3.2.81 \(\int \frac {(g+h \log (f (d+e x)^n)) \text {PolyLog}(2,c (a+b x))}{x^2} \, dx\) [181]

Optimal. Leaf size=2498 \[ \text {result too large to display} \]

[Out]

-b*g*polylog(2,c*(b*x+a))/a-b*g*polylog(2,1-b*c*x/(-a*c+1))/a-(g+h*ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x+b*h
*ln(b*c*x/(-a*c+1))*ln(-b*c*x-a*c+1)*(n*ln(e*x+d)-ln(f*(e*x+d)^n))/a+e*h*n*(ln(1-c*(b*x+a))-ln(-a*(1-c*(b*x+a)
)/b/x))*polylog(2,-b*x/a)/d+e*h*n*ln(x)*polylog(2,c*(b*x+a))/d-e*h*n*ln(e*x+d)*polylog(2,c*(b*x+a))/d-b*h*n*(l
n(e*x+d)-ln((-a*c+1)*(e*x+d)/d/(-b*c*x-a*c+1)))*polylog(2,1-b*c*x/(-a*c+1))/a-b*h*n*ln((-a*c+1)*(e*x+d)/d/(-b*
c*x-a*c+1))*polylog(2,d*(-b*c*x-a*c+1)/(-a*c+1)/(e*x+d))/a+b*h*n*ln((-a*c+1)*(e*x+d)/d/(-b*c*x-a*c+1))*polylog
(2,-e*(-b*c*x-a*c+1)/b/c/(e*x+d))/a+b*h*n*(ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))+ln(1-c*(b*x+a)))*polylog(2,b
*(e*x+d)/(-a*e+b*d))/a-e*h*n*(ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))+ln(1-c*(b*x+a)))*polylog(2,b*(e*x+d)/(-a*
e+b*d))/d-b*h*n*(ln(-b*c*x-a*c+1)+ln((-a*c+1)*(e*x+d)/d/(-b*c*x-a*c+1)))*polylog(2,1+e*x/d)/a+e*h*n*ln(-a*(1-c
*(b*x+a))/b/x)*polylog(2,-b*x/a/(1-c*(b*x+a)))/d-e*h*n*ln(-a*(1-c*(b*x+a))/b/x)*polylog(2,-b*c*x/(1-c*(b*x+a))
)/d+b*h*n*(ln(e*x+d)-ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a))))*polylog(2,1-c*(b*x+a))/a-e*h*n*(ln(e*x+d)-ln(b*(e
*x+d)/(-a*e+b*d)/(1-c*(b*x+a))))*polylog(2,1-c*(b*x+a))/d+e*h*n*(ln(x)+ln(-a*(1-c*(b*x+a))/b/x))*polylog(2,1-c
*(b*x+a))/d-b*h*n*ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))*polylog(2,-e*(1-c*(b*x+a))/b/c/(e*x+d))/a+e*h*n*ln(b*
(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))*polylog(2,-e*(1-c*(b*x+a))/b/c/(e*x+d))/d+b*h*n*ln(b*(e*x+d)/(-a*e+b*d)/(1-c
*(b*x+a)))*polylog(2,(-a*e+b*d)*(1-c*(b*x+a))/b/(e*x+d))/a-e*h*n*ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))*polylo
g(2,(-a*e+b*d)*(1-c*(b*x+a))/b/(e*x+d))/d-1/2*b*h*n*(ln(b*c*x/(-a*c+1))+ln((-a*c*e+b*c*d+e)/b/c/(e*x+d))-ln((-
a*c*e+b*c*d+e)*x/(-a*c+1)/(e*x+d)))*ln((-a*c+1)*(e*x+d)/d/(-b*c*x-a*c+1))^2/a+1/2*b*h*n*(ln(b*c*x/(-a*c+1))-ln
(-e*x/d))*(ln(-b*c*x-a*c+1)+ln((-a*c+1)*(e*x+d)/d/(-b*c*x-a*c+1)))^2/a+1/2*b*h*n*(ln(c*(b*x+a))+ln((-a*c*e+b*c
*d+e)/b/c/(e*x+d))-ln((-a*c*e+b*c*d+e)*(b*x+a)/b/(e*x+d)))*ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))^2/a-1/2*e*h*
n*(ln(c*(b*x+a))+ln((-a*c*e+b*c*d+e)/b/c/(e*x+d))-ln((-a*c*e+b*c*d+e)*(b*x+a)/b/(e*x+d)))*ln(b*(e*x+d)/(-a*e+b
*d)/(1-c*(b*x+a)))^2/d-1/2*b*h*n*(ln(c*(b*x+a))-ln(-e*(b*x+a)/(-a*e+b*d)))*(ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+
a)))+ln(1-c*(b*x+a)))^2/a+1/2*e*h*n*(ln(c*(b*x+a))-ln(-e*(b*x+a)/(-a*e+b*d)))*(ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b
*x+a)))+ln(1-c*(b*x+a)))^2/d+1/2*e*h*n*(ln(1+b*x/a)+ln((-a*c+1)/(1-c*(b*x+a)))-ln((-a*c+1)*(b*x+a)/a/(1-c*(b*x
+a))))*ln(-a*(1-c*(b*x+a))/b/x)^2/d+1/2*e*h*n*(ln(c*(b*x+a))-ln(1+b*x/a))*(ln(x)+ln(-a*(1-c*(b*x+a))/b/x))^2/d
-b*h*n*polylog(3,1-c*(b*x+a))/a-b*h*n*polylog(3,-e*(1-c*(b*x+a))/b/c/(e*x+d))/a+e*h*n*polylog(3,-e*(1-c*(b*x+a
))/b/c/(e*x+d))/d+b*h*n*polylog(3,(-a*e+b*d)*(1-c*(b*x+a))/b/(e*x+d))/a-e*h*n*polylog(3,(-a*e+b*d)*(1-c*(b*x+a
))/b/(e*x+d))/d-b*g*ln(b*c*x/(-a*c+1))*ln(-b*c*x-a*c+1)/a+b*h*(n*ln(e*x+d)-ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a
))/a+b*h*(n*ln(e*x+d)-ln(f*(e*x+d)^n))*polylog(2,1-b*c*x/(-a*c+1))/a+b*h*n*polylog(3,1-b*c*x/(-a*c+1))/a-b*h*n
*polylog(3,d*(-b*c*x-a*c+1)/(-a*c+1)/(e*x+d))/a+b*h*n*polylog(3,-e*(-b*c*x-a*c+1)/b/c/(e*x+d))/a-b*h*n*polylog
(3,b*(e*x+d)/(-a*e+b*d))/a+e*h*n*polylog(3,b*(e*x+d)/(-a*e+b*d))/d+e*h*n*polylog(3,-b*x/a/(1-c*(b*x+a)))/d-e*h
*n*polylog(3,-b*c*x/(1-c*(b*x+a)))/d-e*h*n*polylog(3,-b*x/a)/d+b*h*n*polylog(3,1+e*x/d)/a-b*h*n*ln(b*c*x/(-a*c
+1))*ln(-b*c*x-a*c+1)*ln(e*x+d)/a+e*h*n*ln(x)*ln(1+b*x/a)*ln(1-c*(b*x+a))/d+b*h*n*ln(c*(b*x+a))*ln(e*x+d)*ln(1
-c*(b*x+a))/a-e*h*n*ln(c*(b*x+a))*ln(e*x+d)*ln(1-c*(b*x+a))/d

