3.198 \(\int \frac{(a+b x+c x^2) \log (1-d x) \text{PolyLog}(2,d x)}{x^5} \, dx\)

Optimal. Leaf size=767 \[ -\frac{1}{12} d^2 \text{PolyLog}(3,d x) (d (3 a d+4 b)+6 c)-\frac{1}{6} d^2 \text{PolyLog}(3,1-d x) (d (3 a d+4 b)+6 c)+\frac{1}{12} d^2 \log (1-d x) \text{PolyLog}(2,d x) (d (3 a d+4 b)+6 c)+\frac{1}{6} d^2 \log (1-d x) \text{PolyLog}(2,1-d x) (d (3 a d+4 b)+6 c)-\frac{1}{12} \log (1-d x) \text{PolyLog}(2,d x) \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right )+\frac{d \text{PolyLog}(2,d x) (d (3 a d+4 b)+6 c)}{12 x}+\frac{d (3 a d+4 b) \text{PolyLog}(2,d x)}{24 x^2}-\frac{1}{8} a d^4 \text{PolyLog}(2,d x)+\frac{a d \text{PolyLog}(2,d x)}{12 x^3}-\frac{2}{9} b d^3 \text{PolyLog}(2,d x)-\frac{1}{2} c d^2 \text{PolyLog}(2,d x)+\frac{1}{12} d^2 \log (d x) \log ^2(1-d x) (d (3 a d+4 b)+6 c)-\frac{1}{12} d^2 \log (x) (d (3 a d+4 b)+6 c)+\frac{1}{12} d^2 \log (1-d x) (d (3 a d+4 b)+6 c)-\frac{d \log (1-d x) (d (3 a d+4 b)+6 c)}{12 x}+\frac{d^2 (3 a d+4 b)}{48 x}-\frac{1}{48} d^3 \log (x) (3 a d+4 b)+\frac{1}{48} d^3 (3 a d+4 b) \log (1-d x)-\frac{d (3 a d+4 b) \log (1-d x)}{48 x^2}+\frac{5 a d^2}{144 x^2}-\frac{a d^2 \log (1-d x)}{16 x^2}+\frac{19 a d^3}{144 x}-\frac{1}{16} a d^4 \log ^2(1-d x)-\frac{37}{144} a d^4 \log (x)+\frac{37}{144} a d^4 \log (1-d x)-\frac{a d^3 \log (1-d x)}{8 x}+\frac{a \log ^2(1-d x)}{16 x^4}-\frac{5 a d \log (1-d x)}{72 x^3}+\frac{b d^2}{9 x}-\frac{1}{9} b d^3 \log ^2(1-d x)-\frac{1}{3} b d^3 \log (x)+\frac{1}{3} b d^3 \log (1-d x)-\frac{2 b d^2 \log (1-d x)}{9 x}+\frac{b \log ^2(1-d x)}{9 x^3}-\frac{b d \log (1-d x)}{9 x^2}-\frac{1}{4} c d^2 \log ^2(1-d x)-\frac{1}{2} c d^2 \log (x)+\frac{1}{2} c d^2 \log (1-d x)+\frac{c \log ^2(1-d x)}{4 x^2}-\frac{c d \log (1-d x)}{2 x} \]

[Out]

