Are you sure that your expression has a minimum? It seems to me that the infimum is attained as D tends to -Infinity.
The structure of your expression is clearer if treated this way:
coeffs = Cases[F, _Real, All] // Union;
subst1 = Thread[coeffs -> Table[c[i], {i, Length[coeffs]}]];
c0 = (coeffs[[-4]]/12);
subst5 = Map[Reverse,
Cases[Rationalize[subst1[[1 ;; 9]]], Except[_Real -> _]]];
subst2 = Thread[
Table[c[i], {i, 10, 13}] -> c[0]*Rationalize[coeffs[[-4 ;;]]/c0]];
subst3 = subst1 /. subst2;
F2 = Simplify[F /. subst3 /. subst5]
subst4 = c[0] (-D + c[2]) :> Log[x];
F3 = FullSimplify[F2 /. subst4]
F4 = F3 /. Map[Reverse, subst1]
Plot[F4, {x, -1.98^(1/15), 5}]
where x == Exp[c[0] (-D + c[2])] is a new variable.