There exist a family of quintic polynomials, called Brumer's polynomials (or Kondo-Brumer), which have the form:
$$x^5+(a-3)x^4+(-a+b+3)x^3+(a^2-a-1-2b)x^2+bx+a,~~~a,b \in \mathbb{Q}$$
According to Wikipedia these polynomials are solvable in radicals.
Is there a general formula for roots of these polynomials? Or at least the closed form for some special cases?
I searched the web, but only found papers discussing the group properties (for example here) or other properties of Brumer's polynomials. Nothing about the roots.
Edit
I'm starting a bounty, and I would like either of these things:
General solution (at least one root), depending on $a,b$ - only if such a solution exists, and is short enough to write here in closed form.
Some methods for obtaining this solution - again, if it will lead to a form of the solution more compact and simpler than the general way to solve an arbitrary solvable quintic.
Solutions to some special cases (for some values of $a,b$ with $b \neq 0$)
A proof that no such simple solution is possible for this family of quintics.