(5*a*d^2)/(144*x^2) + (b*d^2)/(9*x) + (19*a*d^3)/(144*x) + (d^2*(4*b + 3*a*d))/(48*x) - (c*d^2*Log[x])/2 - (b*
d^3*Log[x])/3 - (37*a*d^4*Log[x])/144 - (d^3*(4*b + 3*a*d)*Log[x])/48 - (d^2*(6*c + d*(4*b + 3*a*d))*Log[x])/1
2 + (c*d^2*Log[1 - d*x])/2 + (b*d^3*Log[1 - d*x])/3 + (37*a*d^4*Log[1 - d*x])/144 + (d^3*(4*b + 3*a*d)*Log[1 -
d*x])/48 + (d^2*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x])/12 - (5*a*d*Log[1 - d*x])/(72*x^3) - (b*d*Log[1 - d*x])
/(9*x^2) - (a*d^2*Log[1 - d*x])/(16*x^2) - (d*(4*b + 3*a*d)*Log[1 - d*x])/(48*x^2) - (c*d*Log[1 - d*x])/(2*x)
- (2*b*d^2*Log[1 - d*x])/(9*x) - (a*d^3*Log[1 - d*x])/(8*x) - (d*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x])/(12*x)
- (c*d^2*Log[1 - d*x]^2)/4 - (b*d^3*Log[1 - d*x]^2)/9 - (a*d^4*Log[1 - d*x]^2)/16 + (a*Log[1 - d*x]^2)/(16*x^4
) + (b*Log[1 - d*x]^2)/(9*x^3) + (c*Log[1 - d*x]^2)/(4*x^2) + (d^2*(6*c + d*(4*b + 3*a*d))*Log[d*x]*Log[1 - d*
x]^2)/12 - (c*d^2*PolyLog[2, d*x])/2 - (2*b*d^3*PolyLog[2, d*x])/9 - (a*d^4*PolyLog[2, d*x])/8 + (a*d*PolyLog[
2, d*x])/(12*x^3) + (d*(4*b + 3*a*d)*PolyLog[2, d*x])/(24*x^2) + (d*(6*c + d*(4*b + 3*a*d))*PolyLog[2, d*x])/(
12*x) + (d^2*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x]*PolyLog[2, d*x])/12 - (((3*a)/x^4 + (4*b)/x^3 + (6*c)/x^2)*L
og[1 - d*x]*PolyLog[2, d*x])/12 + (d^2*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x]*PolyLog[2, 1 - d*x])/6 - (d^2*(6*c
+ d*(4*b + 3*a*d))*PolyLog[3, d*x])/12 - (d^2*(6*c + d*(4*b + 3*a*d))*PolyLog[3, 1 - d*x])/6

________________________________________________________________________________________

Rubi [A]  time = 1.12465, antiderivative size = 767, normalized size of antiderivative = 1., number of steps used = 61, number of rules used = 19, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.731, Rules used = {6742, 6591, 2395, 44, 14, 6606, 2398, 2410, 36, 29, 31, 2391, 2390, 2301, 6589, 6596, 2396, 2433, 2374} \[ -\frac{1}{12} d^2 \text{PolyLog}(3,d x) (d (3 a d+4 b)+6 c)-\frac{1}{6} d^2 \text{PolyLog}(3,1-d x) (d (3 a d+4 b)+6 c)+\frac{1}{12} d^2 \log (1-d x) \text{PolyLog}(2,d x) (d (3 a d+4 b)+6 c)+\frac{1}{6} d^2 \log (1-d x) \text{PolyLog}(2,1-d x) (d (3 a d+4 b)+6 c)-\frac{1}{12} \log (1-d x) \text{PolyLog}(2,d x) \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right )+\frac{d \text{PolyLog}(2,d x) (d (3 a d+4 b)+6 c)}{12 x}+\frac{d (3 a d+4 b) \text{PolyLog}(2,d x)}{24 x^2}-\frac{1}{8} a d^4 \text{PolyLog}(2,d x)+\frac{a d \text{PolyLog}(2,d x)}{12 x^3}-\frac{2}{9} b d^3 \text{PolyLog}(2,d x)-\frac{1}{2} c d^2 \text{PolyLog}(2,d x)+\frac{1}{12} d^2 \log (d x) \log ^2(1-d x) (d (3 a d+4 b)+6 c)-\frac{1}{12} d^2 \log (x) (d (3 a d+4 b)+6 c)+\frac{1}{12} d^2 \log (1-d x) (d (3 a d+4 b)+6 c)-\frac{d \log (1-d x) (d (3 a d+4 b)+6 c)}{12 x}+\frac{d^2 (3 a d+4 b)}{48 x}-\frac{1}{48} d^3 \log (x) (3 a d+4 b)+\frac{1}{48} d^3 (3 a d+4 b) \log (1-d x)-\frac{d (3 a d+4 b) \log (1-d x)}{48 x^2}+\frac{5 a d^2}{144 x^2}-\frac{a d^2 \log (1-d x)}{16 x^2}+\frac{19 a d^3}{144 x}-\frac{1}{16} a d^4 \log ^2(1-d x)-\frac{37}{144} a d^4 \log (x)+\frac{37}{144} a d^4 \log (1-d x)-\frac{a d^3 \log (1-d x)}{8 x}+\frac{a \log ^2(1-d x)}{16 x^4}-\frac{5 a d \log (1-d x)}{72 x^3}+\frac{b d^2}{9 x}-\frac{1}{9} b d^3 \log ^2(1-d x)-\frac{1}{3} b d^3 \log (x)+\frac{1}{3} b d^3 \log (1-d x)-\frac{2 b d^2 \log (1-d x)}{9 x}+\frac{b \log ^2(1-d x)}{9 x^3}-\frac{b d \log (1-d x)}{9 x^2}-\frac{1}{4} c d^2 \log ^2(1-d x)-\frac{1}{2} c d^2 \log (x)+\frac{1}{2} c d^2 \log (1-d x)+\frac{c \log ^2(1-d x)}{4 x^2}-\frac{c d \log (1-d x)}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x])/x^5,x]

