How can I find the integer numbers a, b, c, m,n so that the funtion f[x]=(a x^2 + b x + c)/(m x + n) with the equation D[f[x], x]] == 0 have two integer solutions x1, x2 and the values f[x1] and f[x2] are also integer numbers?

I tried

Clear["Global`*"]
f[x_] = (a x^2 + b x + c)/(m x + n);
x1 = (-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
x2 = (-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
list = Table[
  If[GCD[a, b, c] == 1 && GCD[m, n] == 1 && 
    a c m^2 - a b m n + a^2 n^2 > 0 && a n - b m  != 0 && 
    a (c m^2 - b m n + a n^2) > 0 && b^2 - 4 * a * c < 0 && 
    IntegerQ[x1] && IntegerQ[x2] && b c m n x1 x2 f[x1]* f[x2] != 0 &&
     IntegerQ[f[x1]] && 
    IntegerQ[f[x2]], {(a x^2 + b x  + c)/(m x + n), {x1, f[x1]}, {x2, 
     f[x2]}}, Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]

How can I remove the empty set in my results?