I do not know Wolfram alpha pro, but the command in Mathematica are
Clear[x, y, z, g];
f = {z^3 - 2 x y^2, z^2 - 2 x^2 y, 3 x z^2 + 2 y z};
Curl[f, {x, y, z}] (*check first if conservative field*)
(* {0, 0, 0} *) (*ok it is*)
DSolve[{D[g[x, y, z], x]== f[[1]], D[g[x, y, z], y]== f[[2]], D[g[x, y, z], z]== f[[3]]},g[x, y, z],{x, y, z}]
(* {{g[x, y, z] -> (-x^2)*y^2 + y*z^2 + x*z^3 + C[1]}} *) (*this is the potential function g(x,y,z) *)