[Out]

(5*a*d^2)/(144*x^2) + (b*d^2)/(9*x) + (19*a*d^3)/(144*x) + (d^2*(4*b + 3*a*d))/(48*x) - (c*d^2*Log[x])/2 - (b*
d^3*Log[x])/3 - (37*a*d^4*Log[x])/144 - (d^3*(4*b + 3*a*d)*Log[x])/48 - (d^2*(6*c + d*(4*b + 3*a*d))*Log[x])/1
2 + (c*d^2*Log[1 - d*x])/2 + (b*d^3*Log[1 - d*x])/3 + (37*a*d^4*Log[1 - d*x])/144 + (d^3*(4*b + 3*a*d)*Log[1 -
d*x])/48 + (d^2*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x])/12 - (5*a*d*Log[1 - d*x])/(72*x^3) - (b*d*Log[1 - d*x])
/(9*x^2) - (a*d^2*Log[1 - d*x])/(16*x^2) - (d*(4*b + 3*a*d)*Log[1 - d*x])/(48*x^2) - (c*d*Log[1 - d*x])/(2*x)
- (2*b*d^2*Log[1 - d*x])/(9*x) - (a*d^3*Log[1 - d*x])/(8*x) - (d*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x])/(12*x)
- (c*d^2*Log[1 - d*x]^2)/4 - (b*d^3*Log[1 - d*x]^2)/9 - (a*d^4*Log[1 - d*x]^2)/16 + (a*Log[1 - d*x]^2)/(16*x^4
) + (b*Log[1 - d*x]^2)/(9*x^3) + (c*Log[1 - d*x]^2)/(4*x^2) + (d^2*(6*c + d*(4*b + 3*a*d))*Log[d*x]*Log[1 - d*
x]^2)/12 - (c*d^2*PolyLog[2, d*x])/2 - (2*b*d^3*PolyLog[2, d*x])/9 - (a*d^4*PolyLog[2, d*x])/8 + (a*d*PolyLog[
2, d*x])/(12*x^3) + (d*(4*b + 3*a*d)*PolyLog[2, d*x])/(24*x^2) + (d*(6*c + d*(4*b + 3*a*d))*PolyLog[2, d*x])/(
12*x) + (d^2*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x]*PolyLog[2, d*x])/12 - (((3*a)/x^4 + (4*b)/x^3 + (6*c)/x^2)*L
og[1 - d*x]*PolyLog[2, d*x])/12 + (d^2*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x]*PolyLog[2, 1 - d*x])/6 - (d^2*(6*c
+ d*(4*b + 3*a*d))*PolyLog[3, d*x])/12 - (d^2*(6*c + d*(4*b + 3*a*d))*PolyLog[3, 1 - d*x])/6

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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 6606

Int[((g_.) + Log[(f_.)*((d_.) + (e_.)*(x_))^(n_.)]*(h_.))*(Px_)*(x_)^(m_.)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_
))], x_Symbol] :> With[{u = IntHide[x^m*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] - Dis
t[e*h*n, Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x], x])] /; FreeQ[{a, b, c, d, e, f, g,
h, n}, x] && PolyQ[Px, x] && IntegerQ[m]

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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 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]

