyLog[3, 1 - c*x])/(6*c^4)

________________________________________________________________________________________

Rubi [A]  time = 0.981897, antiderivative size = 661, normalized size of antiderivative = 1., number of steps used = 52, number of rules used = 17, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.81, Rules used = {6742, 6591, 2395, 43, 6604, 2398, 2410, 2389, 2295, 2390, 2301, 6586, 6596, 2396, 2433, 2374, 6589} \[ -\frac{x^2 (4 a c+3 b) \text{PolyLog}(2,c x)}{24 c^2}-\frac{x (4 a c+3 b) \text{PolyLog}(2,c x)}{12 c^3}+\frac{(4 a c+3 b) \text{PolyLog}(3,1-c x)}{6 c^4}-\frac{(4 a c+3 b) \log (1-c x) \text{PolyLog}(2,c x)}{12 c^4}-\frac{(4 a c+3 b) \log (1-c x) \text{PolyLog}(2,1-c x)}{6 c^4}-\frac{x^3 (4 a c+3 b) \text{PolyLog}(2,c x)}{36 c}+\frac{1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \text{PolyLog}(2,c x)-\frac{1}{16} b x^4 \text{PolyLog}(2,c x)+\frac{13 x^2 (4 a c+3 b)}{864 c^2}-\frac{x^2 (4 a c+3 b) \log (1-c x)}{48 c^2}+\frac{49 x (4 a c+3 b)}{432 c^3}-\frac{(4 a c+3 b) \log (c x) \log ^2(1-c x)}{12 c^4}+\frac{13 (4 a c+3 b) \log (1-c x)}{432 c^4}+\frac{(1-c x) (4 a c+3 b) \log (1-c x)}{12 c^4}+\frac{x^3 (4 a c+3 b)}{324 c}-\frac{x^3 (4 a c+3 b) \log (1-c x)}{108 c}+\frac{11 a x}{27 c^2}-\frac{a \log ^2(1-c x)}{9 c^3}+\frac{2 a (1-c x) \log (1-c x)}{9 c^3}+\frac{5 a \log (1-c x)}{27 c^3}+\frac{5 a x^2}{54 c}+\frac{1}{9} a x^3 \log ^2(1-c x)-\frac{2}{27} a x^3 \log (1-c x)-\frac{a x^2 \log (1-c x)}{9 c}+\frac{2 a x^3}{81}+\frac{29 b x^2}{384 c^2}-\frac{b x^2 \log (1-c x)}{16 c^2}+\frac{53 b x}{192 c^3}-\frac{b \log ^2(1-c x)}{16 c^4}+\frac{b (1-c x) \log (1-c x)}{8 c^4}+\frac{29 b \log (1-c x)}{192 c^4}+\frac{17 b x^3}{576 c}+\frac{1}{16} b x^4 \log ^2(1-c x)-\frac{3}{64} b x^4 \log (1-c x)-\frac{b x^3 \log (1-c x)}{24 c}+\frac{3 b x^4}{256} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(53*b*x)/(192*c^3) + (11*a*x)/(27*c^2) + (49*(3*b + 4*a*c)*x)/(432*c^3) + (29*b*x^2)/(384*c^2) + (5*a*x^2)/(54
*c) + (13*(3*b + 4*a*c)*x^2)/(864*c^2) + (2*a*x^3)/81 + (17*b*x^3)/(576*c) + ((3*b + 4*a*c)*x^3)/(324*c) + (3*
b*x^4)/256 + (29*b*Log[1 - c*x])/(192*c^4) + (5*a*Log[1 - c*x])/(27*c^3) + (13*(3*b + 4*a*c)*Log[1 - c*x])/(43
2*c^4) - (b*x^2*Log[1 - c*x])/(16*c^2) - (a*x^2*Log[1 - c*x])/(9*c) - ((3*b + 4*a*c)*x^2*Log[1 - c*x])/(48*c^2
) - (2*a*x^3*Log[1 - c*x])/27 - (b*x^3*Log[1 - c*x])/(24*c) - ((3*b + 4*a*c)*x^3*Log[1 - c*x])/(108*c) - (3*b*
