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Not a general solution, but works for the case you cited;
(*Definitions*)
an=6;
bn=4;
h[0,x_]:=x; h[1,x_]:=1/x;
h[2,x_]:=1-x; h[3,x_]:=1/(1-x); h[4,x_]:=(x-1)/x;
f[x_] := Sum[a@i x^i,{i,an}] / Sum[b@i x^i,{i,bn}]; (*Proposed form / Rational*)
(*Now find the coefficients a[i] and b[i]*...