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right ) \log (1-d x) \text{Li}_2(d x)}{x^5} \, dx &=-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)+d \int \left (-\frac{a \text{Li}_2(d x)}{4 x^4}+\frac{(-4 b-3 a d) \text{Li}_2(d x)}{12 x^3}+\frac{\left (-6 c-4 b d-3 a d^2\right ) \text{Li}_2(d x)}{12 x^2}+\frac{d (-6 c-d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{d^2 (-6 c-d (4 b+3 a d)) \text{Li}_2(d x)}{12 (1-d x)}\right ) \, dx+\int \left (-\frac{a \log ^2(1-d x)}{4 x^5}-\frac{b \log ^2(1-d x)}{3 x^4}-\frac{c \log ^2(1-d x)}{2 x^3}\right ) \, dx\\ &=-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{4} a \int \frac{\log ^2(1-d x)}{x^5} \, dx-\frac{1}{3} b \int \frac{\log ^2(1-d x)}{x^4} \, dx-\frac{1}{2} c \int \frac{\log ^2(1-d x)}{x^3} \, dx-\frac{1}{4} (a d) \int \frac{\text{Li}_2(d x)}{x^4} \, dx-\frac{1}{12} (d (4 b+3 a d)) \int \frac{\text{Li}_2(d x)}{x^3} \, dx-\frac{1}{12} (d (6 c+d (4 b+3 a d))) \int \frac{\text{Li}_2(d x)}{x^2} \, dx-\frac{1}{12} \left (d^2 (6 c+d (4 b+3 a d))\right ) \int \frac{\text{Li}_2(d x)}{x} \, dx-\frac{1}{12} \left (d^3 (6 c+d (4 b+3 a d))\right ) \int \frac{\text{Li}_2(d x)}{1-d x} \, dx\\ &=\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)+\frac{1}{12} (a d) \int \frac{\log (1-d x)}{x^4} \, dx+\frac{1}{8} (a d) \int \frac{\log (1-d x)}{x^4 (1-d x)} \, dx+\frac{1}{9} (2 b d) \int \frac{\log (1-d x)}{x^3 (1-d x)} \, dx+\frac{1}{2} (c d) \int \frac{\log (1-d x)}{x^2 (1-d x)} \, dx+\frac{1}{24} (d (4 b+3 a d)) \int \frac{\log (1-d x)}{x^3} \, dx+\frac{1}{12} (d (6 c+d (4 b+3 a d))) \int \frac{\log (1-d x)}{x^2} \, dx+\frac{1}{12} \left (d^2 (6 c+d (4 b+3 a d))\right ) \int \frac{\log ^2(1-d x)}{x} \, dx\\ &=-\frac{a d \log (1-d x)}{36 x^3}-\frac{d (4 b+3 a d) \log (1-d x)}{48 x^2}-\frac{d (6 c+d (4 b+3 a d)) \log (1-d x)}{12 x}+\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (d x) \log ^2(1-d x)+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)+\frac{1}{8} (a d) \int \left (\frac{\log (1-d x)}{x^4}+\frac{d \log (1-d x)}{x^3}+\frac{d^2 \log (1-d x)}{x^2}+\frac{d^3 \log (1-d x)}{x}-\frac{d^4 \log (1-d x)}{-1+d x}\right ) \, dx+\frac{1}{9} (2 b d) \int \left (\frac{\log (1-d x)}{x^3}+\frac{d \log (1-d x)}{x^2}+\frac{d^2 \log (1-d x)}{x}-\frac{d^3 \log (1-d x)}{-1+d x}\right ) \, dx+\frac{1}{2} (c d) \int \left (\frac{\log (1-d x)}{x^2}+\frac{d \log (1-d x)}{x}-\frac{d^2 \log (1-d x)}{-1+d x}\right ) \, dx-\frac{1}{36} \left (a d^2\right ) \int \frac{1}{x^3 (1-d x)} \, dx-\frac{1}{48} \left (d^2 (4 b+3 a d)\right ) \int \frac{1}{x^2 (1-d x)} \, dx-\frac{1}{12} \left (d^2 (6 c+d (4 b+3 a d))\right ) \int \frac{1}{x (1-d x)} \, dx+\frac{1}{6} \left (d^3 (6 c+d (4 b+3 a d))\right ) \int \frac{\log (d x) \log (1-d x)}{1-d x} \, dx\\ &=-\frac{a d \log (1-d x)}{36 x^3}-\frac{d (4 b+3 a d) \log (1-d x)}{48 x^2}-\frac{d (6 c+d (4 b+3 a d)) \log (1-d x)}{12 x}+\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (d x) \log ^2(1-d x)+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)+\frac{1}{8} (a d) \int \frac{\log (1-d x)}{x^4} \, dx+\frac{1}{9} (2 b d) \int \frac{\log (1-d x)}{x^3} \, dx+\frac{1}{2} (c d) \int \frac{\log (1-d x)}{x^2} \, dx-\frac{1}{36} \left (a d^2\right ) \int \left (\frac{1}{x^3}+\frac{d}{x^2}+\frac{d^2}{x}-\frac{d^3}{-1+d x}\right ) \, dx+\frac{1}{8} \left (a d^2\right ) \int \frac{\log (1-d x)}{x^3} \, dx+\frac{1}{9} \left (2 b d^2\right ) \int \frac{\log (1-d x)}{x^2} \, dx+\frac{1}{2} \left (c d^2\right ) \int \frac{\log (1-d x)}{x} \, dx+\frac{1}{8} \left (a d^3\right ) \int \frac{\log (1-d x)}{x^2} \, dx+\frac{1}{9} \left (2 b d^3\right ) \int \frac{\log (1-d x)}{x} \, dx-\frac{1}{2} \left (c d^3\right ) \int \frac{\log (1-d x)}{-1+d x} \, dx+\frac{1}{8} \left (a d^4\right ) \int \frac{\log (1-d x)}{x} \, dx-\frac{1}{9} \left (2 b d^4\right ) \int \frac{\log (1-d x)}{-1+d x} \, dx-\frac{1}{8} \left (a d^5\right ) \int \frac{\log (1-d x)}{-1+d x} \, dx-\frac{1}{48} \left (d^2 (4 b+3 a d)\right ) \int \left (\frac{1}{x^2}+\frac{d}{x}-\frac{d^2}{-1+d x}\right ) \, dx-\frac{1}{12} \left (d^2 (6 c+d (4 b+3 a d))\right ) \int \frac{1}{x} \, dx-\frac{1}{6} \left (d^2 (6 c+d (4 b+3 a d))\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 )-\frac{1}{12} \left (d^3 (6 c+d (4 b+3 a d))\right ) \int \frac{1}{1-d x} \, dx\\ &=\frac{a d^2}{72 x^2}+\frac{a d^3}{36 x}+\frac{d^2 (4 b+3 a d)}{48 x}-\frac{1}{36} a d^4 \log (x)-\frac{1}{48} d^3 (4 b+3 a d) \log (x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (x)+\frac{1}{36} a d^4 \log (1-d x)+\frac{1}{48} d^3 (4 b+3 a d) \log (1-d x)+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x)-\frac{5 a d \log (1-d x)}{72 x^3}-\frac{b d \log (1-d x)}{9 x^2}-\frac{a d^2 \log (1-d x)}{16 x^2}-\frac{d (4 b+3 a d) \log (1-d x)}{48 x^2}-\frac{c d \log (1-d x)}{2 x}-\frac{2 b d^2 \log (1-d x)}{9 x}-\frac{a d^3 \log (1-d x)}{8 x}-\frac{d (6 c+d (4 b+3 a d)) \log (1-d x)}{12 x}+\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (d x) \log ^2(1-d x)-\frac{1}{2} c d^2 \text{Li}_2(d x)-\frac{2}{9} b d^3 \text{Li}_2(d x)-\frac{1}{8} a d^4 \text{Li}_2(d x)+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)+\frac{1}{6} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(1-d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)-\frac{1}{24} \left (a d^2\right ) \int \frac{1}{x^3 (1-d x)} \, dx-\frac{1}{9} \left (b d^2\right ) \int \frac{1}{x^2 (1-d x)} \, dx-\frac{1}{2} \left (c d^2\right ) \int \frac{1}{x (1-d x)} \, dx-\frac{1}{2} \left (c d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-d x\right )-\frac{1}{16} \left (a d^3\right ) \int \frac{1}{x^2 (1-d x)} \, dx-\frac{1}{9} \left (2 b d^3\right ) \int \frac{1}{x (1-d x)} \, dx-\frac{1}{9} \left (2 b d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-d x\right )-\frac{1}{8} \left (a d^4\right ) \int \frac{1}{x (1-d x)} \, dx-\frac{1}{8} \left (a d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-d x\right )-\frac{1}{6} \left (d^2 (6 c+d (4 b+3 a d))\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-d x\right )\\ &=\frac{a d^2}{72 x^2}+\frac{a d^3}{36 x}+\frac{d^2 (4 b+3 a d)}{48 x}-\frac{1}{36} a d^4 \log (x)-\frac{1}{48} d^3 (4 b+3 a d) \log (x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (x)+\frac{1}{36} a d^4 \log (1-d x)+\frac{1}{48} d^3 (4 b+3 a d) \log (1-d x)+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x)-\frac{5 a d \log (1-d x)}{72 x^3}-\frac{b d \log (1-d x)}{9 x^2}-\frac{a d^2 \log (1-d x)}{16 x^2}-\frac{d (4 b+3 a d) \log (1-d x)}{48 x^2}-\frac{c d \log (1-d x)}{2 x}-\frac{2 b d^2 \log (1-d x)}{9 x}-\frac{a d^3 \log (1-d x)}{8 x}-\frac{d (6 c+d (4 b+3 a d)) \log (1-d x)}{12 x}-\frac{1}{4} c d^2 \log ^2(1-d x)-\frac{1}{9} b d^3 \log ^2(1-d x)-\frac{1}{16} a d^4 \log ^2(1-d x)+\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (d x) \log ^2(1-d x)-\frac{1}{2} c d^2 \text{Li}_2(d x)-\frac{2}{9} b d^3 \text{Li}_2(d x)-\frac{1}{8} a d^4 \text{Li}_2(d x)+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)+\frac{1}{6} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(1-d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)-\frac{1}{6} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(1-d x)-\frac{1}{24} \left (a d^2\right ) \int \left (\frac{1}{x^3}+\frac{d}{x^2}+\frac{d^2}{x}-\frac{d^3}{-1+d x}\right ) \, dx-\frac{1}{9} \left (b d^2\right ) \int \left (\frac{1}{x^2}+\frac{d}{x}-\frac{d^2}{-1+d x}\right ) \, dx-\frac{1}{2} \left (c d^2\right ) \int \frac{1}{x} \, dx-\frac{1}{16} \left (a d^3\right ) \int \left (\frac{1}{x^2}+\frac{d}{x}-\frac{d^2}{-1+d x}\right ) \, dx-\frac{1}{9} \left (2 b d^3\right ) \int \frac{1}{x} \, dx-\frac{1}{2} \left (c d^3\right ) \int \frac{1}{1-d x} \, dx-\frac{1}{8} \left (a d^4\right ) \int \frac{1}{x} \, dx-\frac{1}{9} \left (2 b d^4\right ) \int \frac{1}{1-d x} \, dx-\frac{1}{8} \left (a d^5\right ) \int \frac{1}{1-d x} \, dx\\ &=\frac{5 a d^2}{144 x^2}+\frac{b d^2}{9 x}+\frac{19 a d^3}{144 x}+\frac{d^2 (4 b+3 a d)}{48 x}-\frac{1}{2} c d^2 \log (x)-\frac{1}{3} b d^3 \log (x)-\frac{37}{144} a d^4 \log (x)-\frac{1}{48} d^3 (4 b+3 a d) \log (x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (x)+\frac{1}{2} c d^2 \log (1-d x)+\frac{1}{3} b d^3 \log (1-d x)+\frac{37}{144} a d^4 \log (1-d x)+\frac{1}{48} d^3 (4 b+3 a d) \log (1-d x)+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x)-\frac{5 a d \log (1-d x)}{72 x^3}-\frac{b d \log (1-d x)}{9 x^2}-\frac{a d^2 \log (1-d x)}{16 x^2}-\frac{d (4 b+3 a d) \log (1-d x)}{48 x^2}-\frac{c d \log (1-d x)}{2 x}-\frac{2 b d^2 \log (1-d x)}{9 x}-\frac{a d^3 \log (1-d x)}{8 x}-\frac{d (6 c+d (4 b+3 a d)) \log (1-d x)}{12 x}-\frac{1}{4} c d^2 \log ^2(1-d x)-\frac{1}{9} b d^3 \log ^2(1-d x)-\frac{1}{16} a d^4 \log ^2(1-d x)+\frac{a \log ^2(1-d x)}{16 x^4}+\frac{b \log ^2(1-d x)}{9 x^3}+\frac{c \log ^2(1-d x)}{4 x^2}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (d x) \log ^2(1-d x)-\frac{1}{2} c d^2 \text{Li}_2(d x)-\frac{2}{9} b d^3 \text{Li}_2(d x)-\frac{1}{8} a d^4 \text{Li}_2(d x)+\frac{a d \text{Li}_2(d x)}{12 x^3}+\frac{d (4 b+3 a d) \text{Li}_2(d x)}{24 x^2}+\frac{d (6 c+d (4 b+3 a d)) \text{Li}_2(d x)}{12 x}+\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(d x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}+\frac{6 c}{x^2}\right ) \log (1-d x) \text{Li}_2(d x)+\frac{1}{6} d^2 (6 c+d (4 b+3 a d)) \log (1-d x) \text{Li}_2(1-d x)-\frac{1}{12} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(d x)-\frac{1}{6} d^2 (6 c+d (4 b+3 a d)) \text{Li}_3(1-d x)\\ \end{align*}

