A diophantine problem from calculus

A univariate polynomial f(x) is said to be nice if all of its coefficients as well as all of the roots of both f(x) and its derivative f′(x) are integers. The known examples of nice polynomials with distinct roots are limited to quadratic polynomials, cubic polynomials, symmetric quartic polynomials and, up to equivalence, only a finite number of nonsymmetric quartic polynomials and one quintic polynomial. In this paper we find parametrized families of nice nonsymmetric quartic polynomials with distinct roots, as well as infinitely many nice quintic and sextic polynomials with distinct roots.