________________________________________________________________________________________

Rubi [A]
time = 1.78, antiderivative size = 2498, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6738, 2488, 2441, 2352, 2487, 2485, 2490, 2438, 6732} \begin {gather*} \text {Too large to display} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/x^2,x]

[Out]

-((b*g*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/a) - (b*h*n*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x]*Lo
g[d + e*x])/a - (b*h*n*(Log[(b*c*x)/(1 - a*c)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e -
a*c*e)*x)/((1 - a*c)*(d + e*x))])*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]^2)/(2*a) + (b*h*n*(Log[(b*c
*x)/(1 - a*c)] - Log[-((e*x)/d)])*(Log[1 - a*c - b*c*x] + Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])^2)
/(2*a) + (b*h*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x]*(n*Log[d + e*x] - Log[f*(d + e*x)^n]))/a + (b*h*n*(L
og[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))
])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(2*a) - (e*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a
*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1
- c*(a + b*x)))]^2)/(2*d) + (e*h*n*Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)])/d + (b*h*n*Log[c*(a + b*x)]*
Log[d + e*x]*Log[1 - c*(a + b*x)])/a - (e*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/d - (b*h*n*(
Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Lo
g[1 - c*(a + b*x)])^2)/(2*a) + (e*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x)
)/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(2*d) + (e*h*n*(Log[1 + (b*x)/a] + Log[(1 - a*c)
/(1 - c*(a + b*x))] - Log[((1 - a*c)*(a + b*x))/(a*(1 - c*(a + b*x)))])*Log[-((a*(1 - c*(a + b*x)))/(b*x))]^2)
/(2*d) + (e*h*n*(Log[c*(a + b*x)] - Log[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2)/(2*d)
+ (e*h*n*(Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, -((b*x)/a)])/d - (b*g*PolyLog
[2, c*(a + b*x)])/a + (e*h*n*Log[x]*PolyLog[2, c*(a + b*x)])/d - (e*h*n*Log[d + e*x]*PolyLog[2, c*(a + b*x)])/
d + (b*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/a - ((g + h*Log[f*(d + e*x)^n])*PolyLo
g[2, c*(a + b*x)])/x - (b*g*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/a - (b*h*n*(Log[d + e*x] - Log[((1 - a*c)*(d +
e*x))/(d*(1 - a*c - b*c*x))])*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/a + (b*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n]
)*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/a - (b*h*n*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]*PolyLog[2, (d
*(1 - a*c - b*c*x))/((1 - a*c)*(d + e*x))])/a + (b*h*n*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]*PolyLo
g[2, -((e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/a + (b*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]
+ Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/a - (e*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1
- c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/d - (b*h*n*(Log[1 - a*c - b*c*
x] + Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])*PolyLog[2, 1 + (e*x)/d])/a + (e*h*n*Log[-((a*(1 - c*(a
+ b*x)))/(b*x))]*PolyLog[2, -((b*x)/(a*(1 - c*(a + b*x))))])/d - (e*h*n*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*Po
lyLog[2, -((b*c*x)/(1 - c*(a + b*x)))])/d + (b*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a +
b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/a - (e*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*
x)))])*PolyLog[2, 1 - c*(a + b*x)])/d + (e*h*n*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, 1 - c
*(a + b*x)])/d - (b*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/
(b*c*(d + e*x)))])/a + (e*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b
*x)))/(b*c*(d + e*x)))])/d + (b*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)
*(1 - c*(a + b*x)))/(b*(d + e*x))])/a - (e*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, (
(b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/d - (e*h*n*PolyLog[3, -((b*x)/a)])/d + (b*h*n*PolyLog[3, 1 - (b
*c*x)/(1 - a*c)])/a - (b*h*n*PolyLog[3, (d*(1 - a*c - b*c*x))/((1 - a*c)*(d + e*x))])/a + (b*h*n*PolyLog[3, -(
(e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/a - (b*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/a + (e*h*n*PolyLog[
3, (b*(d + e*x))/(b*d - a*e)])/d + (b*h*n*PolyLog[3, 1 + (e*x)/d])/a + (e*h*n*PolyLog[3, -((b*x)/(a*(1 - c*(a
+ b*x))))])/d - (e*h*n*PolyLog[3, -((b*c*x)/(1 - c*(a + b*x)))])/d - (b*h*n*PolyLog[3, 1 - c*(a + b*x)])/a - (
b*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/a + (e*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c
*(d + e*x)))])/d + (b*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/a - (e*h*n*PolyLog[3, ((b
*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/d

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 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 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2485

