Minimize[{
  Gamma[x + 1] 100 (9 EulerGamma + Pi^2 x)
    - (657 EulerGamma + 900 EulerGamma^2 x + 73 Pi^2 x),
  0 < x < 1
}, x] // N

(*  {1.77889, {x -> 0.612041}}  *)

r[x_] := (657 EulerGamma + 900 EulerGamma^2 x + 73 Pi^2 x)/(100 (9 EulerGamma + Pi^2 x))
Minimize[{Gamma[x + 1] - r[x], 0 < x < 1}, x] // N

(*  {0.00158229, {x -> 0.613441}}  *)