(∫_(-1)^1) dx(1+(x^2))/(0.42304687500000004` +0.7332812500000001` x+0.5251562500000001` (x^2)-0.09000000000000002` (x^3)-0.1631250000000002` (x^4)-0.09750000000000014` (x^5)-0.25` (x^6)+(x^7))

Clear[n,d]

x0 =- 0.75
x1 = -0.5 + I 0.5
x3 = I 
x5 = 1 + I 0.5

d[x_]=(x - x0)(x - x1)(x - x3)(x - x5)*
(x - Conjugate[x1])(x - Conjugate[x3])(x - Conjugate[x5])//Expand//Chop

n[x] = x^2 + 1

Integrate[n[x]/d[x],{x,-1,1}]

so =Solve[d[x] == 0.,x]

liro = {Re[x],Im[x]}/.so

plroots = ListPlot[liro, AspectRatio -> 1]

pathes = Graphics [{Line[{{-1,0},{-0.25,0.5},{1,0}}],Line[{{-1,0},{-0.25,-0.5},{1,0}}]}]

Show[pathes,plroots,Axes -> True, PlotRange -> All]

xp= N[x /. so[[1]]]

i1 = NIntegrate[n[x]/d[x],{x,-1,-0.25+ I/2,1}]

i2 = NIntegrate[n[x]/d[x],{x,-1,-0.25- I/2, 1}]

vint = (i1 + i2)/2//Chop