The solution is complicated and even cannot understand.
Why can’t Mathematica give a simple expression like this?
(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 *)
The result given has three solutions, all generically correct (that is, correct off a measure-0 set in the parameter space). The one proposed solution, in contrast, is not generically correct.
Thank you for your reply, I am still confused that you mean the Cardano’s formula is not generically correct?
Could you please explain more? Your help would be highly appreciated.
The version show the original post does not account for now-standard branch cuts, hence will not always evaluate numerically to a correct root.