3.2.91 \(\int \frac {(a+b x) \log (1-c x) \text {PolyLog}(2,c x)}{x^5} \, dx\) [191]

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

[Out]

1/12*a*c*polylog(2,c*x)/x^3+1/24*c*(3*a*c+4*b)*polylog(2,c*x)/x^2+1/12*c^2*(3*a*c+4*b)*polylog(2,c*x)/x-5/72*a
*c*ln(-c*x+1)/x^3-1/9*b*c*ln(-c*x+1)/x^2-1/16*a*c^2*ln(-c*x+1)/x^2-1/48*c*(3*a*c+4*b)*ln(-c*x+1)/x^2-2/9*b*c^2
*ln(-c*x+1)/x-1/8*a*c^3*ln(-c*x+1)/x-1/12*c^2*(3*a*c+4*b)*ln(-c*x+1)/x+1/12*c^3*(3*a*c+4*b)*ln(c*x)*ln(-c*x+1)
^2+1/12*c^3*(3*a*c+4*b)*ln(-c*x+1)*polylog(2,c*x)+1/6*c^3*(3*a*c+4*b)*ln(-c*x+1)*polylog(2,-c*x+1)-2/9*b*c^3*p
olylog(2,c*x)-1/8*a*c^4*polylog(2,c*x)-1/12*c^3*(3*a*c+4*b)*polylog(3,c*x)-1/6*c^3*(3*a*c+4*b)*polylog(3,-c*x+
1)+5/144*a*c^2/x^2+1/9*b*c^2/x+19/144*a*c^3/x+1/48*c^2*(3*a*c+4*b)/x-1/9*b*c^3*ln(-c*x+1)^2-1/16*a*c^4*ln(-c*x
+1)^2+1/16*a*ln(-c*x+1)^2/x^4+1/9*b*ln(-c*x+1)^2/x^3-1/12*(3*a/x^4+4*b/x^3)*ln(-c*x+1)*polylog(2,c*x)-1/3*b*c^
3*ln(x)-37/144*a*c^4*ln(x)-5/48*c^3*(3*a*c+4*b)*ln(x)+1/3*b*c^3*ln(-c*x+1)+37/144*a*c^4*ln(-c*x+1)+5/48*c^3*(3
*a*c+4*b)*ln(-c*x+1)

________________________________________________________________________________________

Rubi [A]
time = 0.55, antiderivative size = 584, normalized size of antiderivative = 1.00, 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 = {6874, 6726, 2442, 46, 45, 6741, 2445, 2457, 36, 29, 31, 2438, 2437, 2338, 6724, 6731, 2443, 2481, 2421} \begin {gather*} -\frac {1}{12} c^3 (3 a c+4 b) \text {Li}_3(c x)-\frac {1}{6} c^3 (3 a c+4 b) \text {Li}_3(1-c x)+\frac {1}{12} c^3 (3 a c+4 b) \text {Li}_2(c x) \log (1-c x)+\frac {1}{6} c^3 (3 a c+4 b) \text {Li}_2(1-c x) \log (1-c 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) \text {Li}_2(c x)}{12 x}+\frac {c^2 (3 a c+4 b)}{48 x}-\frac {c^2 (3 a c+4 b) \log (1-c x)}{12 x}+\frac {c (3 a c+4 b) \text {Li}_2(c x)}{24 x^2}-\frac {1}{12} \left (\frac {3 a}{x^4}+\frac {4 b}{x^3}\right ) \text {Li}_2(c x) \log (1-c x)-\frac {c (3 a c+4 b) \log (1-c x)}{48 x^2}-\frac {1}{8} a c^4 \text {Li}_2(c 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 {19 a c^3}{144 x}-\frac {a c^3 \log (1-c x)}{8 x}+\frac {5 a c^2}{144 x^2}-\frac {a c^2 \log (1-c x)}{16 x^2}+\frac {a c \text {Li}_2(c x)}{12 x^3}+\frac {a \log ^2(1-c x)}{16 x^4}-\frac {5 a c \log (1-c x)}{72 x^3}-\frac {2}{9} b c^3 \text {Li}_2(c 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 {b c^2}{9 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} \end {gather*}

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 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 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 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 46

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] && Lt
Q[m + n + 2, 0])

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 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 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 2438

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 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 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 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 6741

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 6874

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

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 1.06, size = 505, normalized size = 0.86 \begin {gather*} -\frac {-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 ^2(1-c x)-16 b x \log ^2(1-c x)+16 b c^3 x^4 \log ^2(1-c x)+9 a c^4 x^4 \log ^2(1-c x)-48 b c^3 x^4 \log (c x) \log ^2(1-c x)-36 a c^4 x^4 \log (c x) \log ^2(1-c x)-6 \left (c x \left (4 b x (1+2 c x)+a \left (2+3 c x+6 c^2 x^2\right )\right )+\left (8 b x \left (-1+c^3 x^3\right )+6 a \left (-1+c^4 x^4\right )\right ) \log (1-c x)\right ) \text {PolyLog}(2,c x)-2 c^3 x^4 (16 b+9 a c+12 (4 b+3 a c) \log (1-c x)) \text {PolyLog}(2,1-c x)+48 b c^3 x^4 \text {PolyLog}(3,c x)+36 a c^4 x^4 \text {PolyLog}(3,c x)+96 b c^3 x^4 \text {PolyLog}(3,1-c x)+72 a c^4 x^4 \text {PolyLog}(3,1-c x)}{144 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/144*(-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])*P
olyLog[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])/x^4

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right ) \ln \left (-c x +1\right ) \polylog \left (2, c x \right )}{x^{5}}\, dx\]

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 = 0.31, size = 341, normalized size = 0.58 \begin {gather*} \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 {gather*}

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 = 0.00, 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((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)] 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((b*x+a)*ln(-c*x+1)*polylog(2,c*x)/x**5,x)

[Out]

Timed out

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Giac [F]
time = 0.00, 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((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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (1-c\,x\right )\,\mathrm {polylog}\left (2,c\,x\right )\,\left (a+b\,x\right )}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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