x^4*Log[1 - c*x])/64 + (b*(1 - c*x)*Log[1 - c*x])/(8*c^4) + (2*a*(1 - c*x)*Log[1 - c*x])/(9*c^3) + ((3*b + 4*a
*c)*(1 - c*x)*Log[1 - c*x])/(12*c^4) - (b*Log[1 - c*x]^2)/(16*c^4) - (a*Log[1 - c*x]^2)/(9*c^3) + (a*x^3*Log[1
- c*x]^2)/9 + (b*x^4*Log[1 - c*x]^2)/16 - ((3*b + 4*a*c)*Log[c*x]*Log[1 - c*x]^2)/(12*c^4) - ((3*b + 4*a*c)*x
*PolyLog[2, c*x])/(12*c^3) - ((3*b + 4*a*c)*x^2*PolyLog[2, c*x])/(24*c^2) - ((3*b + 4*a*c)*x^3*PolyLog[2, c*x]
)/(36*c) - (b*x^4*PolyLog[2, c*x])/16 - ((3*b + 4*a*c)*Log[1 - c*x]*PolyLog[2, c*x])/(12*c^4) + ((4*a*x^3 + 3*
b*x^4)*Log[1 - c*x]*PolyLog[2, c*x])/12 - ((3*b + 4*a*c)*Log[1 - c*x]*PolyLog[2, 1 - c*x])/(6*c^4) + ((3*b + 4
*a*c)*PolyLog[3, 1 - c*x])/(6*c^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 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 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 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 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 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^2 (a+b x) \log (1-c x) \text{Li}_2(c x) \, dx &=\frac{1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \text{Li}_2(c x)+c \int \left (\frac{(-3 b-4 a c) \text{Li}_2(c x)}{12 c^4}-\frac{(3 b+4 a c) x \text{Li}_2(c x)}{12 c^3}-\frac{(3 b+4 a c) x^2 \text{Li}_2(c x)}{12 c^2}-\frac{b x^3 \text{Li}_2(c x)}{4 c}+\frac{(-3 b-4 a c) \text{Li}_2(c x)}{12 c^4 (-1+c x)}\right ) \, dx+\int \left (\frac{1}{3} a x^2 \log ^2(1-c x)+\frac{1}{4} b x^3 \log ^2(1-c x)\right ) \, dx\\ &=\frac{1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \text{Li}_2(c x)+\frac{1}{3} a \int x^2 \log ^2(1-c x) \, dx+\frac{1}{4} b \int x^3 \log ^2(1-c x) \, dx-\frac{1}{4} b \int x^3 \text{Li}_2(c x) \, dx-\frac{(3 b+4 a c) \int \text{Li}_2(c x) \, dx}{12 c^3}-\frac{(3 b+4 a c) \int \frac{\text{Li}_2(c x)}{-1+c x} \, dx}{12 c^3}-\frac{(3 b+4 a c) \int x \text{Li}_2(c x) \, dx}{12 c^2}-\frac{(3 b+4 a c) \int x^2 \text{Li}_2(c x) \, dx}{12 c}\\ &=\frac{1}{9} a x^3 \log ^2(1-c x)+\frac{1}{16} b x^4 \log ^2(1-c x)-\frac{(3 b+4 a c) x \text{Li}_2(c x)}{12 c^3}-\frac{(3 b+4 a c) x^2 \text{Li}_2(c x)}{24 c^2}-\frac{(3 b+4 a c) x^3 \text{Li}_2(c x)}{36 c}-\frac{1}{16} b x^4 \text{Li}_2(c x)-\frac{(3 b+4 a c) \log (1-c x) \text{Li}_2(c x)}{12 c^4}+\frac{1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \text{Li}_2(c x)-\frac{1}{16} b \int x^3 \log (1-c x) \, dx+\frac{1}{9} (2 a c) \int \frac{x^3 \log (1-c x)}{1-c x} \, dx+\frac{1}{8} (b c) \int \frac{x^4 \log (1-c x)}{1-c x} \, dx-\frac{(3 b+4 a c) \int \frac{\log ^2(1-c x)}{x} \, dx}{12 c^4}-\frac{(3 b+4 a c) \int \log (1-c x) \, dx}{12 c^3}-\frac{(3 b+4 a c) \int x \log (1-c x) \, dx}{24 c^2}-\frac{(3 b+4 a c) \int x^2 \log (1-c x) \, dx}{36 c}\\ &=-\frac{(3 b+4 a c) x^2 \log (1-c x)}{48 c^2}-\frac{(3 b+4 a c) x^3 \log (1-c x)}{108 c}-\frac{1}{64} b x^4 \log (1-c x)+\frac{1}{9} a x^3 \log ^2(1-c x)+\frac{1}{16} b x^4 \log ^2(1-c x)-\frac{(3 b+4 a c) \log (c x) \log ^2(1-c x)}{12 c^4}-\frac{(3 b+4 a c) x \text{Li}_2(c x)}{12 c^3}-\frac{(3 b+4 a c) x^2 \text{Li}_2(c x)}{24 c^2}-\frac{(3 b+4 a c) x^3 \text{Li}_2(c x)}{36 c}-\frac{1}{16} b x^4 \text{Li}_2(c x)-\frac{(3 b+4 a c) \log (1-c x) \text{Li}_2(c x)}{12 c^4}+\frac{1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \text{Li}_2(c x)+\frac{1}{9} (2 a c) \int \left (-\frac{\log (1-c x)}{c^3}-\frac{x \log (1-c x)}{c^2}-\frac{x^2 \log (1-c x)}{c}-\frac{\log (1-c x)}{c^3 (-1+c x)}\right ) \, dx-\frac{1}{64} (b c) \int \frac{x^4}{1-c x} \, dx+\frac{1}{8} (b c) \int \left (-\frac{\log (1-c x)}{c^4}-\frac{x \log (1-c x)}{c^3}-\frac{x^2 \log (1-c x)}{c^2}-\frac{x^3 \log (1-c x)}{c}-\frac{\log (1-c x)}{c^4 (-1+c x)}\right ) \, dx-\frac{1}{108} (3 b+4 a c) \int \frac{x^3}{1-c x} \, dx+\frac{(3 b+4 a c) \operatorname{Subst}(\int \log (x) \, dx,x,1-c x)}{12 c^4}-\frac{(3 b+4 a c) \int \frac{\log (c x) \log (1-c x)}{1-c x} \, dx}{6 c^3}-\frac{(3 b+4 a c) \int \frac{x^2}{1-c x} \, dx}{48 c}\\ &=\frac{(3 b+4 a c) x}{12 c^3}-\frac{(3 b+4 a c) x^2 \log (1-c x)}{48 c^2}-\frac{(3 b+4 a c) x^3 \log (1-c x)}{108 c}-\frac{1}{64} b x^4 \log (1-c x)+\frac{(3 b+4 a c) (1-c x) \log (1-c x)}{12 c^4}+\frac{1}{9} a x^3 \log ^2(1-c x)+\frac{1}{16} b x^4 \log ^2(1-c x)-\frac{(3 b+4 a c) \log (c x) \log ^2(1-c x)}{12 c^4}-\frac{(3 b+4 a c) x \text{Li}_2(c x)}{12 c^3}-\frac{(3 b+4 a c) x^2 \text{Li}_2(c x)}{24 c^2}-\frac{(3 b+4 a c) x^3 \text{Li}_2(c x)}{36 c}-\frac{1}{16} b x^4 \text{Li}_2(c x)-\frac{(3 b+4 a c) \log (1-c x) \text{Li}_2(c x)}{12 c^4}+\frac{1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \text{Li}_2(c x)-\frac{1}{9} (2 a) \int x^2 \log (1-c x) \, dx-\frac{1}{8} b \int x^3 \log (1-c x) \, dx-\frac{b \int \log (1-c x) \, dx}{8 c^3}-\frac{b \int \frac{\log (1-c x)}{-1+c x} \, dx}{8 c^3}-\frac{(2 a) \int \log (1-c x) \, dx}{9 c^2}-\frac{(2 a) \int \frac{\log (1-c x)}{-1+c x} \, dx}{9 c^2}-\frac{b \int x \log (1-c x) \, dx}{8 c^2}-\frac{(2 a) \int x \log (1-c x) \, dx}{9 c}-\frac{b \int x^2 \log (1-c x) \, dx}{8 c}-\frac{1}{64} (b c) \int \left (-\frac{1}{c^4}-\frac{x}{c^3}-\frac{x^2}{c^2}-\frac{x^3}{c}-\frac{1}{c^4 (-1+c x)}\right ) \, dx-\frac{1}{108} (3 b+4 a c) \int \left (-\frac{1}{c^3}-\frac{x}{c^2}-\frac{x^2}{c}-\frac{1}{c^3 (-1+c x)}\right ) \, dx+\frac{(3 b+4 a c) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (c \left (\frac{1}{c}-\frac{x}{c}\right )\right )}{x} \, dx,x,1-c x\right )}{6 c^4}-\frac{(3 b+4 a c) \int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx}{48 c}\\ &=\frac{b x}{64 c^3}+\frac{49 (3 b+4 a c) x}{432 c^3}+\frac{b x^2}{128 c^2}+\frac{13 (3 b+4 a c) x^2}{864 c^2}+\frac{b x^3}{192 c}+\frac{(3 b+4 a c) x^3}{324 c}+\frac{b x^4}{256}+\frac{b \log (1-c x)}{64 c^4}+\frac{13 (3 b+4 a c) \log (1-c x)}{432 c^4}-\frac{b x^2 \log (1-c x)}{16 c^2}-\frac{a x^2 \log (1-c x)}{9 c}-\frac{(3 b+4 a c) x^2 \log (1-c x)}{48 c^2}-\frac{2}{27} a x^3 \log (1-c x)-\frac{b x^3 \log (1-c x)}{24 c}-\frac{(3 b+4 a c) x^3 \log (1-c x)}{108 c}-\frac{3}{64} b x^4 \log (1-c x)+\frac{(3 b+4 a c) (1-c x) \log (1-c x)}{12 c^4}+\frac{1}{9} a x^3 \log ^2(1-c x)+\frac{1}{16} b x^4 \log ^2(1-c x)-\frac{(3 b+4 a c) \log (c x) \log ^2(1-c x)}{12 c^4}-\frac{(3 b+4 a c) x \text{Li}_2(c x)}{12 c^3}-\frac{(3 b+4 a c) x^2 \text{Li}_2(c x)}{24 c^2}-\frac{(3 b+4 a c) x^3 \text{Li}_2(c x)}{36 c}-\frac{1}{16} b x^4 \text{Li}_2(c x)-\frac{(3 b+4 a c) \log (1-c x) \text{Li}_2(c x)}{12 c^4}+\frac{1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \text{Li}_2(c x)-\frac{(3 b+4 a c) \log (1-c x) \text{Li}_2(1-c x)}{6 c^4}-\frac{1}{9} a \int \frac{x^2}{1-c x} \, dx-\frac{1}{24} b \int \frac{x^3}{1-c x} \, dx+\frac{b \operatorname{Subst}(\int \log (x) \, dx,x,1-c x)}{8 c^4}-\frac{b \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x\right )}{8 c^4}+\frac{(2 a) \operatorname{Subst}(\int \log (x) \, dx,x,1-c x)}{9 c^3}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x\right )}{9 c^3}-\frac{b \int \frac{x^2}{1-c x} \, dx}{16 c}-\frac{1}{27} (2 a c) \int \frac{x^3}{1-c x} \, dx-\frac{1}{32} (b c) \int \frac{x^4}{1-c x} \, dx+\frac{(3 b+4 a c) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-c x\right )}{6 c^4}\\ &=\frac{9 b x}{64 c^3}+\frac{2 a x}{9 c^2}+\frac{49 (3 b+4 a c) x}{432 c^3}+\frac{b x^2}{128 c^2}+\frac{13 (3 b+4 a c) x^2}{864 c^2}+\frac{b x^3}{192 c}+\frac{(3 b+4 a c) x^3}{324 c}+\frac{b x^4}{256}+\frac{b \log (1-c x)}{64 c^4}+\frac{13 (3 b+4 a c) \log (1-c x)}{432 c^4}-\frac{b x^2 \log (1-c x)}{16 c^2}-\frac{a x^2 \log (1-c x)}{9 c}-\frac{(3 b+4 a c) x^2 \log (1-c x)}{48 c^2}-\frac{2}{27} a x^3 \log (1-c x)-\frac{b x^3 \log (1-c x)}{24 c}-\frac{(3 b+4 a c) x^3 \log (1-c x)}{108 c}-\frac{3}{64} b x^4 \log (1-c x)+\frac{b (1-c x) \log (1-c x)}{8 c^4}+\frac{2 a (1-c x) \log (1-c x)}{9 c^3}+\frac{(3 b+4 a c) (1-c x) \log (1-c x)}{12 c^4}-\frac{b \log ^2(1-c x)}{16 c^4}-\frac{a \log ^2(1-c x)}{9 c^3}+\frac{1}{9} a x^3 \log ^2(1-c x)+\frac{1}{16} b x^4 \log ^2(1-c x)-\frac{(3 b+4 a c) \log (c x) \log ^2(1-c x)}{12 c^4}-\frac{(3 b+4 a c) x \text{Li}_2(c x)}{12 c^3}-\frac{(3 b+4 a c) x^2 \text{Li}_2(c x)}{24 c^2}-\frac{(3 b+4 a c) x^3 \text{Li}_2(c x)}{36 c}-\frac{1}{16} b x^4 \text{Li}_2(c x)-\frac{(3 b+4 a c) \log (1-c x) \text{Li}_2(c x)}{12 c^4}+\frac{1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \text{Li}_2(c x)-\frac{(3 b+4 a c) \log (1-c x) \text{Li}_2(1-c x)}{6 c^4}+\frac{(3 b+4 a c) \text{Li}_3(1-c x)}{6 c^4}-\frac{1}{9} a \int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx-\frac{1}{24} b \int \left (-\frac{1}{c^3}-\frac{x}{c^2}-\frac{x^2}{c}-\frac{1}{c^3 (-1+c x)}\right ) \, dx-\frac{b \int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx}{16 c}-\frac{1}{27} (2 a c) \int \left (-\frac{1}{c^3}-\frac{x}{c^2}-\frac{x^2}{c}-\frac{1}{c^3 (-1+c x)}\right ) \, dx-\frac{1}{32} (b c) \int \left (-\frac{1}{c^4}-\frac{x}{c^3}-\frac{x^2}{c^2}-\frac{x^3}{c}-\frac{1}{c^4 (-1+c x)}\right ) \, dx\\ &=\frac{53 b x}{192 c^3}+\frac{11 a x}{27 c^2}+\frac{49 (3 b+4 a c) x}{432 c^3}+\frac{29 b x^2}{384 c^2}+\frac{5 a x^2}{54 c}+\frac{13 (3 b+4 a c) x^2}{864 c^2}+\frac{2 a x^3}{81}+\frac{17 b x^3}{576 c}+\frac{(3 b+4 a c) x^3}{324 c}+\frac{3 b x^4}{256}+\frac{29 b \log (1-c x)}{192 c^4}+\frac{5 a \log (1-c x)}{27 c^3}+\frac{13 (3 b+4 a c) \log (1-c x)}{432 c^4}-\frac{b x^2 \log (1-c x)}{16 c^2}-\frac{a x^2 \log (1-c x)}{9 c}-\frac{(3 b+4 a c) x^2 \log (1-c x)}{48 c^2}-\frac{2}{27} a x^3 \log (1-c x)-\frac{b x^3 \log (1-c x)}{24 c}-\frac{(3 b+4 a c) x^3 \log (1-c x)}{108 c}-\frac{3}{64} b x^4 \log (1-c x)+\frac{b (1-c x) \log (1-c x)}{8 c^4}+\frac{2 a (1-c x) \log (1-c x)}{9 c^3}+\frac{(3 b+4 a c) (1-c x) \log (1-c x)}{12 c^4}-\frac{b \log ^2(1-c x)}{16 c^4}-\frac{a \log ^2(1-c x)}{9 c^3}+\frac{1}{9} a x^3 \log ^2(1-c x)+\frac{1}{16} b x^4 \log ^2(1-c x)-\frac{(3 b+4 a c) \log (c x) \log ^2(1-c x)}{12 c^4}-\frac{(3 b+4 a c) x \text{Li}_2(c x)}{12 c^3}-\frac{(3 b+4 a c) x^2 \text{Li}_2(c x)}{24 c^2}-\frac{(3 b+4 a c) x^3 \text{Li}_2(c x)}{36 c}-\frac{1}{16} b x^4 \text{Li}_2(c x)-\frac{(3 b+4 a c) \log (1-c x) \text{Li}_2(c x)}{12 c^4}+\frac{1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \text{Li}_2(c x)-\frac{(3 b+4 a c) \log (1-c x) \text{Li}_2(1-c x)}{6 c^4}+\frac{(3 b+4 a c) \text{Li}_3(1-c x)}{6 c^4}\\ \end{align*}