Mathematica [A]  time = 1.95317, size = 621, normalized size = 0.81 \[ \frac{1}{144} \left (\frac{6 \text{PolyLog}(2,d x) \left (d x \left (a \left (6 d^2 x^2+3 d x+2\right )+4 x (2 b d x+b+3 c x)\right )+2 \log (1-d x) \left (3 a \left (d^4 x^4-1\right )+4 b d^3 x^4-4 b x+6 c d^2 x^4-6 c x^2\right )\right )}{x^4}+2 d^2 \text{PolyLog}(2,1-d x) \left (12 \log (1-d x) \left (3 a d^2+4 b d+6 c\right )+9 a d^2+16 b d+36 c\right )-36 a d^4 \text{PolyLog}(3,d x)-72 a d^4 \text{PolyLog}(3,1-d x)-48 b d^3 \text{PolyLog}(3,d x)-96 b d^3 \text{PolyLog}(3,1-d x)-72 c d^2 \text{PolyLog}(3,d x)-144 c d^2 \text{PolyLog}(3,1-d x)+\frac{5 a d^2}{x^2}-\frac{18 a d^2 \log (1-d x)}{x^2}+\frac{28 a d^3}{x}-9 a d^4 \log ^2(1-d x)+36 a d^4 \log (d x) \log ^2(1-d x)-82 a d^4 \log (d x)+82 a d^4 \log (1-d x)+18 a d^4 \log (d x) \log (1-d x)-\frac{54 a d^3 \log (1-d x)}{x}-33 a d^4+\frac{9 a \log ^2(1-d x)}{x^4}-\frac{10 a d \log (1-d x)}{x^3}+\frac{28 b d^2}{x}-16 b d^3 \log ^2(1-d x)+48 b d^3 \log (d x) \log ^2(1-d x)-108 b d^3 \log (d x)+108 b d^3 \log (1-d x)+32 b d^3 \log (d x) \log (1-d x)-\frac{80 b d^2 \log (1-d x)}{x}-28 b d^3+\frac{16 b \log ^2(1-d x)}{x^3}-\frac{28 b d \log (1-d x)}{x^2}-36 c d^2 \log ^2(1-d x)+72 c d^2 \log (d x) \log ^2(1-d x)-144 c d^2 \log (d x)+144 c d^2 \log (1-d x)+72 c d^2 \log (d x) \log (1-d x)+\frac{36 c \log ^2(1-d x)}{x^2}-\frac{144 c d \log (1-d x)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x])/x^5,x]