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp[Log[(-b)*(x/a)]*Log[a + b*x]*Log[c
+ d*x], x] + (Simp[(1/2)*(Log[(-b)*(x/a)] - Log[(-(b*c - a*d))*(x/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d
*x))])*Log[a*((c + d*x)/(c*(a + b*x)))]^2, x] - Simp[(1/2)*(Log[(-b)*(x/a)] - Log[(-d)*(x/c)])*(Log[a + b*x] +
Log[a*((c + d*x)/(c*(a + b*x)))])^2, x] + Simp[(Log[c + d*x] - Log[a*((c + d*x)/(c*(a + b*x)))])*PolyLog[2, 1
+ b*(x/a)], x] + Simp[(Log[a + b*x] + Log[a*((c + d*x)/(c*(a + b*x)))])*PolyLog[2, 1 + d*(x/c)], x] + Simp[Lo
g[a*((c + d*x)/(c*(a + b*x)))]*PolyLog[2, c*((a + b*x)/(a*(c + d*x)))], x] - Simp[Log[a*((c + d*x)/(c*(a + b*x
)))]*PolyLog[2, d*((a + b*x)/(b*(c + d*x)))], x] - Simp[PolyLog[3, 1 + b*(x/a)], x] - Simp[PolyLog[3, 1 + d*(x
/c)], x] + Simp[PolyLog[3, c*((a + b*x)/(a*(c + d*x)))], x] - Simp[PolyLog[3, d*((a + b*x)/(b*(c + d*x)))], x]
) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2487

Int[(Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)])/(x_), x_Symbol] :> Dist[m, In
t[Log[i + j*x]*(Log[c*(d + e*x)^n]/x), x], x] - Dist[m*Log[i + j*x] - Log[h*(i + j*x)^m], Int[Log[c*(d + e*x)^
n]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]

Rule 2488

Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.) + (f_))
)/(x_), x_Symbol] :> Dist[f, Int[(a + b*Log[c*(d + e*x)^n])/x, x], x] + Dist[g, Int[Log[h*(i + j*x)^m]*((a + b
*Log[c*(d + e*x)^n])/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0]

Rule 2490

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))
*((k_) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/l, Subst[Int[x^r*(a + b*Log[c*(-(e*k - d*l)/l + e*(x/l))^n])*(
f + g*Log[h*(-(j*k - i*l)/l + j*(x/l))^m]), x], x, k + l*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l,
m, n}, x] && IntegerQ[r]

Rule 6732

Int[PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[Log[d + e*x]*(PolyLog[2, c*
(a + b*x)]/e), x] + Dist[b/e, Int[Log[d + e*x]*(Log[1 - a*c - b*c*x]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d,
e}, x] && NeQ[c*(b*d - a*e) + e, 0]