Mathematica [A]  time = 0.755005, size = 425, normalized size = 0.64 \[ \frac{48 \text{PolyLog}(2,c x) \left (12 \log (1-c x) \left (4 a c \left (c^3 x^3-1\right )+3 b \left (c^4 x^4-1\right )\right )-c x \left (8 a c \left (2 c^2 x^2+3 c x+6\right )+3 b \left (3 c^3 x^3+4 c^2 x^2+6 c x+12\right )\right )\right )-1152 (4 a c+3 b) \log (1-c x) \text{PolyLog}(2,1-c x)+4608 a c \text{PolyLog}(3,1-c x)+3456 b \text{PolyLog}(3,1-c x)+256 a c^4 x^3+1056 a c^3 x^2+768 a c^4 x^3 \log ^2(1-c x)-768 a c^4 x^3 \log (1-c x)-1344 a c^3 x^2 \log (1-c x)+5952 a c^2 x-3840 a c^2 x \log (1-c x)-768 a c \log ^2(1-c x)-2304 a c \log (c x) \log ^2(1-c x)+5952 a c \log (1-c x)+81 b c^4 x^4+268 b c^3 x^3+834 b c^2 x^2+432 b c^4 x^4 \log ^2(1-c x)-324 b c^4 x^4 \log (1-c x)-480 b c^3 x^3 \log (1-c x)-864 b c^2 x^2 \log (1-c x)+4260 b c x-432 b \log ^2(1-c x)-1728 b \log (c x) \log ^2(1-c x)-2592 b c x \log (1-c x)+4260 b \log (1-c x)}{6912 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4260*b*c*x + 5952*a*c^2*x + 834*b*c^2*x^2 + 1056*a*c^3*x^2 + 268*b*c^3*x^3 + 256*a*c^4*x^3 + 81*b*c^4*x^4 + 4
260*b*Log[1 - c*x] + 5952*a*c*Log[1 - c*x] - 2592*b*c*x*Log[1 - c*x] - 3840*a*c^2*x*Log[1 - c*x] - 864*b*c^2*x
^2*Log[1 - c*x] - 1344*a*c^3*x^2*Log[1 - c*x] - 480*b*c^3*x^3*Log[1 - c*x] - 768*a*c^4*x^3*Log[1 - c*x] - 324*
b*c^4*x^4*Log[1 - c*x] - 432*b*Log[1 - c*x]^2 - 768*a*c*Log[1 - c*x]^2 + 768*a*c^4*x^3*Log[1 - c*x]^2 + 432*b*
c^4*x^4*Log[1 - c*x]^2 - 1728*b*Log[c*x]*Log[1 - c*x]^2 - 2304*a*c*Log[c*x]*Log[1 - c*x]^2 + 48*(-(c*x*(8*a*c*
(6 + 3*c*x + 2*c^2*x^2) + 3*b*(12 + 6*c*x + 4*c^2*x^2 + 3*c^3*x^3))) + 12*(4*a*c*(-1 + c^3*x^3) + 3*b*(-1 + c^
4*x^4))*Log[1 - c*x])*PolyLog[2, c*x] - 1152*(3*b + 4*a*c)*Log[1 - c*x]*PolyLog[2, 1 - c*x] + 3456*b*PolyLog[3
, 1 - c*x] + 4608*a*c*PolyLog[3, 1 - c*x])/(6912*c^4)

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Maple [F]  time = 0.007, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( bx+a \right ) \ln \left ( -cx+1 \right ){\it polylog} \left ( 2,cx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04004, size = 560, normalized size = 0.85 \begin{align*} -\frac{1}{6912} \, c{\left (\frac{576 \,{\left (\log \left (c x\right ) \log \left (-c x + 1\right )^{2} + 2 \,{\rm Li}_2\left (-c x + 1\right ) \log \left (-c x + 1\right ) - 2 \,{\rm Li}_{3}(-c x + 1)\right )}{\left (4 \, a c + 3 \, b\right )}}{c^{5}} - \frac{81 \, b c^{4} x^{4} + 4 \,{\left (64 \, a c^{4} + 67 \, b c^{3}\right )} x^{3} + 6 \,{\left (176 \, a c^{3} + 139 \, b c^{2}\right )} x^{2} + 12 \,{\left (496 \, a c^{2} + 355 \, b c\right )} x - 48 \,{\left (9 \, b c^{4} x^{4} + 4 \,{\left (4 \, a c^{4} + 3 \, b c^{3}\right )} x^{3} + 6 \,{\left (4 \, a c^{3} + 3 \, b c^{2}\right )} x^{2} + 12 \,{\left (4 \, a c^{2} + 3 \, b c\right )} x + 12 \,{\left (4 \, a c + 3 \, b\right )} \log \left (-c x + 1\right )\right )}{\rm Li}_2\left (c x\right ) - 4 \,{\left (54 \, b c^{4} x^{4} + 4 \,{\left (32 \, a c^{4} + 21 \, b c^{3}\right )} x^{3} + 6 \,{\left (40 \, a c^{3} + 27 \, b c^{2}\right )} x^{2} - 1488 \, a c + 12 \,{\left (64 \, a c^{2} + 45 \, b c\right )} x - 1065 \, b\right )} \log \left (-c x + 1\right )}{c^{5}}\right )} + \frac{1}{1728} \,{\left (\frac{32 \,{\left (18 \, c^{3} x^{3}{\rm Li}_2\left (c x\right ) - 2 \, c^{3} x^{3} - 3 \, c^{2} x^{2} - 6 \, c x + 6 \,{\left (c^{3} x^{3} - 1\right )} \log \left (-c x + 1\right )\right )} a}{c^{3}} + \frac{9 \,{\left (48 \, c^{4} x^{4}{\rm Li}_2\left (c x\right ) - 3 \, c^{4} x^{4} - 4 \, c^{3} x^{3} - 6 \, c^{2} x^{2} - 12 \, c x + 12 \,{\left (c^{4} x^{4} - 1\right )} \log \left (-c x + 1\right )\right )} b}{c^{4}}\right )} \log \left (-c x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6912*c*(576*(log(c*x)*log(-c*x + 1)^2 + 2*dilog(-c*x + 1)*log(-c*x + 1) - 2*polylog(3, -c*x + 1))*(4*a*c +
3*b)/c^5 - (81*b*c^4*x^4 + 4*(64*a*c^4 + 67*b*c^3)*x^3 + 6*(176*a*c^3 + 139*b*c^2)*x^2 + 12*(496*a*c^2 + 355*b
*c)*x - 48*(9*b*c^4*x^4 + 4*(4*a*c^4 + 3*b*c^3)*x^3 + 6*(4*a*c^3 + 3*b*c^2)*x^2 + 12*(4*a*c^2 + 3*b*c)*x + 12*
(4*a*c + 3*b)*log(-c*x + 1))*dilog(c*x) - 4*(54*b*c^4*x^4 + 4*(32*a*c^4 + 21*b*c^3)*x^3 + 6*(40*a*c^3 + 27*b*c
^2)*x^2 - 1488*a*c + 12*(64*a*c^2 + 45*b*c)*x - 1065*b)*log(-c*x + 1))/c^5) + 1/1728*(32*(18*c^3*x^3*dilog(c*x
) - 2*c^3*x^3 - 3*c^2*x^2 - 6*c*x + 6*(c^3*x^3 - 1)*log(-c*x + 1))*a/c^3 + 9*(48*c^4*x^4*dilog(c*x) - 3*c^4*x^
4 - 4*c^3*x^3 - 6*c^2*x^2 - 12*c*x + 12*(c^4*x^4 - 1)*log(-c*x + 1))*b/c^4)*log(-c*x + 1)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{3} + a x^{2}\right )}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )} x^{2}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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