3.2.77 \(\int x^2 (g+h \log (f (d+e x)^n)) \text {PolyLog}(2,c (a+b x)) \, dx\) [177]

Optimal. Leaf size=2995 \[ \text {result too large to display} \]

[Out]

-1/3*a^2*g*x/b^2+1/3*a^3*g*polylog(2,c*(b*x+a))/b^3-1/9*h*n*x^3*polylog(2,c*(b*x+a))+1/6*a*(-a*c+1)*h*(e*x+d)*
ln(f*(e*x+d)^n)/b^2/c/e-5/36*(-a*c+1)^2*d*h*n*ln(-b*c*x-a*c+1)/b^2/c^2/e+4/9*d^2*h*n*(-b*c*x-a*c+1)*ln(-b*c*x-
a*c+1)/b/c/e^2+1/18*(-a*c+1)*d^2*h*n*ln(e*x+d)/b/c/e^2+1/6*a*d^2*h*n*ln(-b*c*x-a*c+1)*ln(b*c*(e*x+d)/(-a*c*e+b
*c*d+e))/b/e^2+1/3*a^2*d*h*n*ln(-b*c*x-a*c+1)*ln(b*c*(e*x+d)/(-a*c*e+b*c*d+e))/b^2/e-2/27*h*n*x^3*ln(-b*c*x-a*
c+1)+1/12*a*x^2*(g+h*ln(f*(e*x+d)^n))/b+1/27*h*n*x^3+1/9*x^3*ln(-b*c*x-a*c+1)*(g+h*ln(f*(e*x+d)^n))+1/3*x^3*(g
+h*ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a))-1/9*a^3*h*n*polylog(2,c*(b*x+a))/b^3+1/9*d^3*h*n*polylog(2,e*(-b*c*x-
a*c+1)/(-a*c*e+b*c*d+e))/e^3+1/3*a^3*h*n*polylog(3,b*(e*x+d)/(-a*e+b*d))/b^3-1/3*d^3*h*n*polylog(3,b*(e*x+d)/(
-a*e+b*d))/e^3+1/3*a^3*h*n*polylog(3,1-c*(b*x+a))/b^3-1/3*d^3*h*n*polylog(3,1-c*(b*x+a))/e^3+1/3*a^3*h*n*polyl
og(3,-e*(1-c*(b*x+a))/b/c/(e*x+d))/b^3-1/3*d^3*h*n*polylog(3,-e*(1-c*(b*x+a))/b/c/(e*x+d))/e^3-1/3*a^3*h*n*pol
ylog(3,(-a*e+b*d)*(1-c*(b*x+a))/b/(e*x+d))/b^3+1/3*d^3*h*n*polylog(3,(-a*e+b*d)*(1-c*(b*x+a))/b/(e*x+d))/e^3-1
/9*(-a*c+1)^2*g*x/b^2/c^2+7/9*a^2*h*n*x/b^2+13/27*d^2*h*n*x/e^2-1/9*a*h*n*x^2/b-19/216*d*h*n*x^2/e+1/3*a*d*h*n
*(-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)/b^2/c/e-1/27*x^3*(g+h*ln(f*(e*x+d)^n))-1/3*a^3*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))/b^3+1/3*d^3*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))/e^3+2/27*(-a*c+1)^3*h*n*ln(-b*c*x-a*c+1)/b^3/c^3+5/36*a*h*n*
x^2*ln(-b*c*x-a*c+1)/b+5/36*d*h*n*x^2*ln(-b*c*x-a*c+1)/e+1/9*d^3*h*n*ln(-b*c*x-a*c+1)*ln(b*c*(e*x+d)/(-a*c*e+b
*c*d+e))/e^3-1/3*a^2*h*(e*x+d)*ln(f*(e*x+d)^n)/b^2/e-1/3*a^2*(-a*c+1)*ln(e*(-b*c*x-a*c+1)/(-a*c*e+b*c*d+e))*(g
+h*ln(f*(e*x+d)^n))/b^3/c+1/6*a*(-a*c+1)^2*ln(e*(-b*c*x-a*c+1)/(-a*c*e+b*c*d+e))*(g+h*ln(f*(e*x+d)^n))/b^3/c^2
-1/6*a^3*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/b^3+1/6*d^3*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/e^3+1/6*a^3*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/b^3-1/6*d^3*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/e^3+1/3*d^3*h*n*ln(e*x+d)*pol
ylog(2,c*(b*x+a))/e^3-1/3*a^3*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))/b^3+1/3*d^3*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))/e^3-1/3*a^3*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))/b^3+1/3*d^3*
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))/e^3+1/3*a^3*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))/b^3-1/3*d^3*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))/e^3-1/3*a^3*h*(n*ln(e*x+d)-ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a
))/b^3-1/27*d^3*h*n*ln(e*x+d)/e^3-1/18*(-a*c+1)*x^2*(g+h*ln(f*(e*x+d)^n))/b/c+1/3*a^2*x*ln(-b*c*x-a*c+1)*(g+h*
ln(f*(e*x+d)^n))/b^2-1/6*a*x^2*ln(-b*c*x-a*c+1)*(g+h*ln(f*(e*x+d)^n))/b-1/9*(-a*c+1)^3*ln(e*(-b*c*x-a*c+1)/(-a
*c*e+b*c*d+e))*(g+h*ln(f*(e*x+d)^n))/b^3/c^3-7/36*(-a*c+1)*d*h*n*x/b/c/e+5/12*a*d*h*n*x/b/e-1/3*a*d^2*h*n*poly
log(2,c*(b*x+a))/b/e^2-1/6*a^2*d*h*n*polylog(2,c*(b*x+a))/b^2/e+1/6*a*d^2*h*n*polylog(2,e*(-b*c*x-a*c+1)/(-a*c
*e+b*c*d+e))/b/e^2+1/3*a^2*d*h*n*polylog(2,e*(-b*c*x-a*c+1)/(-a*c*e+b*c*d+e))/b^2/e-1/3*a^2*(-a*c+1)*h*n*polyl
og(2,b*c*(e*x+d)/(-a*c*e+b*c*d+e))/b^3/c+1/6*a*(-a*c+1)^2*h*n*polylog(2,b*c*(e*x+d)/(-a*c*e+b*c*d+e))/b^3/c^2+
1/6*a*(-a*c+1)*g*x/b^2/c+5/27*(-a*c+1)^2*h*n*x/b^2/c^2+7/108*(-a*c+1)*h*n*x^2/b/c-1/3*d^2*h*n*x*polylog(2,c*(b
*x+a))/e^2+1/6*d*h*n*x^2*polylog(2,c*(b*x+a))/e-1/9*(-a*c+1)^3*h*n*polylog(2,b*c*(e*x+d)/(-a*c*e+b*c*d+e))/b^3
/c^3-5/36*a*(-a*c+1)^2*h*n*ln(-b*c*x-a*c+1)/b^3/c^2+4/9*a^2*h*n*(-b*c*x-a*c+1)*ln(-b*c*x-a*c+1)/b^3/c-1/12*a*d
^2*h*n*ln(e*x+d)/b/e^2-1/9*(-a*c+1)^2*h*(e*x+d)*ln(f*(e*x+d)^n)/b^2/c^2/e-1/3*a^3*h*n*ln(c*(b*x+a))*ln(e*x+d)*
ln(1-c*(b*x+a))/b^3+1/3*d^3*h*n*ln(c*(b*x+a))*ln(e*x+d)*ln(1-c*(b*x+a))/e^3-11/36*a*(-a*c+1)*h*n*x/b^2/c