Rule 6738

Int[((g_.) + Log[(f_.)*((d_.) + (e_.)*(x_))^(n_.)]*(h_.))*(x_)^(m_.)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x
_Symbol] :> Simp[x^(m + 1)*(g + h*Log[f*(d + e*x)^n])*(PolyLog[2, c*(a + b*x)]/(m + 1)), x] + (Dist[b/(m + 1),
Int[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], x^(m + 1)/(a + b*x), x], x], x] - Dist[e
*h*(n/(m + 1)), Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], x^(m + 1)/(d + e*x), x], x], x]) /; FreeQ[{a, b,
c, d, e, f, g, h, n}, x] && IntegerQ[m] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x^2} \, dx &=-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}-b \int \left (\frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a x}-\frac {b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a (a+b x)}\right ) \, dx+(e h n) \int \left (\frac {\text {Li}_2(c (a+b x))}{d x}-\frac {e \text {Li}_2(c (a+b x))}{d (d+e x)}\right ) \, dx\\ &=-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}-\frac {b \int \frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{x} \, dx}{a}+\frac {b^2 \int \frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a+b x} \, dx}{a}+\frac {(e h n) \int \frac {\text {Li}_2(c (a+b x))}{x} \, dx}{d}-\frac {\left (e^2 h n\right ) \int \frac {\text {Li}_2(c (a+b x))}{d+e x} \, dx}{d}\\ &=\frac {e h n \log (x) \text {Li}_2(c (a+b x))}{d}-\frac {e h n \log (d+e x) \text {Li}_2(c (a+b x))}{d}-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}+\frac {b \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \left (g+h \log \left (f \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )^n\right )\right )}{x} \, dx,x,a+b x\right )}{a}-\frac {(b g) \int \frac {\log (1-a c-b c x)}{x} \, dx}{a}-\frac {(b h) \int \frac {\log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{x} \, dx}{a}+\frac {(b e h n) \int \frac {\log (x) \log (1-a c-b c x)}{a+b x} \, dx}{d}-\frac {(b e h n) \int \frac {\log (1-a c-b c x) \log (d+e x)}{a+b x} \, dx}{d}\\ &=-\frac {b g \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{a}+\frac {e h n \log (x) \text {Li}_2(c (a+b x))}{d}-\frac {e h n \log (d+e x) \text {Li}_2(c (a+b x))}{d}-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}+\frac {(b g) \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )}{a}-\frac {\left (b^2 c g\right ) \int \frac {\log \left (-\frac {b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{a}+\frac {(b h) \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \log \left (f \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )^n\right )}{x} \, dx,x,a+b x\right )}{a}-\frac {(b h n) \int \frac {\log (1-a c-b c x) \log (d+e x)}{x} \, dx}{a}+\frac {(e h n) \text {Subst}\left (\int \frac {\log \left (-\frac {a}{b}+\frac {x}{b}\right ) \log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )}{d}-\frac {(e h n) \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \log \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )}{x} \, dx,x,a+b x\right )}{d}+\frac {\left (b h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\right ) \int \frac {\log (1-a c-b c x)}{x} \, dx}{a}\\ &=-\frac {b g \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{a}-\frac {b h n \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x) \log (d+e x)}{a}-\frac {b h n \left (\log \left (\frac {b c x}{1-a c}\right )+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) x}{(1-a c) (d+e x)}\right )\right ) \log ^2\left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )}{2 a}+\frac {b h n \left (\log \left (\frac {b c x}{1-a c}\right )-\log \left (-\frac {e x}{d}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right )^2}{2 a}+\frac {b h \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x) \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )}{a}-\frac {e h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 d}+\frac {e h n \log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))}{d}-\frac {e h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{d}+\frac {e h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 d}+\frac {e h n \left (\log \left (1+\frac {b x}{a}\right )+\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )}{2 d}+\frac {e h n \left (\log (c (a+b x))-\log \left (1+\frac {b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )^2}{2 d}+\frac {e h n \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2\left (-\frac {b x}{a}\right )}{d}-\frac {b g \text {Li}_2(c (a+b x))}{a}+\frac {e h n \log (x) \text {Li}_2(c (a+b x))}{d}-\frac {e h n \log (d+e x) \text {Li}_2(c (a+b x))}{d}-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}-\frac {b g \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \left (\log (d+e x)-\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text {Li}_2\left (\frac {d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{a}+\frac {b h n \log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text {Li}_2\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{a}-\frac {e h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {b h n \left (\log (1-a c-b c x)+\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{a}+\frac {e h n \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {Li}_2\left (-\frac {b x}{a (1-c (a+b x))}\right )}{d}-\frac {e h n \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {Li}_2\left (-\frac {b c x}{1-c (a+b x)}\right )}{d}-\frac {e h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{d}+\frac {e h n \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2(1-c (a+b x))}{d}+\frac {e h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{d}-\frac {e h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{d}-\frac {e h n \text {Li}_3\left (-\frac {b x}{a}\right )}{d}+\frac {b h n \text {Li}_3\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \text {Li}_3\left (\frac {d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{a}+\frac {b h n \text {Li}_3\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{a}+\frac {e h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {b h n \text {Li}_3\left (1+\frac {e x}{d}\right )}{a}+\frac {e h n \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )}{d}-\frac {e h n \text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )}{d}+\frac {e h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{d}-\frac {e h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{d}+\frac {(b h n) \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \log \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )}{x} \, dx,x,a+b x\right )}{a}-\frac {\left (b h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )}{a}+\frac {\left (b^2 c h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\right ) \int \frac {\log \left (-\frac {b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{a}\\ &=-\frac {b g \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{a}-\frac {b h n \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x) \log (d+e x)}{a}-\frac {b h n \left (\log \left (\frac {b c x}{1-a c}\right )+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) x}{(1-a c) (d+e x)}\right )\right ) \log ^2\left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )}{2 a}+\frac {b h n \left (\log \left (\frac {b c x}{1-a c}\right )-\log \left (-\frac {e x}{d}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right )^2}{2 a}+\frac {b h \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x) \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )}{a}+\frac {b h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 a}-\frac {e h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 d}+\frac {e h n \log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))}{d}+\frac {b h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{a}-\frac {e h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{d}-\frac {b h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 a}+\frac {e h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 d}+\frac {e h n \left (\log \left (1+\frac {b x}{a}\right )+\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )}{2 d}+\frac {e h n \left (\log (c (a+b x))-\log \left (1+\frac {b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )^2}{2 d}+\frac {e h n \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2\left (-\frac {b x}{a}\right )}{d}-\frac {b g \text {Li}_2(c (a+b x))}{a}+\frac {e h n \log (x) \text {Li}_2(c (a+b x))}{d}-\frac {e h n \log (d+e x) \text {Li}_2(c (a+b x))}{d}+\frac {b h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{a}-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}-\frac {b g \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \left (\log (d+e x)-\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}+\frac {b h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text {Li}_2\left (\frac {d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{a}+\frac {b h n \log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text {Li}_2\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{a}+\frac {b h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{a}-\frac {e h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {b h n \left (\log (1-a c-b c x)+\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{a}+\frac {e h n \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {Li}_2\left (-\frac {b x}{a (1-c (a+b x))}\right )}{d}-\frac {e h n \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {Li}_2\left (-\frac {b c x}{1-c (a+b x)}\right )}{d}+\frac {b h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{a}-\frac {e h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{d}+\frac {e h n \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2(1-c (a+b x))}{d}-\frac {b h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{a}+\frac {e h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{d}+\frac {b h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{a}-\frac {e h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{d}-\frac {e h n \text {Li}_3\left (-\frac {b x}{a}\right )}{d}+\frac {b h n \text {Li}_3\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \text {Li}_3\left (\frac {d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{a}+\frac {b h n \text {Li}_3\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{a}-\frac {b h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{a}+\frac {e h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {b h n \text {Li}_3\left (1+\frac {e x}{d}\right )}{a}+\frac {e h n \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )}{d}-\frac {e h n \text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )}{d}-\frac {b h n \text {Li}_3(1-c (a+b x))}{a}-\frac {b h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{a}+\frac {e h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{d}+\frac {b h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{a}-\frac {e h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 7.16, size = 2247, normalized size = 0.90 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/x^2,x]

