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

Optimal. Leaf size=584 \[ -\frac{1}{12} c^3 (3 a c+4 b) \text{PolyLog}(3,c x)-\frac{1}{6} c^3 (3 a c+4 b) \text{PolyLog}(3,1-c x)+\frac{c^2 (3 a c+4 b) \text{PolyLog}(2,c x)}{12 x}+\frac{1}{12} c^3 (3 a c+4 b) \log (1-c x) \text{PolyLog}(2,c x)+\frac{1}{6} c^3 (3 a c+4 b) \log (1-c x) \text{PolyLog}(2,1-c x)+\frac{c (3 a c+4 b) \text{PolyLog}(2,c x)}{24 x^2}-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}\right ) \log (1-c x) \text{PolyLog}(2,c x)-\frac{1}{8} a c^4 \text{PolyLog}(2,c x)+\frac{a c \text{PolyLog}(2,c x)}{12 x^3}-\frac{2}{9} b c^3 \text{PolyLog}(2,c x)+\frac{c^2 (3 a c+4 b)}{48 x}+\frac{1}{12} c^3 (3 a c+4 b) \log (c x) \log ^2(1-c x)-\frac{5}{48} c^3 \log (x) (3 a c+4 b)+\frac{5}{48} c^3 (3 a c+4 b) \log (1-c x)-\frac{c^2 (3 a c+4 b) \log (1-c x)}{12 x}-\frac{c (3 a c+4 b) \log (1-c x)}{48 x^2}+\frac{5 a c^2}{144 x^2}-\frac{a c^2 \log (1-c x)}{16 x^2}+\frac{19 a c^3}{144 x}-\frac{1}{16} a c^4 \log ^2(1-c x)-\frac{37}{144} a c^4 \log (x)+\frac{37}{144} a c^4 \log (1-c x)-\frac{a c^3 \log (1-c x)}{8 x}+\frac{a \log ^2(1-c x)}{16 x^4}-\frac{5 a c \log (1-c x)}{72 x^3}+\frac{b c^2}{9 x}-\frac{1}{9} b c^3 \log ^2(1-c x)-\frac{1}{3} b c^3 \log (x)+\frac{1}{3} b c^3 \log (1-c x)-\frac{2 b c^2 \log (1-c x)}{9 x}+\frac{b \log ^2(1-c x)}{9 x^3}-\frac{b c \log (1-c x)}{9 x^2} \]

[Out]

(5*a*c^2)/(144*x^2) + (b*c^2)/(9*x) + (19*a*c^3)/(144*x) + (c^2*(4*b + 3*a*c))/(48*x) - (b*c^3*Log[x])/3 - (37
*a*c^4*Log[x])/144 - (5*c^3*(4*b + 3*a*c)*Log[x])/48 + (b*c^3*Log[1 - c*x])/3 + (37*a*c^4*Log[1 - c*x])/144 +
(5*c^3*(4*b + 3*a*c)*Log[1 - c*x])/48 - (5*a*c*Log[1 - c*x])/(72*x^3) - (b*c*Log[1 - c*x])/(9*x^2) - (a*c^2*Lo
g[1 - c*x])/(16*x^2) - (c*(4*b + 3*a*c)*Log[1 - c*x])/(48*x^2) - (2*b*c^2*Log[1 - c*x])/(9*x) - (a*c^3*Log[1 -
c*x])/(8*x) - (c^2*(4*b + 3*a*c)*Log[1 - c*x])/(12*x) - (b*c^3*Log[1 - c*x]^2)/9 - (a*c^4*Log[1 - c*x]^2)/16
+ (a*Log[1 - c*x]^2)/(16*x^4) + (b*Log[1 - c*x]^2)/(9*x^3) + (c^3*(4*b + 3*a*c)*Log[c*x]*Log[1 - c*x]^2)/12 -
(2*b*c^3*PolyLog[2, c*x])/9 - (a*c^4*PolyLog[2, c*x])/8 + (a*c*PolyLog[2, c*x])/(12*x^3) + (c*(4*b + 3*a*c)*Po
lyLog[2, c*x])/(24*x^2) + (c^2*(4*b + 3*a*c)*PolyLog[2, c*x])/(12*x) + (c^3*(4*b + 3*a*c)*Log[1 - c*x]*PolyLog
[2, c*x])/12 - (((3*a)/x^4 + (4*b)/x^3)*Log[1 - c*x]*PolyLog[2, c*x])/12 + (c^3*(4*b + 3*a*c)*Log[1 - c*x]*Pol