________________________________________________________________________________________

Rubi [A]
time = 3.10, antiderivative size = 2995, normalized size of antiderivative = 1.00, number of steps used = 108, number of rules used = 20, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.741, Rules used = {6738, 2479, 45, 2463, 2436, 2332, 2441, 2440, 2438, 2489, 2442, 2490, 2488, 2487, 2485, 6730, 2494, 2468, 6733, 6732} \begin {gather*} \text {Too large to display} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(a^2*g*x)/b^2 + (a*(1 - a*c)*g*x)/(6*b^2*c) - ((1 - a*c)^2*g*x)/(9*b^2*c^2) + (7*a^2*h*n*x)/(9*b^2) - (11
*a*(1 - a*c)*h*n*x)/(36*b^2*c) + (5*(1 - a*c)^2*h*n*x)/(27*b^2*c^2) + (13*d^2*h*n*x)/(27*e^2) + (5*a*d*h*n*x)/
(12*b*e) - (7*(1 - a*c)*d*h*n*x)/(36*b*c*e) - (a*h*n*x^2)/(9*b) + (7*(1 - a*c)*h*n*x^2)/(108*b*c) - (19*d*h*n*
x^2)/(216*e) + (h*n*x^3)/27 - (5*a*(1 - a*c)^2*h*n*Log[1 - a*c - b*c*x])/(36*b^3*c^2) + (2*(1 - a*c)^3*h*n*Log
[1 - a*c - b*c*x])/(27*b^3*c^3) - (5*(1 - a*c)^2*d*h*n*Log[1 - a*c - b*c*x])/(36*b^2*c^2*e) + (5*a*h*n*x^2*Log
[1 - a*c - b*c*x])/(36*b) + (5*d*h*n*x^2*Log[1 - a*c - b*c*x])/(36*e) - (2*h*n*x^3*Log[1 - a*c - b*c*x])/27 +
(4*a^2*h*n*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(9*b^3*c) + (4*d^2*h*n*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*
x])/(9*b*c*e^2) + (a*d*h*n*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(3*b^2*c*e) - (d^3*h*n*Log[d + e*x])/(27*e^
3) - (a*d^2*h*n*Log[d + e*x])/(12*b*e^2) + ((1 - a*c)*d^2*h*n*Log[d + e*x])/(18*b*c*e^2) + (d^3*h*n*Log[1 - a*
c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(9*e^3) + (a*d^2*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e
*x))/(b*c*d + e - a*c*e)])/(6*b*e^2) + (a^2*d*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)
])/(3*b^2*e) - (a^2*h*(d + e*x)*Log[f*(d + e*x)^n])/(3*b^2*e) + (a*(1 - a*c)*h*(d + e*x)*Log[f*(d + e*x)^n])/(
6*b^2*c*e) - ((1 - a*c)^2*h*(d + e*x)*Log[f*(d + e*x)^n])/(9*b^2*c^2*e) + (a*x^2*(g + h*Log[f*(d + e*x)^n]))/(
12*b) - ((1 - a*c)*x^2*(g + h*Log[f*(d + e*x)^n]))/(18*b*c) - (x^3*(g + h*Log[f*(d + e*x)^n]))/27 + (a^2*x*Log
[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]))/(3*b^2) - (a*x^2*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n])
)/(6*b) + (x^3*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]))/9 - (a^2*(1 - a*c)*Log[(e*(1 - a*c - b*c*x))/(
b*c*d + e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(3*b^3*c) + (a*(1 - a*c)^2*Log[(e*(1 - a*c - b*c*x))/(b*c*d +
e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(6*b^3*c^2) - ((1 - a*c)^3*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*
e)]*(g + h*Log[f*(d + e*x)^n]))/(9*b^3*c^3) - (a^3*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)/(6*b^3) + (d^3*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)/(6*e^3) - (a^3*h*n*Log[c*(a
+ b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/(3*b^3) + (d^3*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x
)])/(3*e^3) + (a^3*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)/(6*b^3) - (d^3*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)/(6*e^3) + (a^3*g*Pol
yLog[2, c*(a + b*x)])/(3*b^3) - (a^3*h*n*PolyLog[2, c*(a + b*x)])/(9*b^3) - (a*d^2*h*n*PolyLog[2, c*(a + b*x)]
)/(3*b*e^2) - (a^2*d*h*n*PolyLog[2, c*(a + b*x)])/(6*b^2*e) - (d^2*h*n*x*PolyLog[2, c*(a + b*x)])/(3*e^2) + (d
*h*n*x^2*PolyLog[2, c*(a + b*x)])/(6*e) - (h*n*x^3*PolyLog[2, c*(a + b*x)])/9 + (d^3*h*n*Log[d + e*x]*PolyLog[
2, c*(a + b*x)])/(3*e^3) - (a^3*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/(3*b^3) + (x^
3*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/3 + (d^3*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e
- a*c*e)])/(9*e^3) + (a*d^2*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(6*b*e^2) + (a^2*d*h*n
*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(3*b^2*e) - (a^3*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)])/(3*b^3) + (d^3*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)])/(3*e
^3) - (a^2*(1 - a*c)*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(3*b^3*c) + (a*(1 - a*c)^2*h*n*PolyL
og[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(6*b^3*c^2) - ((1 - a*c)^3*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d +
e - a*c*e)])/(9*b^3*c^3) - (a^3*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyL
og[2, 1 - c*(a + b*x)])/(3*b^3) + (d^3*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)])/(3*e^3) + (a^3*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)))])/(3*b^3) - (d^3*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)))])/(3*e^3) - (a^3*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))])/(3*b^3) + (d^3*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))])/(3*e^3) + (a^3
*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/(3*b^3) - (d^3*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/(3*e^3)
+ (a^3*h*n*PolyLog[3, 1 - c*(a + b*x)])/(3*b^3)...