[Out]

-(((g - h*n*Log[d + e*x] + h*Log[f*(d + e*x)^n])*((a + b*x)*PolyLog[2, c*(a + b*x)] + b*x*(Log[(b*c*x)/(1 - a*
c)]*Log[1 - a*c - b*c*x] + PolyLog[2, (-1 + a*c + b*c*x)/(-1 + a*c)])))/(a*x)) + (h*n*(a*(e*x*Log[x] - (d + e*
x)*Log[d + e*x])*PolyLog[2, c*(a + b*x)] + x*(a*e*(Log[x]*Log[1 + (b*x)/a]*Log[1 - a*c - b*c*x] + ((-Log[c*(a
+ b*x)] + Log[1 + (b*x)/a])*Log[1 - a*c - b*c*x]*(-2*Log[x] + Log[1 - a*c - b*c*x]))/2 + (Log[c*(a + b*x)] - L
og[1 + (b*x)/a])*Log[1 - a*c - b*c*x]*Log[(a*(-1 + a*c + b*c*x))/(b*x)] + ((Log[(1 - a*c)/(b*c*x)] - Log[((1 -
a*c)*(a + b*x))/(b*x)] + Log[1 + (b*x)/a])*Log[(a*(-1 + a*c + b*c*x))/(b*x)]^2)/2 + (Log[1 - a*c - b*c*x] - L
og[(a*(-1 + a*c + b*c*x))/(b*x)])*PolyLog[2, -((b*x)/a)] + (Log[x] + Log[(a*(-1 + a*c + b*c*x))/(b*x)])*PolyLo
g[2, 1 - a*c - b*c*x] + Log[(a*(-1 + a*c + b*c*x))/(b*x)]*(-PolyLog[2, (a*(-1 + a*c + b*c*x))/(b*x)] + PolyLog
[2, (-1 + a*c + b*c*x)/(b*c*x)]) - PolyLog[3, -((b*x)/a)] - PolyLog[3, 1 - a*c - b*c*x] + PolyLog[3, (a*(-1 +
a*c + b*c*x))/(b*x)] - PolyLog[3, (-1 + a*c + b*c*x)/(b*c*x)]) - a*e*(Log[c*(a + b*x)]*Log[1 - a*c - b*c*x]*Lo
g[d + e*x] + ((Log[c*(a + b*x)] - Log[(e*(a + b*x))/(-(b*d) + a*e)])*Log[(b*(d + e*x))/(b*d - a*e)]*(-2*Log[1
- a*c - b*c*x] + Log[(b*(d + e*x))/(b*d - a*e)]))/2 + (-Log[c*(a + b*x)] + Log[(e*(a + b*x))/(-(b*d) + a*e)])*
Log[(b*(d + e*x))/(b*d - a*e)]*Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))] + (Log[-((b*(d + e*x))/(
(b*d - a*e)*(-1 + a*c + b*c*x)))]^2*(Log[c*(a + b*x)] - Log[((b*c*d + e - a*c*e)*(a + b*x))/((b*d - a*e)*(-1 +
a*c + b*c*x))] + Log[(b*c*d + e - a*c*e)/(e - a*c*e - b*c*e*x)]))/2 + (Log[d + e*x] - Log[-((b*(d + e*x))/((b
*d - a*e)*(-1 + a*c + b*c*x)))])*PolyLog[2, 1 - a*c - b*c*x] + (Log[1 - a*c - b*c*x] + Log[-((b*(d + e*x))/((b
*d - a*e)*(-1 + a*c + b*c*x)))])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)] + Log[-((b*(d + e*x))/((b*d - a*e)*(-1
+ a*c + b*c*x)))]*(PolyLog[2, (b*c*(d + e*x))/(e*(-1 + a*c + b*c*x))] - PolyLog[2, -((b*(d + e*x))/((b*d - a*e
)*(-1 + a*c + b*c*x)))]) - PolyLog[3, 1 - a*c - b*c*x] - PolyLog[3, (b*(d + e*x))/(b*d - a*e)] - PolyLog[3, (b
