1/(1-f+a(x-1/2)^2)

Integrate[%,x]

(%/.x->ze)-(%/.x->0)//FullSimplify

z=ze+f %

ParametricPlot[Evaluate[Table[{z,1-f+a(ze-1/2)^2}/.a->0.5,{f,0,0.9,0.1}]],{ze,0,1},PlotRange->All,AspectRatio->1/GoldenRatio,AxesOrigin->{0,0},AxesLabel->{"z","A"},PlotLabel->"Here α = 0.5",PlotLegends->Join[{f==0},Table[f,{f,0.1,0.9,0.1}]]]

z/.ze->1//FullSimplify

s=%

p1=ParametricPlot[Evaluate[{s,f}]/.a->0.5,{f,0,0.99},AspectRatio->1/GoldenRatio,PlotRange->{{0,20},{0,1}},AxesLabel->{"s","f"},PlotLabel->"α = 0.5"]

s-ss/.f->1-Pi^2/a/ss^2-f3/ss^3-f4/ss^4//Simplify//PowerExpand

Assuming[a>0&&f2>0&&ss>0,Series[%,{ss,Infinity,1}]//Simplify]

Solve[%==0,{f3,f4}]//Simplify

1-Pi^2/a/ss^2-f3/ss^3-f4/ss^4/.%[[1]]

Plot[Evaluate[%/.a->0.5],{ss,10,20},PlotStyle->Dashing[{0.01,0.01}]]

Show[p1,%,PlotLabel->"Comparison with asymptotics"]