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

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

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 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 2440

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

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 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 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2468

Int[((a_.) + Log[(c_.)*(v_)^(n_.)]*(b_.))^(p_.)*(u_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^q*(a + b*Log[c*
ExpandToSum[v, x]^n])^p, x] /; FreeQ[{a, b, c, n, p, q}, x] && BinomialQ[u, x] && LinearQ[v, x] &&  !(Binomial
MatchQ[u, x] && LinearMatchQ[v, x])

Rule 2479

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.)), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[g*j*m, Int[x*
((a + b*Log[c*(d + e*x)^n])^p/(i + j*x)), x], x] - Dist[b*e*n*p, Int[x*(a + b*Log[c*(d + e*x)^n])^(p - 1)*((f
+ g*Log[h*(i + j*x)^m])/(d + e*x)), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 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 2489

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*(x_)^(r_.), x_Symbol] :> Simp[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p*((f + g*Log[h*(i + j*x)^m])/(r + 1
)), x] + (-Dist[g*j*(m/(r + 1)), Int[x^(r + 1)*((a + b*Log[c*(d + e*x)^n])^p/(i + j*x)), x], x] - Dist[b*e*n*(
p/(r + 1)), Int[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)*((f + g*Log[h*(i + j*x)^m])/(d + e*x)), x], x]) /
; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (EqQ[p, 1] || GtQ[r, 0]) && N
eQ[r, -1]

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 2494

Int[((a_.) + Log[(c_.)*(v_)^(n_.)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Int[u*(a + b*Log[c*ExpandToSum[v, x]^n])^p
, x] /; FreeQ[{a, b, c, n, p}, x] && LinearQ[v, x] &&  !LinearMatchQ[v, x] &&  !(EqQ[n, 1] && MatchQ[c*v, (e_.
)*((f_) + (g_.)*x) /; FreeQ[{e, f, g}, x]])

Rule 6730

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)], x_Symbol] :> Simp[x*PolyLog[n, c*(a + b*x)^p], x] + (-Dist[
p, Int[PolyLog[n - 1, c*(a + b*x)^p], x], x] + Dist[a*p, Int[PolyLog[n - 1, c*(a + b*x)^p]/(a + b*x), x], x])
/; FreeQ[{a, b, c, p}, x] && GtQ[n, 0]