yLog[2, 1 - c*x])/6 - (c^3*(4*b + 3*a*c)*PolyLog[3, c*x])/12 - (c^3*(4*b + 3*a*c)*PolyLog[3, 1 - c*x])/6

________________________________________________________________________________________

Rubi [A]  time = 0.844662, antiderivative size = 584, normalized size of antiderivative = 1., number of steps used = 51, number of rules used = 19, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.905, Rules used = {6742, 6591, 2395, 44, 43, 6606, 2398, 2410, 36, 29, 31, 2391, 2390, 2301, 6589, 6596, 2396, 2433, 2374} \[ -\frac{1}{12} c^3 (3 a c+4 b) \text{PolyLog}(3,c x)-\frac{1}{6} c^3 (3 a c+4 b) \text{PolyLog}(3,1-c x)+\frac{c^2 (3 a c+4 b) \text{PolyLog}(2,c x)}{12 x}+\frac{1}{12} c^3 (3 a c+4 b) \log (1-c x) \text{PolyLog}(2,c x)+\frac{1}{6} c^3 (3 a c+4 b) \log (1-c x) \text{PolyLog}(2,1-c x)+\frac{c (3 a c+4 b) \text{PolyLog}(2,c x)}{24 x^2}-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}\right ) \log (1-c x) \text{PolyLog}(2,c x)-\frac{1}{8} a c^4 \text{PolyLog}(2,c x)+\frac{a c \text{PolyLog}(2,c x)}{12 x^3}-\frac{2}{9} b c^3 \text{PolyLog}(2,c x)+\frac{c^2 (3 a c+4 b)}{48 x}+\frac{1}{12} c^3 (3 a c+4 b) \log (c x) \log ^2(1-c x)-\frac{5}{48} c^3 \log (x) (3 a c+4 b)+\frac{5}{48} c^3 (3 a c+4 b) \log (1-c x)-\frac{c^2 (3 a c+4 b) \log (1-c x)}{12 x}-\frac{c (3 a c+4 b) \log (1-c x)}{48 x^2}+\frac{5 a c^2}{144 x^2}-\frac{a c^2 \log (1-c x)}{16 x^2}+\frac{19 a c^3}{144 x}-\frac{1}{16} a c^4 \log ^2(1-c x)-\frac{37}{144} a c^4 \log (x)+\frac{37}{144} a c^4 \log (1-c x)-\frac{a c^3 \log (1-c x)}{8 x}+\frac{a \log ^2(1-c x)}{16 x^4}-\frac{5 a c \log (1-c x)}{72 x^3}+\frac{b c^2}{9 x}-\frac{1}{9} b c^3 \log ^2(1-c x)-\frac{1}{3} b c^3 \log (x)+\frac{1}{3} b c^3 \log (1-c x)-\frac{2 b c^2 \log (1-c x)}{9 x}+\frac{b \log ^2(1-c x)}{9 x^3}-\frac{b c \log (1-c x)}{9 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a*c^2)/(144*x^2) + (b*c^2)/(9*x) + (19*a*c^3)/(144*x) + (c^2*(4*b + 3*a*c))/(48*x) - (b*c^3*Log[x])/3 - (37
*a*c^4*Log[x])/144 - (5*c^3*(4*b + 3*a*c)*Log[x])/48 + (b*c^3*Log[1 - c*x])/3 + (37*a*c^4*Log[1 - c*x])/144 +
(5*c^3*(4*b + 3*a*c)*Log[1 - c*x])/48 - (5*a*c*Log[1 - c*x])/(72*x^3) - (b*c*Log[1 - c*x])/(9*x^2) - (a*c^2*Lo
g[1 - c*x])/(16*x^2) - (c*(4*b + 3*a*c)*Log[1 - c*x])/(48*x^2) - (2*b*c^2*Log[1 - c*x])/(9*x) - (a*c^3*Log[1 -
