The version show the original post does not account for now-standard branch cuts, hence will not always evaluate numerically to a correct root.

(1) Don't you expect three solutions, not one? (Fundamental Theorem of Algebra.)

(2) Mathematica does have a command Simplify[], which helps.

(3) There will be some limitations on simplification because of branch cuts of the principal roots. If there are restrictions on c and d, including them in the assumptions of Simplify[] may help.

Another thing: Your formula is wrong for some values of the parameters c and d:

x^3 + 3 c x == 2 d /. 
   {x -> Power[d/2 + Sqrt[c^3/27 + d^2/4], (3)^-1] +
     Power[d/2 - Sqrt[c^3/27 + d^2/4], (3)^-1]} /.
  {c -> 2, d -> 1}

(*  False  *)