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 6733

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

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 x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x)) \, dx &=\frac {1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {1}{3} b \int \left (\frac {a^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^3}-\frac {a x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^2}+\frac {x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b}-\frac {a^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^3 (a+b x)}\right ) \, dx-\frac {1}{3} (e h n) \int \left (\frac {d^2 \text {Li}_2(c (a+b x))}{e^3}-\frac {d x \text {Li}_2(c (a+b x))}{e^2}+\frac {x^2 \text {Li}_2(c (a+b x))}{e}-\frac {d^3 \text {Li}_2(c (a+b x))}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {1}{3} \int x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx+\frac {a^2 \int \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{3 b^2}-\frac {a^3 \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}{3 b^2}-\frac {a \int x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{3 b}-\frac {1}{3} (h n) \int x^2 \text {Li}_2(c (a+b x)) \, dx-\frac {\left (d^2 h n\right ) \int \text {Li}_2(c (a+b x)) \, dx}{3 e^2}+\frac {\left (d^3 h n\right ) \int \frac {\text {Li}_2(c (a+b x))}{d+e x} \, dx}{3 e^2}+\frac {(d h n) \int x \text {Li}_2(c (a+b x)) \, dx}{3 e}\\ &=\frac {a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac {a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac {1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac {d^2 h n x \text {Li}_2(c (a+b x))}{3 e^2}+\frac {d h n x^2 \text {Li}_2(c (a+b x))}{6 e}-\frac {1}{9} h n x^3 \text {Li}_2(c (a+b x))+\frac {d^3 h n \log (d+e x) \text {Li}_2(c (a+b x))}{3 e^3}+\frac {1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))-\frac {a^3 \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 )}{3 b^3}-\frac {1}{6} (a c) \int \frac {x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{1-a c-b c x} \, dx+\frac {\left (a^2 c\right ) \int \frac {x \left (g+h \log \left (f (d+e x)^n\right )\right )}{1-a c-b c x} \, dx}{3 b}+\frac {1}{9} (b c) \int \frac {x^3 \left (g+h \log \left (f (d+e x)^n\right )\right )}{1-a c-b c x} \, dx-\frac {1}{9} (b h n) \int \frac {x^3 \log (1-a c-b c x)}{a+b x} \, dx+\frac {\left (b d^3 h n\right ) \int \frac {\log (1-a c-b c x) \log (d+e x)}{a+b x} \, dx}{3 e^3}-\frac {\left (d^2 h n\right ) \int \log (1-c (a+b x)) \, dx}{3 e^2}+\frac {\left (a d^2 h n\right ) \int \frac {\log (1-c (a+b x))}{a+b x} \, dx}{3 e^2}+\frac {(b d h n) \int \frac {x^2 \log (1-a c-b c x)}{a+b x} \, dx}{6 e}-\frac {1}{9} (e h n) \int \frac {x^3 \log (1-a c-b c x)}{d+e x} \, dx-\frac {\left (a^2 e h n\right ) \int \frac {x \log (1-a c-b c x)}{d+e x} \, dx}{3 b^2}+\frac {(a e h n) \int \frac {x^2 \log (1-a c-b c x)}{d+e x} \, dx}{6 b}\\ &=\frac {a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac {a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac {1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac {d^2 h n x \text {Li}_2(c (a+b x))}{3 e^2}+\frac {d h n x^2 \text {Li}_2(c (a+b x))}{6 e}-\frac {1}{9} h n x^3 \text {Li}_2(c (a+b x))+\frac {d^3 h n \log (d+e x) \text {Li}_2(c (a+b x))}{3 e^3}+\frac {1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))-\frac {1}{6} (a c) \int \left (\frac {(-1+a c) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^2 c^2}-\frac {x \left (g+h \log \left (f (d+e x)^n\right )\right )}{b c}-\frac {(-1+a c)^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx+\frac {\left (a^2 c\right ) \int \left (-\frac {g+h \log \left (f (d+e x)^n\right )}{b c}+\frac {(-1+a c) \left (g+h \log \left (f (d+e x)^n\right )\right )}{b c (-1+a c+b c x)}\right ) \, dx}{3 b}+\frac {1}{9} (b c) \int \left (-\frac {(-1+a c)^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^3 c^3}+\frac {(-1+a c) x \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^2 c^2}-\frac {x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{b c}+\frac {(-1+a c)^3 \left (g+h \log \left (f (d+e x)^n\right )\right )}{b^3 c^3 (-1+a c+b c x)}\right ) \, dx-\frac {\left (a^3 g\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 )}{3 b^3}-\frac {\left (a^3 h\right ) \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 )}{3 b^3}-\frac {1}{9} (b h n) \int \left (\frac {a^2 \log (1-a c-b c x)}{b^3}-\frac {a x \log (1-a c-b c x)}{b^2}+\frac {x^2 \log (1-a c-b c x)}{b}-\frac {a^3 \log (1-a c-b c x)}{b^3 (a+b x)}\right ) \, dx+\frac {\left (d^3 h n\right ) \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 )}{3 e^3}-\frac {\left (d^2 h n\right ) \int \log (1-a c-b c x) \, dx}{3 e^2}+\frac {\left (a d^2 h n\right ) \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{3 e^2}+\frac {(b d h n) \int \left (-\frac {a \log (1-a c-b c x)}{b^2}+\frac {x \log (1-a c-b c x)}{b}+\frac {a^2 \log (1-a c-b c x)}{b^2 (a+b x)}\right ) \, dx}{6 e}-\frac {1}{9} (e h n) \int \left (\frac {d^2 \log (1-a c-b c x)}{e^3}-\frac {d x \log (1-a c-b c x)}{e^2}+\frac {x^2 \log (1-a c-b c x)}{e}-\frac {d^3 \log (1-a c-b c x)}{e^3 (d+e x)}\right ) \, dx-\frac {\left (a^2 e h n\right ) \int \left (\frac {\log (1-a c-b c x)}{e}-\frac {d \log (1-a c-b c x)}{e (d+e x)}\right ) \, dx}{3 b^2}+\frac {(a e h n) \int \left (-\frac {d \log (1-a c-b c x)}{e^2}+\frac {x \log (1-a c-b c x)}{e}+\frac {d^2 \log (1-a c-b c x)}{e^2 (d+e x)}\right ) \, dx}{6 b}\\ &=\frac {a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac {a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac {1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac {d^3 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 )}{6 e^3}+\frac {d^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 e^3}-\frac {d^3 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}{6 e^3}+\frac {a^3 g \text {Li}_2(c (a+b x))}{3 b^3}-\frac {d^2 h n x \text {Li}_2(c (a+b x))}{3 e^2}+\frac {d h n x^2 \text {Li}_2(c (a+b x))}{6 e}-\frac {1}{9} h n x^3 \text {Li}_2(c (a+b x))+\frac {d^3 h n \log (d+e x) \text {Li}_2(c (a+b x))}{3 e^3}+\frac {1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {d^3 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 )}{3 e^3}+\frac {d^3 