c*x])/(8*x) - (c^2*(4*b + 3*a*c)*Log[1 - c*x])/(12*x) - (b*c^3*Log[1 - c*x]^2)/9 - (a*c^4*Log[1 - c*x]^2)/16
+ (a*Log[1 - c*x]^2)/(16*x^4) + (b*Log[1 - c*x]^2)/(9*x^3) + (c^3*(4*b + 3*a*c)*Log[c*x]*Log[1 - c*x]^2)/12 -
(2*b*c^3*PolyLog[2, c*x])/9 - (a*c^4*PolyLog[2, c*x])/8 + (a*c*PolyLog[2, c*x])/(12*x^3) + (c*(4*b + 3*a*c)*Po
lyLog[2, c*x])/(24*x^2) + (c^2*(4*b + 3*a*c)*PolyLog[2, c*x])/(12*x) + (c^3*(4*b + 3*a*c)*Log[1 - c*x]*PolyLog
[2, c*x])/12 - (((3*a)/x^4 + (4*b)/x^3)*Log[1 - c*x]*PolyLog[2, c*x])/12 + (c^3*(4*b + 3*a*c)*Log[1 - c*x]*Pol
yLog[2, 1 - c*x])/6 - (c^3*(4*b + 3*a*c)*PolyLog[3, c*x])/12 - (c^3*(4*b + 3*a*c)*PolyLog[3, 1 - c*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 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 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{(a+b x) \log (1-c x) \text{Li}_2(c x)}{x^5} \, dx &=-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}\right ) \log (1-c x) \text{Li}_2(c x)+c \int \left (-\frac{a \text{Li}_2(c x)}{4 x^4}+\frac{(-4 b-3 a c) \text{Li}_2(c x)}{12 x^3}-\frac{c (4 b+3 a c) \text{Li}_2(c x)}{12 x^2}-\frac{c^2 (4 b+3 a c) \text{Li}_2(c x)}{12 x}+\frac{c^3 (4 b+3 a c) \text{Li}_2(c x)}{12 (-1+c x)}\right ) \, dx+\int \left (-\frac{a \log ^2(1-c x)}{4 x^5}-\frac{b \log ^2(1-c x)}{3 x^4}\right ) \, dx\\ &=-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}\right ) \log (1-c x) \text{Li}_2(c x)-\frac{1}{4} a \int \frac{\log ^2(1-c x)}{x^5} \, dx-\frac{1}{3} b \int \frac{\log ^2(1-c x)}{x^4} \, dx-\frac{1}{4} (a c) \int \frac{\text{Li}_2(c x)}{x^4} \, dx-\frac{1}{12} (c (4 b+3 a c)) \int \frac{\text{Li}_2(c x)}{x^3} \, dx-\frac{1}{12} \left (c^2 (4 b+3 a c)\right ) \int \frac{\text{Li}_2(c x)}{x^2} \, dx-\frac{1}{12} \left (c^3 (4 b+3 a c)\right ) \int \frac{\text{Li}_2(c x)}{x} \, dx+\frac{1}{12} \left (c^4 (4 b+3 a c)\right ) \int \frac{\text{Li}_2(c x)}{-1+c x} \, dx\\ &=\frac{a \log ^2(1-c x)}{16 x^4}+\frac{b \log ^2(1-c x)}{9 x^3}+\frac{a c \text{Li}_2(c x)}{12 x^3}+\frac{c (4 b+3 a c) \text{Li}_2(c x)}{24 x^2}+\frac{c^2 (4 b+3 a c) \text{Li}_2(c x)}{12 x}+\frac{1}{12} c^3 (4 b+3 a c) \log (1-c x) \text{Li}_2(c x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}\right ) \log (1-c x) \text{Li}_2(c x)-\frac{1}{12} c^3 (4 b+3 a c) \text{Li}_3(c x)+\frac{1}{12} (a c) \int \frac{\log (1-c x)}{x^4} \, dx+\frac{1}{8} (a c) \int \frac{\log (1-c x)}{x^4 (1-c x)} \, dx+\frac{1}{9} (2 b