[Out]

(-28*b*d^3 - 33*a*d^4 + (5*a*d^2)/x^2 + (28*b*d^2)/x + (28*a*d^3)/x - 144*c*d^2*Log[d*x] - 108*b*d^3*Log[d*x]
- 82*a*d^4*Log[d*x] + 144*c*d^2*Log[1 - d*x] + 108*b*d^3*Log[1 - d*x] + 82*a*d^4*Log[1 - d*x] - (10*a*d*Log[1
- d*x])/x^3 - (28*b*d*Log[1 - d*x])/x^2 - (18*a*d^2*Log[1 - d*x])/x^2 - (144*c*d*Log[1 - d*x])/x - (80*b*d^2*L
og[1 - d*x])/x - (54*a*d^3*Log[1 - d*x])/x + 72*c*d^2*Log[d*x]*Log[1 - d*x] + 32*b*d^3*Log[d*x]*Log[1 - d*x] +
18*a*d^4*Log[d*x]*Log[1 - d*x] - 36*c*d^2*Log[1 - d*x]^2 - 16*b*d^3*Log[1 - d*x]^2 - 9*a*d^4*Log[1 - d*x]^2 +
(9*a*Log[1 - d*x]^2)/x^4 + (16*b*Log[1 - d*x]^2)/x^3 + (36*c*Log[1 - d*x]^2)/x^2 + 72*c*d^2*Log[d*x]*Log[1 -
d*x]^2 + 48*b*d^3*Log[d*x]*Log[1 - d*x]^2 + 36*a*d^4*Log[d*x]*Log[1 - d*x]^2 + (6*(d*x*(4*x*(b + 3*c*x + 2*b*d
*x) + a*(2 + 3*d*x + 6*d^2*x^2)) + 2*(-4*b*x - 6*c*x^2 + 6*c*d^2*x^4 + 4*b*d^3*x^4 + 3*a*(-1 + d^4*x^4))*Log[1
- d*x])*PolyLog[2, d*x])/x^4 + 2*d^2*(36*c + 16*b*d + 9*a*d^2 + 12*(6*c + 4*b*d + 3*a*d^2)*Log[1 - d*x])*Poly
Log[2, 1 - d*x] - 72*c*d^2*PolyLog[3, d*x] - 48*b*d^3*PolyLog[3, d*x] - 36*a*d^4*PolyLog[3, d*x] - 144*c*d^2*P
olyLog[3, 1 - d*x] - 96*b*d^3*PolyLog[3, 1 - d*x] - 72*a*d^4*PolyLog[3, 1 - d*x])/144