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))}{3 e^3}-\frac {d^3 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 )}{3 e^3}+\frac {d^3 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 )}{3 e^3}-\frac {d^3 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{3 e^3}-\frac {d^3 h n \text {Li}_3(1-c (a+b x))}{3 e^3}-\frac {d^3 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{3 e^3}+\frac {d^3 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 e^3}-\frac {1}{9} \int x^2 \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx-\frac {a^2 \int \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{3 b^2}+\frac {a \int x \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{6 b}-\frac {\left (a^2 (1-a c)\right ) \int \frac {g+h \log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx}{3 b^2}+\frac {(a (1-a c)) \int \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{6 b^2 c}-\frac {(1-a c) \int x \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{9 b c}-\frac {(1-a c)^2 \int \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx}{9 b^2 c^2}+\frac {\left (a (1-a c)^2\right ) \int \frac {g+h \log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx}{6 b^2 c}-\frac {(1-a c)^3 \int \frac {g+h \log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx}{9 b^2 c^2}-2 \left (\frac {1}{9} (h n) \int x^2 \log (1-a c-b c x) \, dx\right )-\frac {\left (a^3 h n\right ) \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 )}{3 b^3}-\frac {\left (a^2 h n\right ) \int \log (1-a c-b c x) \, dx}{9 b^2}-\frac {\left (a^2 h n\right ) \int \log (1-a c-b c x) \, dx}{3 b^2}+\frac {\left (a^3 h n\right ) \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{9 b^2}+\frac {(a h n) \int x \log (1-a c-b c x) \, dx}{9 b}+\frac {(a h n) \int x \log (1-a c-b c x) \, dx}{6 b}+\frac {\left (a^2 d h n\right ) \int \frac {\log (1-a c-b c x)}{d+e x} \, dx}{3 b^2}-\frac {\left (d^2 h n\right ) \int \log (1-a c-b c x) \, dx}{9 e^2}+\frac {\left (a d^2 h n\right ) \text {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{3 b e^2}+\frac {\left (d^2 h n\right ) \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{3 b c e^2}+\frac {\left (d^3 h n\right ) \int \frac {\log (1-a c-b c x)}{d+e x} \, dx}{9 e^2}+\frac {(d h n) \int x \log (1-a c-b c x) \, dx}{9 e}+\frac {(d h n) \int x \log (1-a c-b c x) \, dx}{6 e}-2 \frac {(a d h n) \int \log (1-a c-b c x) \, dx}{6 b e}+\frac {\left (a^2 d h n\right ) \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{6 b e}+\frac {\left (a d^2 h n\right ) \int \frac {\log (1-a c-b c x)}{d+e x} \, dx}{6 b e}+\frac {\left (a^3 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 )}{3 b^3}\\ &=-\frac {a^2 g x}{3 b^2}+\frac {a (1-a c) g x}{6 b^2 c}-\frac {(1-a c)^2 g x}{9 b^2 c^2}+\frac {d^2 h n x}{3 e^2}+\frac {5 a h n x^2 \log (1-a c-b c x)}{36 b}+\frac {5 d h n x^2 \log (1-a c-b c x)}{36 e}+\frac {d^2 h n (1-a c-b c x) \log (1-a c-b c x)}{3 b c e^2}+\frac {d^3 h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{9 e^3}+\frac {a d^2 h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{6 b e^2}+\frac {a^2 d h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 b^2 e}+\frac {a x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{12 b}-\frac {(1-a c) x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{18 b c}-\frac {1}{27} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac {a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac {a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac {1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac {a^2 (1-a c) \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^3 c}+\frac {a (1-a c)^2 \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b^3 c^2}-\frac {(1-a c)^3 \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{9 b^3 c^3}-\frac {a^3 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 )}{6 b^3}+\frac {d^3 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 )}{6 e^3}-\frac {a^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 b^3}+\frac {d^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 e^3}+\frac {a^3 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}{6 b^3}-\frac {d^3 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}{6 e^3}+\frac {a^3 g \text {Li}_2(c (a+b x))}{3 b^3}-\frac {a d^2 h n \text {Li}_2(c (a+b x))}{3 b e^2}-\frac {d^2 h n x \text {Li}_2(c (a+b x))}{3 e^2}+\frac {d h n x^2 \text {Li}_2(c (a+b x))}{6 e}-\frac {1}{9} h n x^3 \text {Li}_2(c (a+b x))+\frac {d^3 h n \log (d+e x) \text {Li}_2(c (a+b x))}{3 e^3}-\frac {a^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{3 b^3}+\frac {1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))-\frac {a^3 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 )}{3 b^3}+\frac {d^3 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 )}{3 e^3}-\frac {a^3 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))}{3 b^3}+\frac {d^3 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))}{3 e^3}+\frac {a^3 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 )}{3 b^3}-\frac {d^3 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 )}{3 e^3}-\frac {a^3 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 )}{3 b^3}+\frac {d^3 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 )}{3 e^3}+\frac {a^3 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{3 b^3}-\frac {d^3 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{3 e^3}+\frac {a^3 h n \text {Li}_3(1-c (a+b x))}{3 b^3}-\frac {d^3 h n \text {Li}_3(1-c (a+b x))}{3 e^3}+\frac {a^3 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{3 b^3}-\frac {d^3 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{3 e^3}-\frac {a^3 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 b^3}+\frac {d^3 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 e^3}-\frac {\left (a^2 h\right ) \int \log \left (f (d+e x)^n\right ) \, dx}{3 b^2}+\frac {(a (1-a c) h) \int \log \left (f (d+e x)^n\right ) \, dx}{6 b^2 c}-\frac {\left ((1-a c)^2 h\right ) \int \log \left (f (d+e x)^n\right ) \, dx}{9 b^2 c^2}+\frac {\left (a^3 h n\right ) \text {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{9 b^3}+\frac {\left (a^2 h n\right ) \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{9 b^3 c}+\frac {\left (a^2 h n\right ) \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{3 b^3 c}+\frac {1}{18} (a c h n) \int \frac {x^2}{1-a c-b c x} \, dx+\frac {1}{12} (a c h n) \int \frac {x^2}{1-a c-b c x} \, dx-2 \left (\frac {1}{27} h n x^3 \log (1-a c-b c x)+\frac {1}{27} (b c h n) \int \frac {x^3}{1-a c-b c x} \, dx\right )+\frac {\left (b c d^3 h n\right ) \int \frac {\log \left (-\frac {b c (d+e x)}{-b c d-(1-a c) e}\right )}{1-a c-b c x} \, dx}{9 e^3}+\frac {\left (d^2 h n\right ) \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{9 b c e^2}+\frac {\left (a c d^2 h n\right ) \int \frac {\log \left (-\frac {b c (d+e x)}{-b c d-(1-a c) e}\right )}{1-a c-b c x} \, dx}{6 e^2}+\frac {\left (a^2 d h n\right ) \text {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{6 b^2 e}+2 \frac {(a d h n) \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{6 b^2 c e}+\frac {\left (a^2 c d h n\right ) \int \frac {\log \left (-\frac {b c (d+e x)}{-b c d-(1-a c) e}\right )}{1-a c-b c x} \, dx}{3 b e}+\frac {(b c d h n) \int \frac {x^2}{1-a c-b c x} \, dx}{18 e}+\frac {(b c d h n) \int \frac {x^2}{1-a c-b c x} \, dx}{12 e}+\frac {1}{27} (e h n) \int \frac {x^3}{d+e x} \, dx-\frac {(a e h n) \int \frac {x^2}{d+e x} \, dx}{12 b}+\frac {\left (a^2 (1-a c) e h n\right ) \int \frac {\log \left (\frac {e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )}{d+e x} \, dx}{3 b^3 c}+\frac {((1-a c) e h n) \int \frac {x^2}{d+e x} \, dx}{18 b c}-\frac {\left (a (1-a c)^2 e h n\right ) \int \frac {\log \left (\frac {e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )}{d+e x} \, dx}{6 b^3 c^2}+\frac {\left ((1-a c)^3 e h n\right ) \int \frac {\log \left (\frac {e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )}{d+e x} \, dx}{9 b^3 c^3}\\ &=-\frac {a^2 g x}{3 b^2}+\frac {a (1-a c) g x}{6 b^2 c}-\frac {(1-a c)^2 g x}{9 b^2 c^2}+\frac {4 a^2 h n x}{9 b^2}+\frac {4 d^2 h n x}{9 e^2}+\frac {5 a h n x^2 \log (1-a c-b c x)}{36 b}+\frac {5 d h n x^2 \log (1-a c-b c x)}{36 e}+\frac {4 a^2 h n (1-a c-b c x) \log (1-a c-b c x)}{9 b^3 c}+\frac {4 d^2 h n (1-a c-b c x) \log (1-a c-b c x)}{9 b c e^2}+2 \left (\frac {a d h n x}{6 b e}+\frac {a d h n (1-a c-b c x) \log (1-a c-b c x)}{6 b^2 c e}\right )+\frac {d^3 h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{9 e^3}+\frac {a d^2 h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{6 b e^2}+\frac {a^2 d h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 b^2 e}+\frac {a x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{12 b}-\frac {(1-a c) x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{18 b c}-\frac {1}{27} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac {a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac {a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac {1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac {a^2 (1-a c) \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^3 c}+\frac {a (1-a c)^2 \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b^3 c^2}-\frac {(1-a c)^3 \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{9 b^3 c^3}-\frac {a^3 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 )}{6 b^3}+\frac {d^3 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 )}{6 e^3}-\frac {a^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 b^3}+\frac {d^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 e^3}+\frac {a^3 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}{6 b^3}-\frac {d^3 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}{6 e^3}+\frac {a^3 g \text {Li}_2(c (a+b x))}{3 b^3}-\frac {a^3 h n \text {Li}_2(c (a+b x))}{9 b^3}-\frac {a d^2 h n \text {Li}_2(c (a+b x))}{3 b e^2}-\frac {a^2 d h n \text {Li}_2(c (a+b x))}{6 b^2 e}-\frac {d^2 h n x \text {Li}_2(c (a+b x))}{3 e^2}+\frac {d h n x^2 \text {Li}_2(c (a+b x))}{6 e}-\frac {1}{9} h n x^3 \text {Li}_2(c (a+b x))+\frac {d^3 h n \log (d+e x) \text {Li}_2(c (a+b x))}{3 e^3}-\frac {a^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{3 b^3}+\frac {1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))-\frac {a^3 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 )}{3 b^3}+\frac {d^3 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 )}{3 e^3}-\frac {a^3 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))}{3 b^3}+\frac {d^3 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))}{3 e^3}+\frac {a^3 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 )}{3 b^3}-\frac {d^3 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 )}{3 e^3}-\frac {a^3 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 )}{3 b^3}+\frac {d^3 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 )}{3 e^3}+\frac {a^3 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{3 b^3}-\frac {d^3 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{3 e^3}+\frac {a^3 h n \text {Li}_3(1-c (a+b x))}{3 b^3}-\frac {d^3 h n \text {Li}_3(1-c (a+b x))}{3 e^3}+\frac {a^3 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{3 b^3}-\frac {d^3 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{3 e^3}-\frac {a^3 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 b^3}+\frac {d^3 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 e^3}-\frac {\left (a^2 h\right ) \text {Subst}\left (\int \log \left (f x^n\right ) \, dx,x,d+e x\right )}{3 b^2 e}+\frac {(a (1-a c) h) \text {Subst}\left (\int \log \left (f x^n\right ) \, dx,x,d+e x\right )}{6 b^2 c e}-\frac {\left ((1-a c)^2 h\right ) \text {Subst}\left (\int \log \left (f x^n\right ) \, dx,x,d+e x\right )}{9 b^2 c^2 e}+\frac {1}{18} (a c h n) \int \left (\frac {-1+a c}{b^2 c^2}-\frac {x}{b c}-\frac {(-1+a c)^2}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx+\frac {1}{12} (a c h n) \int \left (\frac {-1+a c}{b^2 c^2}-\frac {x}{b c}-\frac {(-1+a c)^2}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx-2 \left (\frac {1}{27} h n x^3 \log (1-a c-b c x)+\frac {1}{27} (b c h n) \int \left (-\frac {(-1+a c)^2}{b^3 c^3}+\frac {(-1+a c) x}{b^2 c^2}-\frac {x^2}{b c}+\frac {(-1+a c)^3}{b^3 c^3 (-1+a c+b c x)}\right ) \, dx\right )+\frac {\left (a^2 (1-a c) h n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b c x}{-b c d+(-1+a c) e}\right )}{x} \, dx,x,d+e x\right )}{3 b^3 c}-\frac {\left (a (1-a c)^2 h n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b c x}{-b c d+(-1+a