c) \int \frac{\log (1-c x)}{x^3 (1-c x)} \, dx+\frac{1}{24} (c (4 b+3 a c)) \int \frac{\log (1-c x)}{x^3} \, dx+\frac{1}{12} \left (c^2 (4 b+3 a c)\right ) \int \frac{\log (1-c x)}{x^2} \, dx+\frac{1}{12} \left (c^3 (4 b+3 a c)\right ) \int \frac{\log ^2(1-c x)}{x} \, dx\\ &=-\frac{a c \log (1-c x)}{36 x^3}-\frac{c (4 b+3 a c) \log (1-c x)}{48 x^2}-\frac{c^2 (4 b+3 a c) \log (1-c x)}{12 x}+\frac{a \log ^2(1-c x)}{16 x^4}+\frac{b \log ^2(1-c x)}{9 x^3}+\frac{1}{12} c^3 (4 b+3 a c) \log (c x) \log ^2(1-c x)+\frac{a c \text{Li}_2(c x)}{12 x^3}+\frac{c (4 b+3 a c) \text{Li}_2(c x)}{24 x^2}+\frac{c^2 (4 b+3 a c) \text{Li}_2(c x)}{12 x}+\frac{1}{12} c^3 (4 b+3 a c) \log (1-c x) \text{Li}_2(c x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}\right ) \log (1-c x) \text{Li}_2(c x)-\frac{1}{12} c^3 (4 b+3 a c) \text{Li}_3(c x)+\frac{1}{8} (a c) \int \left (\frac{\log (1-c x)}{x^4}+\frac{c \log (1-c x)}{x^3}+\frac{c^2 \log (1-c x)}{x^2}+\frac{c^3 \log (1-c x)}{x}-\frac{c^4 \log (1-c x)}{-1+c x}\right ) \, dx+\frac{1}{9} (2 b c) \int \left (\frac{\log (1-c x)}{x^3}+\frac{c \log (1-c x)}{x^2}+\frac{c^2 \log (1-c x)}{x}-\frac{c^3 \log (1-c x)}{-1+c x}\right ) \, dx-\frac{1}{36} \left (a c^2\right ) \int \frac{1}{x^3 (1-c x)} \, dx-\frac{1}{48} \left (c^2 (4 b+3 a c)\right ) \int \frac{1}{x^2 (1-c x)} \, dx-\frac{1}{12} \left (c^3 (4 b+3 a c)\right ) \int \frac{1}{x (1-c x)} \, dx+\frac{1}{6} \left (c^4 (4 b+3 a c)\right ) \int \frac{\log (c x) \log (1-c x)}{1-c x} \, dx\\ &=-\frac{a c \log (1-c x)}{36 x^3}-\frac{c (4 b+3 a c) \log (1-c x)}{48 x^2}-\frac{c^2 (4 b+3 a c) \log (1-c x)}{12 x}+\frac{a \log ^2(1-c x)}{16 x^4}+\frac{b \log ^2(1-c x)}{9 x^3}+\frac{1}{12} c^3 (4 b+3 a c) \log (c x) \log ^2(1-c x)+\frac{a c \text{Li}_2(c x)}{12 x^3}+\frac{c (4 b+3 a c) \text{Li}_2(c x)}{24 x^2}+\frac{c^2 (4 b+3 a c) \text{Li}_2(c x)}{12 x}+\frac{1}{12} c^3 (4 b+3 a c) \log (1-c x) \text{Li}_2(c x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}\right ) \log (1-c x) \text{Li}_2(c x)-\frac{1}{12} c^3 (4 b+3 a c) \text{Li}_3(c x)+\frac{1}{8} (a c) \int \frac{\log (1-c x)}{x^4} \, dx+\frac{1}{9} (2 b c) \int \frac{\log (1-c x)}{x^3} \, dx-\frac{1}{36} \left (a c^2\right ) \int \left (\frac{1}{x^3}+\frac{c}{x^2}+\frac{c^2}{x}-\frac{c^3}{-1+c x}\right ) \, dx+\frac{1}{8} \left (a c^2\right ) \int \frac{\log (1-c x)}{x^3} \, dx+\frac{1}{9} \left (2 b c^2\right ) \int \frac{\log (1-c x)}{x^2} \, dx+\frac{1}{8} \left (a c^3\right ) \int \frac{\log (1-c x)}{x^2} \, dx+\frac{1}{9} \left (2 b c^3\right ) \int \frac{\log (1-c x)}{x} \, dx+\frac{1}{8} \left (a c^4\right ) \int \frac{\log (1-c x)}{x} \, dx-\frac{1}{9} \left (2 b c^4\right ) \int \frac{\log (1-c x)}{-1+c x} \, dx-\frac{1}{8} \left (a c^5\right ) \int \frac{\log (1-c x)}{-1+c x} \, dx-\frac{1}{48} \left (c^2 (4 b+3 a c)\right ) \int \left (\frac{1}{x^2}+\frac{c}{x}-\frac{c^2}{-1+c x}\right ) \, dx-\frac{1}{12} \left (c^3 (4 b+3 a c)\right ) \int \frac{1}{x} \, dx-\frac{1}{6} \left (c^3 (4 b+3 a c)\right ) \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 )-\frac{1}{12} \left (c^4 (4 b+3 a c)\right ) \int \frac{1}{1-c x} \, dx\\ &=\frac{a c^2}{72 x^2}+\frac{a c^3}{36 x}+\frac{c^2 (4 b+3 a c)}{48 x}-\frac{1}{36} a c^4 \log (x)-\frac{5}{48} c^3 (4 b+3 a c) \log (x)+\frac{1}{36} a c^4 \log (1-c x)+\frac{5}{48} c^3 (4 b+3 a c) \log (1-c x)-\frac{5 a c \log (1-c x)}{72 x^3}-\frac{b c \log (1-c x)}{9 x^2}-\frac{a c^2 \log (1-c x)}{16 x^2}-\frac{c (4 b+3 a c) \log (1-c x)}{48 x^2}-\frac{2 b c^2 \log (1-c x)}{9 x}-\frac{a c^3 \log (1-c x)}{8 x}-\frac{c^2 (4 b+3 a c) \log (1-c x)}{12 x}+\frac{a \log ^2(1-c x)}{16 x^4}+\frac{b \log ^2(1-c x)}{9 x^3}+\frac{1}{12} c^3 (4 b+3 a c) \log (c x) \log ^2(1-c x)-\frac{2}{9} b c^3 \text{Li}_2(c x)-\frac{1}{8} a c^4 \text{Li}_2(c x)+\frac{a c \text{Li}_2(c x)}{12 x^3}+\frac{c (4 b+3 a c) \text{Li}_2(c x)}{24 x^2}+\frac{c^2 (4 b+3 a c) \text{Li}_2(c x)}{12 x}+\frac{1}{12} c^3 (4 b+3 a c) \log (1-c x) \text{Li}_2(c x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}\right ) \log (1-c x) \text{Li}_2(c x)+\frac{1}{6} c^3 (4 b+3 a c) \log (1-c x) \text{Li}_2(1-c x)-\frac{1}{12} c^3 (4 b+3 a c) \text{Li}_3(c x)-\frac{1}{24} \left (a c^2\right ) \int \frac{1}{x^3 (1-c x)} \, dx-\frac{1}{9} \left (b c^2\right ) \int \frac{1}{x^2 (1-c x)} \, dx-\frac{1}{16} \left (a c^3\right ) \int \frac{1}{x^2 (1-c x)} \, dx-\frac{1}{9} \left (2 b c^3\right ) \int \frac{1}{x (1-c x)} \, dx-\frac{1}{9} \left (2 b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x\right )-\frac{1}{8} \left (a c^4\right ) \int \frac{1}{x (1-c x)} \, dx-\frac{1}{8} \left (a c^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x\right )-\frac{1}{6} \left (c^3 (4 b+3 a