*c*(d + e*x))/(e*(-1 + a*c + b*c*x))] + PolyLog[3, -((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))]) - b*d*(
Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x]*Log[d + e*x] - Log[c*(a + b*x)]*Log[1 - a*c - b*c*x]*Log[d + e*x]
- ((Log[c*(a + b*x)] - Log[(e*(a + b*x))/(-(b*d) + a*e)])*Log[(b*(d + e*x))/(b*d - a*e)]*(-2*Log[1 - a*c - b*c
*x] + Log[(b*(d + e*x))/(b*d - a*e)]))/2 + (Log[c*(a + b*x)] - Log[(e*(a + b*x))/(-(b*d) + a*e)])*Log[(b*(d +
e*x))/(b*d - a*e)]*Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))] + (Log[((-1 + a*c)*(d + e*x))/(d*(-1
+ a*c + b*c*x))]^2*(Log[(b*c*x)/(1 - a*c)] - Log[((b*c*d + e - a*c*e)*x)/(d*(-1 + a*c + b*c*x))] + Log[(b*c*d
+ e - a*c*e)/(e - a*c*e - b*c*e*x)]))/2 - (Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))]^2*(Log[c*(a
+ b*x)] - Log[((b*c*d + e - a*c*e)*(a + b*x))/((b*d - a*e)*(-1 + a*c + b*c*x))] + Log[(b*c*d + e - a*c*e)/(e
- a*c*e - b*c*e*x)]))/2 + (-Log[(b*c*x)/(1 - a*c)] + Log[-((e*x)/d)])*Log[((-1 + a*c)*(d + e*x))/(d*(-1 + a*c
+ b*c*x))]*Log[1 + (e*x)/d] + ((Log[(b*c*x)/(1 - a*c)] - Log[-((e*x)/d)])*Log[1 + (e*x)/d]*(-2*Log[1 - a*c - b
*c*x] + Log[1 + (e*x)/d]))/2 - (Log[d + e*x] - Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))])*PolyLog
[2, 1 - a*c - b*c*x] + (Log[d + e*x] - Log[((-1 + a*c)*(d + e*x))/(d*(-1 + a*c + b*c*x))])*PolyLog[2, (-1 + a*
c + b*c*x)/(-1 + a*c)] - (Log[1 - a*c - b*c*x] + Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))])*PolyL
og[2, (b*(d + e*x))/(b*d - a*e)] + Log[((-1 + a*c)*(d + e*x))/(d*(-1 + a*c + b*c*x))]*(-PolyLog[2, ((-1 + a*c)
*(d + e*x))/(d*(-1 + a*c + b*c*x))] + PolyLog[2, (b*c*(d + e*x))/(e*(-1 + a*c + b*c*x))]) - Log[-((b*(d + e*x)
)/((b*d - a*e)*(-1 + a*c + b*c*x)))]*(PolyLog[2, (b*c*(d + e*x))/(e*(-1 + a*c + b*c*x))] - PolyLog[2, -((b*(d
+ e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))]) + (Log[1 - a*c - b*c*x] + Log[((-1 + a*c)*(d + e*x))/(d*(-1 + a*c
+ b*c*x))])*PolyLog[2, 1 + (e*x)/d] + PolyLog[3, 1 - a*c - b*c*x] - PolyLog[3, (-1 + a*c + b*c*x)/(-1 + a*c)]
+ PolyLog[3, (b*(d + e*x))/(b*d - a*e)] + PolyLog[3, ((-1 + a*c)*(d + e*x))/(d*(-1 + a*c + b*c*x))] - PolyLog[
3, -((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))] - PolyLog[3, 1 + (e*x)/d]))))/(a*d*x)