c) e}\right )}{x} \, dx,x,d+e x\right )}{6 b^3 c^2}+\frac {\left ((1-a c)^3 h n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b c x}{-b c d+(-1+a c) e}\right )}{x} \, dx,x,d+e x\right )}{9 b^3 c^3}-\frac {\left (d^3 h n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-b c d-(1-a c) e}\right )}{x} \, dx,x,1-a c-b c x\right )}{9 e^3}-\frac {\left (a d^2 h n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-b c d-(1-a c) e}\right )}{x} \, dx,x,1-a c-b c x\right )}{6 b e^2}-\frac {\left (a^2 d h n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-b c d-(1-a c) e}\right )}{x} \, dx,x,1-a c-b c x\right )}{3 b^2 e}+\frac {(b c d h n) \int \left (\frac {-1+a c}{b^2 c^2}-\frac {x}{b c}-\frac {(-1+a c)^2}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx}{18 e}+\frac {(b c d h n) \int \left (\frac {-1+a c}{b^2 c^2}-\frac {x}{b c}-\frac {(-1+a c)^2}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx}{12 e}+\frac {1}{27} (e h n) \int \left (\frac {d^2}{e^3}-\frac {d x}{e^2}+\frac {x^2}{e}-\frac {d^3}{e^3 (d+e x)}\right ) \, dx-\frac {(a e h n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx}{12 b}+\frac {((1-a c) e h n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx}{18 b c}\\ &=-\frac {a^2 g x}{3 b^2}+\frac {a (1-a c) g x}{6 b^2 c}-\frac {(1-a c)^2 g x}{9 b^2 c^2}+\frac {7 a^2 h n x}{9 b^2}-\frac {11 a (1-a c) h n x}{36 b^2 c}+\frac {(1-a c)^2 h n x}{9 b^2 c^2}+\frac {13 d^2 h n x}{27 e^2}+\frac {a d h n x}{12 b e}-\frac {7 (1-a c) d h n x}{36 b c e}-\frac {a h n x^2}{9 b}+\frac {(1-a c) h n x^2}{36 b c}-\frac {19 d h n x^2}{216 e}+\frac {1}{81} h n x^3-\frac {5 a (1-a c)^2 h n \log (1-a c-b c x)}{36 b^3 c^2}-\frac {5 (1-a c)^2 d h n \log (1-a c-b c x)}{36 b^2 c^2 e}+\frac {5 a h n x^2 \log (1-a c-b c x)}{36 b}+\frac {5 d h n x^2 \log (1-a c-b c x)}{36 e}+\frac {4 a^2 h n (1-a c-b c x) \log (1-a c-b c x)}{9 b^3 c}+\frac {4 d^2 h n (1-a c-b c x) \log (1-a c-b c x)}{9 b c e^2}-2 \left (-\frac {(1-a c)^2 h n x}{27 b^2 c^2}-\frac {(1-a c) h n x^2}{54 b c}-\frac {1}{81} h n x^3-\frac {(1-a c)^3 h n \log (1-a c-b c x)}{27 b^3 c^3}+\frac {1}{27} h n x^3 \log (1-a c-b c x)\right )+2 \left (\frac {a d h n x}{6 b e}+\frac {a d h n (1-a c-b c x) \log (1-a c-b c x)}{6 b^2 c e}\right )-\frac {d^3 h n \log (d+e x)}{27 e^3}-\frac {a d^2 h n \log (d+e x)}{12 b e^2}+\frac {(1-a c) d^2 h n \log (d+e x)}{18 b c e^2}+\frac {d^3 h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{9 e^3}+\frac {a d^2 h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{6 b e^2}+\frac {a^2 d h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 b^2 e}-\frac {a^2 h (d+e x) \log \left (f (d+e x)^n\right )}{3 b^2 e}+\frac {a (1-a c) h (d+e x) \log \left (f (d+e x)^n\right )}{6 b^2 c e}-\frac {(1-a c)^2 h (d+e x) \log \left (f (d+e x)^n\right )}{9 b^2 c^2 e}+\frac {a x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{12 b}-\frac {(1-a c) x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{18 b c}-\frac {1}{27} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right )+\frac {a^2 x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^2}-\frac {a x^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b}+\frac {1}{9} x^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )-\frac {a^2 (1-a c) \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 b^3 c}+\frac {a (1-a c)^2 \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 b^3 c^2}-\frac {(1-a c)^3 \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \left (g+h \log \left (f (d+e x)^n\right )\right )}{9 b^3 c^3}-\frac {a^3 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 )}{6 b^3}+\frac {d^3 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 )}{6 e^3}-\frac {a^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 b^3}+\frac {d^3 h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{3 e^3}+\frac {a^3 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}{6 b^3}-\frac {d^3 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}{6 e^3}+\frac {a^3 g \text {Li}_2(c (a+b x))}{3 b^3}-\frac {a^3 h n \text {Li}_2(c (a+b x))}{9 b^3}-\frac {a d^2 h n \text {Li}_2(c (a+b x))}{3 b e^2}-\frac {a^2 d h n \text {Li}_2(c (a+b x))}{6 b^2 e}-\frac {d^2 h n x \text {Li}_2(c (a+b x))}{3 e^2}+\frac {d h n x^2 \text {Li}_2(c (a+b x))}{6 e}-\frac {1}{9} h n x^3 \text {Li}_2(c (a+b x))+\frac {d^3 h n \log (d+e x) \text {Li}_2(c (a+b x))}{3 e^3}-\frac {a^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{3 b^3}+\frac {1}{3} x^3 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {d^3 h n \text {Li}_2\left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{9 e^3}+\frac {a d^2 h n \text {Li}_2\left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{6 b e^2}+\frac {a^2 d h n \text {Li}_2\left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{3 b^2 e}-\frac {a^3 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 )}{3 b^3}+\frac {d^3 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 )}{3 e^3}-\frac {a^2 (1-a c) h n \text {Li}_2\left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 b^3 c}+\frac {a (1-a c)^2 h n \text {Li}_2\left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{6 b^3 c^2}-\frac {(1-a c)^3 h n \text {Li}_2\left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{9 b^3 c^3}-\frac {a^3 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))}{3 b^3}+\frac {d^3 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))}{3 e^3}+\frac {a^3 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 )}{3 b^3}-\frac {d^3 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 )}{3 e^3}-\frac {a^3 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 )}{3 b^3}+\frac {d^3 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 )}{3 e^3}+\frac {a^3 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{3 b^3}-\frac {d^3 h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{3 e^3}+\frac {a^3 h n \text {Li}_3(1-c (a+b x))}{3 b^3}-\frac {d^3 h n \text {Li}_3(1-c (a+b x))}{3 e^3}+\frac {a^3 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{3 b^3}-\frac {d^3 h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{3 e^3}-\frac {a^3 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 b^3}+\frac {d^3 h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{3 e^3}\\ \end {align*}

