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n = Table[{140, 80, 57, 41, 31, 24, 20, 16, 13, 11, 10, 8, 7, 6, 5, 4,
     4, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1}];
f[j_, r_, x_, a_] :=
  n[[j]] Log[r x] + LogGamma[a + 1] + LogGamma[n[[j]]] +
   Log[Hypergeometric2F1[n[[j]], n[[j]], n[[j]] + a, x - r x]] -
   Log[x] - Log[n[[j]]] - LogGamma[n[[j]] + a] -
   Log[HypergeometricPFQ[{1, 1, 1}, {2, 1 + a},
      x] - (1 - r) HypergeometricPFQ[{1, 1, 1}, {2, 1 + a}, x - r x]];
L[r_, x_, a_] := Sum[N[f[j, r, x, a]], {j, 1, Length[n]}];
NMaximize[{L[x, r, a], 0 < x, x < 1, 0 < r, r < 0.2, a > 0}, {x, r, a}]

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