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c{x}^{2}+bx+a \right ) \ln \left ( -dx+1 \right ){\it polylog} \left ( 2,dx \right ) }{{x}^{5}}}\, 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^5,x)

[Out]

int((c*x^2+b*x+a)*ln(-d*x+1)*polylog(2,d*x)/x^5,x)

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Maxima [A]  time = 1.23365, size = 544, normalized size = 0.71 \begin{align*} \frac{1}{12} \,{\left (3 \, a d^{4} + 4 \, b d^{3} + 6 \, c d^{2}\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 )} + \frac{1}{72} \,{\left (9 \, a d^{4} + 16 \, b d^{3} + 36 \, c d^{2}\right )}{\left (\log \left (d x\right ) \log \left (-d x + 1\right ) +{\rm Li}_2\left (-d x + 1\right )\right )} - \frac{1}{72} \,{\left (41 \, a d^{4} + 54 \, b d^{3} + 72 \, c d^{2}\right )} \log \left (x\right ) - \frac{1}{12} \,{\left (3 \, a d^{4} + 4 \, b d^{3} + 6 \, c d^{2}\right )}{\rm Li}_{3}(d x) + \frac{5 \, a d^{2} x^{2} + 28 \,{\left (a d^{3} + b d^{2}\right )} x^{3} -{\left ({\left (9 \, a d^{4} + 16 \, b d^{3} + 36 \, c d^{2}\right )} x^{4} - 36 \, c x^{2} - 16 \, b x - 9 \, a\right )} \log \left (-d x + 1\right )^{2} + 6 \,{\left (2 \,{\left (3 \, a d^{3} + 4 \, b d^{2} + 6 \, c d\right )} x^{3} + 2 \, a d x +{\left (3 \, a d^{2} + 4 \, b d\right )} x^{2} + 2 \,{\left ({\left (3 \, a d^{4} + 4 \, b d^{3} + 6 \, c d^{2}\right )} x^{4} - 6 \, c x^{2} - 4 \, b x - 3 \, a\right )} \log \left (-d x + 1\right )\right )}{\rm Li}_2\left (d x\right ) + 2 \,{\left ({\left (41 \, a d^{4} + 54 \, b d^{3} + 72 \, c d^{2}\right )} x^{4} -{\left (27 \, a d^{3} + 40 \, b d^{2} + 72 \, c d\right )} x^{3} - 5 \, a d x -{\left (9 \, a d^{2} + 14 \, b d\right )} x^{2}\right )} \log \left (-d x + 1\right )}{144 \, x^{4}} \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^5,x, algorithm="maxima")

[Out]

1/12*(3*a*d^4 + 4*b*d^3 + 6*c*d^2)*(log(d*x)*log(-d*x + 1)^2 + 2*dilog(-d*x + 1)*log(-d*x + 1) - 2*polylog(3,
-d*x + 1)) + 1/72*(9*a*d^4 + 16*b*d^3 + 36*c*d^2)*(log(d*x)*log(-d*x + 1) + dilog(-d*x + 1)) - 1/72*(41*a*d^4
+ 54*b*d^3 + 72*c*d^2)*log(x) - 1/12*(3*a*d^4 + 4*b*d^3 + 6*c*d^2)*polylog(3, d*x) + 1/144*(5*a*d^2*x^2 + 28*(
a*d^3 + b*d^2)*x^3 - ((9*a*d^4 + 16*b*d^3 + 36*c*d^2)*x^4 - 36*c*x^2 - 16*b*x - 9*a)*log(-d*x + 1)^2 + 6*(2*(3
*a*d^3 + 4*b*d^2 + 6*c*d)*x^3 + 2*a*d*x + (3*a*d^2 + 4*b*d)*x^2 + 2*((3*a*d^4 + 4*b*d^3 + 6*c*d^2)*x^4 - 6*c*x
^2 - 4*b*x - 3*a)*log(-d*x + 1))*dilog(d*x) + 2*((41*a*d^4 + 54*b*d^3 + 72*c*d^2)*x^4 - (27*a*d^3 + 40*b*d^2 +
72*c*d)*x^3 - 5*a*d*x - (9*a*d^2 + 14*b*d)*x^2)*log(-d*x + 1))/x^4

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}{\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{5}}, 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^5,x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)*dilog(d*x)*log(-d*x + 1)/x^5, 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**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}{\rm Li}_2\left (d x\right ) \log \left (-d x + 1\right )}{x^{5}}\,{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^5,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)*dilog(d*x)*log(-d*x + 1)/x^5, x)