An irreducible quintic $f(x)\in\mathbb{Q}[x]$, is solvable by radicals if and only if its sextic resolvent $\theta_f (y)=(y^3+py^2+qy+r)^2-2^{10}\Delta(f)y$ has a rational root ($\Delta$ is the discriminant of $f$, and $a,b,c$ are some defined rationals, see the Cox reference).

In many cases, this associated sextic is not pretty. More troublesome is that in my work I am often coming across quintics with the coefficients in terms of some parameter, where in general we cannot use the Rational Root Test on the sextic.

Any advice on proving the (non)existence of a rational root of the sextic resolvent of an irreducible rational quintic (especially sufficient criteria for non-existence)? Examples or references to special cases are fine. Is the usual technique just ad-hoc tricks?

References:

Cox, Galois Theory, Theorem 13.2.6, p372

Dummit & Foote, Abstract Algebra, Problem 14.7.21, p639