________________________________________________________________________________________

Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (g +h \ln \left (f \left (e x +d \right )^{n}\right )\right ) \polylog \left (2, c \left (b x +a \right )\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g+h*ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^2,x)

[Out]

int((g+h*ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^2,x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g+h*log(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^2,x, algorithm="maxima")

[Out]

integrate((h*log((x*e + d)^n*f) + g)*dilog((b*x + a)*c)/x^2, x)

________________________________________________________________________________________

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((g+h*log(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^2,x, algorithm="fricas")

[Out]

integral((h*dilog(b*c*x + a*c)*log((x*e + d)^n*f) + g*dilog(b*c*x + a*c))/x^2, x)

________________________________________________________________________________________

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((g+h*ln(f*(e*x+d)**n))*polylog(2,c*(b*x+a))/x**2,x)

[Out]

Timed out

________________________________________________________________________________________

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((g+h*log(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^2,x, algorithm="giac")

[Out]

integrate((h*log((e*x + d)^n*f) + g)*dilog((b*x + a)*c)/x^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((polylog(2, c*(a + b*x))*(g + h*log(f*(d + e*x)^n)))/x^2,x)

[Out]

int((polylog(2, c*(a + b*x))*(g + h*log(f*(d + e*x)^n)))/x^2, x)

________________________________________________________________________________________