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Mathematica [A]
time = 9.81, size = 2610, normalized size = 0.87 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((g - h*n*Log[d + e*x] + h*Log[f*(d + e*x)^n])*(-(b*c*x*(12 + 66*a^2*c^2 + 6*b*c*x + 4*b^2*c^2*x^2 - 3*a*c*(14
+ 5*b*c*x))) + 6*(-2 + 11*a^3*c^3 + 2*b^3*c^3*x^3 + 6*a^2*c^2*(-3 + b*c*x) + a*(9*c - 3*b^2*c^3*x^2))*Log[1 -
a*c - b*c*x] + 36*c^3*(a^3 + b^3*x^3)*PolyLog[2, c*(a + b*x)]))/(108*b^3*c^3) + (h*n*(36*b^3*c^3*(e*x*(-6*d^2
+ 3*d*e*x - 2*e^2*x^2) + 6*(d^3 + e^3*x^3)*Log[d + e*x])*PolyLog[2, c*(a + b*x)] - 216*b^2*c^2*d^2*e*(1 - a*c
- b*c*x + (-1 + a*c + b*c*x - a*c*Log[c*(a + b*x)])*Log[1 - a*c - b*c*x] - a*c*PolyLog[2, 1 - a*c - b*c*x]) -
27*b*c*d*e^2*(c*(-4*a^2*c + a*(4 - 6*b*c*x) + b*x*(2 + b*c*x)) + (2 + 6*a^2*c^2 - 2*b^2*c^2*x^2 + 4*a*c*(-2 +
b*c*x) - 4*a^2*c^2*Log[c*(a + b*x)])*Log[1 - a*c - b*c*x] - 4*a^2*c^2*PolyLog[2, 1 - a*c - b*c*x]) - 2*e^3*(-
(c*(36*a^3*c^2 - 3*a*b*c*x*(14 + 5*b*c*x) + 6*a^2*c*(-6 + 11*b*c*x) + 2*b*x*(6 + 3*b*c*x + 2*b^2*c^2*x^2))) -
6*(2 - 11*a^3*c^3 - 2*b^3*c^3*x^3 - 6*a^2*c^2*(-3 + b*c*x) + 3*a*c*(-3 + b^2*c^2*x^2) + 6*a^3*c^3*Log[c*(a + b
*x)])*Log[1 - a*c - b*c*x] - 36*a^3*c^3*PolyLog[2, 1 - a*c - b*c*x]) + 216*b^3*c^3*d^3*(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
[-((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)))]) + 2*(b*c*(e*(48*(-1 + a*c)^2*e^2*x + 3*b*c*(-1 + a*c)*(12*d^2 + 12*d*e*x - 5*e^2*x^2) + b^2*c^2*x*(
48*d^2 - 15*d*e*x + 8*e^2*x^2)) - 6*(d + e*x)*(6*(-1 + a*c)^2*e^2 + 3*b*c*(-1 + a*c)*e*(d - e*x) + 2*b^2*c^2*(
d^2 - d*e*x + e^2*x^2))*Log[d + e*x]) + 6*Log[1 - a*c - b*c*x]*(-(e*(-1 + a*c + b*c*x)*(2*(-1 + a*c)^2*e^2 + b
*c*(-1 + a*c)*e*(3*d - 2*e*x) + b^2*c^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2))) + 6*e^3*(-1 + 3*a*c - 3*a^2*c^2 + a^3*
c^3 + b^3*c^3*x^3)*Log[d + e*x] + 6*(b^3*c^3*d^3 - (-1 + a*c)^3*e^3)*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])
+ 36*(b^3*c^3*d^3 - (-1 + a*c)^3*e^3)*PolyLog[2, (e*(-1 + a*c + b*c*x))/(-(b*c*d) + (-1 + a*c)*e)] - 108*a^2*
c^2*e^2*(e - a*c*e - 2*b*c*e*x + b*c*d*Log[d + e*x] + b*c*e*x*Log[d + e*x] - Log[1 - a*c - b*c*x]*(-(e*(-1 + a
*c + b*c*x)) + e*(-1 + a*c + b*c*x)*Log[d + e*x] + (b*c*d + e - a*c*e)*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)
]) - (b*c*d + e - a*c*e)*PolyLog[2, (e*(-1 + a*c + b*c*x))/(-(b*c*d) + (-1 + a*c)*e)]) - 27*a*c*e*(b*c*(e*(d*(
2 - 2*a*c - 3*b*c*x) + e*x*(3 - 3*a*c + b*c*x)) + (d + e*x)*(2*(-1 + a*c)*e + b*c*(d - e*x))*Log[d + e*x]) + L
og[1 - a*c - b*c*x]*(e*(-1 + a*c + b*c*x)*((-1 + a*c)*e + b*c*(2*d - e*x)) - 2*e^2*(1 - 2*a*c + a^2*c^2 - b^2*
c^2*x^2)*Log[d + e*x] + 2*(-(b^2*c^2*d^2) + (-1 + a*c)^2*e^2)*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)]) + 2*(-
(b^2*c^2*d^2) + (-1 + a*c)^2*e^2)*PolyLog[2, (e*(-1 + a*c + b*c*x))/(-(b*c*d) + (-1 + a*c)*e)]) - 108*a^3*c^3*
e^3*(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[-((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] + (L
og[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)))]))))/(648*b^3*c^3*e^3)

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int x^{2} \left (g +h \ln \left (f \left (e x +d \right )^{n}\right )\right ) \polylog \left (2, c \left (b x +a \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

1/18*(3*d*h*n*x^2*e^2 - 6*d^2*h*n*x*e + 6*d^3*h*n*log(x*e + d) + 6*h*x^3*e^3*log((x*e + d)^n) - 2*(h*n*e^3 - 3
*h*e^3*log(f) - 3*g*e^3)*x^3)*dilog(b*c*x + a*c)*e^(-3) + integrate(1/18*(6*b*h*x^3*e^3*log(-b*c*x - a*c + 1)*
log((x*e + d)^n) + (3*b*d*h*n*x^2*e^2 - 6*b*d^2*h*n*x*e + 6*b*d^3*h*n*log(x*e + d) - 2*(b*h*n*e^3 - 3*b*h*e^3*
log(f) - 3*b*g*e^3)*x^3)*log(-b*c*x - a*c + 1))/(b*x*e^3 + a*e^3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\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 ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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