3.2.92 \(\int x (a+b x+c x^2) \log (1-d x) \text {PolyLog}(2,d x) \, dx\) [192]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.75, 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 = {6874, 6726, 2442, 45, 14, 6739, 2448, 2436, 2333, 2332, 2437, 2342, 2341, 2445, 2457, 2338, 6721, 6731, 2443, 2481, 2421, 6724} \begin {gather*} \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} \end {gather*}

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 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 45

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 2332

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

Rule 2333

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 2338

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 2341

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 2342

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2436

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 2437

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 2442

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 2443

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 2445

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 2448

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 2457

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 2481

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 6721

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 6724

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]

Rule 6726

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 6731

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 6739

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 6874

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

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 ) \text {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 \text {Subst}\left (\int \log ^2(x) \, dx,x,1-d x\right )}{2 d^2}+\frac {a \text {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 ) \text {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 \text {Subst}(\int \log (x) \, dx,x,1-d x)}{8 d^4}-\frac {c \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-d x\right )}{8 d^4}+\frac {(2 b) \text {Subst}(\int \log (x) \, dx,x,1-d x)}{9 d^3}-\frac {(2 b) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-d x\right )}{9 d^3}-\frac {a \text {Subst}(\int x \log (x) \, dx,x,1-d x)}{2 d^2}+\frac {a \text {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 ) \text {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 = 0.85, size = 583, normalized size = 0.65 \begin {gather*} \frac {\frac {355 c d x}{4}+124 b d^2 x+198 a d^3 x+\frac {139}{8} c d^2 x^2+22 b d^3 x^2+27 a d^4 x^2+\frac {67}{12} c d^3 x^3+\frac {16}{3} b d^4 x^3+\frac {27}{16} c d^4 x^4+\frac {355}{4} c \log (1-d x)+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)-\frac {27}{4} c d^4 x^4 \log (1-d x)-9 c \log ^2(1-d x)-16 b d \log ^2(1-d x)-36 a d^2 \log ^2(1-d x)+36 a d^4 x^2 \log ^2(1-d x)+16 b d^4 x^3 \log ^2(1-d x)+9 c d^4 x^4 \log ^2(1-d x)-36 c \log (d x) \log ^2(1-d x)-48 b d \log (d x) \log ^2(1-d x)-72 a d^2 \log (d x) \log ^2(1-d x)+\left (-d x \left (3 c \left (12+6 d x+4 d^2 x^2+3 d^3 x^3\right )+4 d \left (9 a d (2+d x)+2 b \left (6+3 d x+2 d^2 x^2\right )\right )\right )+12 \left (-4 b d-6 a d^2+6 a d^4 x^2+4 b d^4 x^3+3 c \left (-1+d^4 x^4\right )\right ) \log (1-d x)\right ) \text {PolyLog}(2,d x)-24 \left (3 c+4 b d+6 a d^2\right ) \log (1-d x) \text {PolyLog}(2,1-d x)+24 \left (3 c+4 b d+6 a d^2\right ) \text {PolyLog}(3,1-d x)}{144 d^4} \end {gather*}

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 = 0.04, 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.

[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 = 0.27, size = 518, normalized size = 0.58 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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