c)\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-c x\right )\\ &=\frac{a c^2}{72 x^2}+\frac{a c^3}{36 x}+\frac{c^2 (4 b+3 a c)}{48 x}-\frac{1}{36} a c^4 \log (x)-\frac{5}{48} c^3 (4 b+3 a c) \log (x)+\frac{1}{36} a c^4 \log (1-c x)+\frac{5}{48} c^3 (4 b+3 a c) \log (1-c x)-\frac{5 a c \log (1-c x)}{72 x^3}-\frac{b c \log (1-c x)}{9 x^2}-\frac{a c^2 \log (1-c x)}{16 x^2}-\frac{c (4 b+3 a c) \log (1-c x)}{48 x^2}-\frac{2 b c^2 \log (1-c x)}{9 x}-\frac{a c^3 \log (1-c x)}{8 x}-\frac{c^2 (4 b+3 a c) \log (1-c x)}{12 x}-\frac{1}{9} b c^3 \log ^2(1-c x)-\frac{1}{16} a c^4 \log ^2(1-c x)+\frac{a \log ^2(1-c x)}{16 x^4}+\frac{b \log ^2(1-c x)}{9 x^3}+\frac{1}{12} c^3 (4 b+3 a c) \log (c x) \log ^2(1-c x)-\frac{2}{9} b c^3 \text{Li}_2(c x)-\frac{1}{8} a c^4 \text{Li}_2(c x)+\frac{a c \text{Li}_2(c x)}{12 x^3}+\frac{c (4 b+3 a c) \text{Li}_2(c x)}{24 x^2}+\frac{c^2 (4 b+3 a c) \text{Li}_2(c x)}{12 x}+\frac{1}{12} c^3 (4 b+3 a c) \log (1-c x) \text{Li}_2(c x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}\right ) \log (1-c x) \text{Li}_2(c x)+\frac{1}{6} c^3 (4 b+3 a c) \log (1-c x) \text{Li}_2(1-c x)-\frac{1}{12} c^3 (4 b+3 a c) \text{Li}_3(c x)-\frac{1}{6} c^3 (4 b+3 a c) \text{Li}_3(1-c x)-\frac{1}{24} \left (a c^2\right ) \int \left (\frac{1}{x^3}+\frac{c}{x^2}+\frac{c^2}{x}-\frac{c^3}{-1+c x}\right ) \, dx-\frac{1}{9} \left (b c^2\right ) \int \left (\frac{1}{x^2}+\frac{c}{x}-\frac{c^2}{-1+c x}\right ) \, dx-\frac{1}{16} \left (a c^3\right ) \int \left (\frac{1}{x^2}+\frac{c}{x}-\frac{c^2}{-1+c x}\right ) \, dx-\frac{1}{9} \left (2 b c^3\right ) \int \frac{1}{x} \, dx-\frac{1}{8} \left (a c^4\right ) \int \frac{1}{x} \, dx-\frac{1}{9} \left (2 b c^4\right ) \int \frac{1}{1-c x} \, dx-\frac{1}{8} \left (a c^5\right ) \int \frac{1}{1-c x} \, dx\\ &=\frac{5 a c^2}{144 x^2}+\frac{b c^2}{9 x}+\frac{19 a c^3}{144 x}+\frac{c^2 (4 b+3 a c)}{48 x}-\frac{1}{3} b c^3 \log (x)-\frac{37}{144} a c^4 \log (x)-\frac{5}{48} c^3 (4 b+3 a c) \log (x)+\frac{1}{3} b c^3 \log (1-c x)+\frac{37}{144} a c^4 \log (1-c x)+\frac{5}{48} c^3 (4 b+3 a c) \log (1-c x)-\frac{5 a c \log (1-c x)}{72 x^3}-\frac{b c \log (1-c x)}{9 x^2}-\frac{a c^2 \log (1-c x)}{16 x^2}-\frac{c (4 b+3 a c) \log (1-c x)}{48 x^2}-\frac{2 b c^2 \log (1-c x)}{9 x}-\frac{a c^3 \log (1-c x)}{8 x}-\frac{c^2 (4 b+3 a c) \log (1-c x)}{12 x}-\frac{1}{9} b c^3 \log ^2(1-c x)-\frac{1}{16} a c^4 \log ^2(1-c x)+\frac{a \log ^2(1-c x)}{16 x^4}+\frac{b \log ^2(1-c x)}{9 x^3}+\frac{1}{12} c^3 (4 b+3 a c) \log (c x) \log ^2(1-c x)-\frac{2}{9} b c^3 \text{Li}_2(c x)-\frac{1}{8} a c^4 \text{Li}_2(c x)+\frac{a c \text{Li}_2(c x)}{12 x^3}+\frac{c (4 b+3 a c) \text{Li}_2(c x)}{24 x^2}+\frac{c^2 (4 b+3 a c) \text{Li}_2(c x)}{12 x}+\frac{1}{12} c^3 (4 b+3 a c) \log (1-c x) \text{Li}_2(c x)-\frac{1}{12} \left (\frac{3 a}{x^4}+\frac{4 b}{x^3}\right ) \log (1-c x) \text{Li}_2(c x)+\frac{1}{6} c^3 (4 b+3 a c) \log (1-c x) \text{Li}_2(1-c x)-\frac{1}{12} c^3 (4 b+3 a c) \text{Li}_3(c x)-\frac{1}{6} c^3 (4 b+3 a c) \text{Li}_3(1-c x)\\ \end{align*}

Mathematica [A]  time = 1.51805, size = 505, normalized size = 0.86 \[ -\frac{-2 c^3 x^4 \text{PolyLog}(2,1-c x) (12 (3 a c+4 b) \log (1-c x)+9 a c+16 b)-6 \text{PolyLog}(2,c x) \left (c x \left (a \left (6 c^2 x^2+3 c x+2\right )+4 b x (2 c x+1)\right )+\log (1-c x) \left (6 a \left (c^4 x^4-1\right )+8 b x \left (c^3 x^3-1\right )\right )\right )+36 a c^4 x^4 \text{PolyLog}(3,c x)+72 a c^4 x^4 \text{PolyLog}(3,1-c x)+48 b c^3 x^4 \text{PolyLog}(3,c x)+96 b c^3 x^4 \text{PolyLog}(3,1-c x)+33 a c^4 x^4-28 a c^3 x^3-5 a c^2 x^2+9 a c^4 x^4 \log ^2(1-c x)-36 a c^4 x^4 \log (c x) \log ^2(1-c x)+82 a c^4 x^4 \log (c x)-82 a c^4 x^4 \log (1-c x)-18 a c^4 x^4 \log (c x) \log (1-c x)+54 a c^3 x^3 \log (1-c x)+18 a c^2 x^2 \log (1-c x)-9 a \log ^2(1-c x)+10 a c x \log (1-c x)+28 b c^3 x^4-28 b c^2 x^3+16 b c^3 x^4 \log ^2(1-c x)-48 b c^3 x^4 \log (c x) \log ^2(1-c x)+108 b c^3 x^4 \log (c x)-108 b c^3 x^4 \log (1-c x)-32 b c^3 x^4 \log (c x) \log (1-c x)+80 b c^2 x^3 \log (1-c x)+28 b c x^2 \log (1-c x)-16 b x \log ^2(1-c x)}{144 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-5*a*c^2*x^2 - 28*b*c^2*x^3 - 28*a*c^3*x^3 + 28*b*c^3*x^4 + 33*a*c^4*x^4 + 108*b*c^3*x^4*Log[c*x] + 82*a*c^4
*x^4*Log[c*x] + 10*a*c*x*Log[1 - c*x] + 28*b*c*x^2*Log[1 - c*x] + 18*a*c^2*x^2*Log[1 - c*x] + 80*b*c^2*x^3*Log
[1 - c*x] + 54*a*c^3*x^3*Log[1 - c*x] - 108*b*c^3*x^4*Log[1 - c*x] - 82*a*c^4*x^4*Log[1 - c*x] - 32*b*c^3*x^4*
Log[c*x]*Log[1 - c*x] - 18*a*c^4*x^4*Log[c*x]*Log[1 - c*x] - 9*a*Log[1 - c*x]^2 - 16*b*x*Log[1 - c*x]^2 + 16*b
*c^3*x^4*Log[1 - c*x]^2 + 9*a*c^4*x^4*Log[1 - c*x]^2 - 48*b*c^3*x^4*Log[c*x]*Log[1 - c*x]^2 - 36*a*c^4*x^4*Log
[c*x]*Log[1 - c*x]^2 - 6*(c*x*(4*b*x*(1 + 2*c*x) + a*(2 + 3*c*x + 6*c^2*x^2)) + (8*b*x*(-1 + c^3*x^3) + 6*a*(-
1 + c^4*x^4))*Log[1 - c*x])*PolyLog[2, c*x] - 2*c^3*x^4*(16*b + 9*a*c + 12*(4*b + 3*a*c)*Log[1 - c*x])*PolyLog
[2, 1 - c*x] + 48*b*c^3*x^4*PolyLog[3, c*x] + 36*a*c^4*x^4*PolyLog[3, c*x] + 96*b*c^3*x^4*PolyLog[3, 1 - c*x]
+ 72*a*c^4*x^4*PolyLog[3, 1 - c*x])/(144*x^4)

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Maple [F]  time = 0.007, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) \ln \left ( -cx+1 \right ){\it polylog} \left ( 2,cx \right ) }{{x}^{5}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.18925, size = 460, normalized size = 0.79 \begin{align*} \frac{1}{12} \,{\left (3 \, a c^{4} + 4 \, b c^{3}\right )}{\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 )} + \frac{1}{72} \,{\left (9 \, a c^{4} + 16 \, b c^{3}\right )}{\left (\log \left (c x\right ) \log \left (-c x + 1\right ) +{\rm Li}_2\left (-c x + 1\right )\right )} - \frac{1}{72} \,{\left (41 \, a c^{4} + 54 \, b c^{3}\right )} \log \left (x\right ) - \frac{1}{12} \,{\left (3 \, a c^{4} + 4 \, b c^{3}\right )}{\rm Li}_{3}(c x) + \frac{5 \, a c^{2} x^{2} + 28 \,{\left (a c^{3} + b c^{2}\right )} x^{3} -{\left ({\left (9 \, a c^{4} + 16 \, b c^{3}\right )} x^{4} - 16 \, b x - 9 \, a\right )} \log \left (-c x + 1\right )^{2} + 6 \,{\left (2 \,{\left (3 \, a c^{3} + 4 \, b c^{2}\right )} x^{3} + 2 \, a c x +{\left (3 \, a c^{2} + 4 \, b c\right )} x^{2} + 2 \,{\left ({\left (3 \, a c^{4} + 4 \, b c^{3}\right )} x^{4} - 4 \, b x - 3 \, a\right )} \log \left (-c x + 1\right )\right )}{\rm Li}_2\left (c x\right ) + 2 \,{\left ({\left (41 \, a c^{4} + 54 \, b c^{3}\right )} x^{4} -{\left (27 \, a c^{3} + 40 \, b c^{2}\right )} x^{3} - 5 \, a c x -{\left (9 \, a c^{2} + 14 \, b c\right )} x^{2}\right )} \log \left (-c x + 1\right )}{144 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*log(-c*x+1)*polylog(2,c*x)/x^5,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*log(-c*x+1)*polylog(2,c*x)/x^5,x, algorithm="fricas")

[Out]

integral((b*x + a)*dilog(c*x)*log(-c*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((b*x+a)*ln(-c*x+1)*polylog(2,c*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 (b x + a\right )}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*log(-c*x+1)*polylog(2,c*x)/x^5,x, algorithm="giac")

[Out]

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