A bring radical is a series solution for a trinomial. A trinomial has a nice single-summation solution via the Lagrange inversion theorem. A 4-term expression requires a second summation. A general quintic can be made into in a trinomial through solving a quartic, but a sextic cannot. The sextic will be of the form x^6+ax^2+bx+d=0. Removing the x^2 term will requires solving a quintic, so you end up having to apply a series to an infinite series: First for the bring radical and then for the sextic trinomial. Likewise, trying to invert the sextic x^6+ax^2+bx+d=0 will require a double summation. One can check that the Lagrange inversion theorem will not generate a single series for a general 4-term expression.

A bring radical is a series solution for a quintic that can be derived via the Lagrange inversion theorem on the Bring-Jerrard quintic form. A trinomial has a nice single-summation solution via the Lagrange inversion theorem. A 4-term expression requires a second summation. A general quintic can be made into in a trinomial through solving a quartic, but a sextic cannot. The sextic will be of the form x^6+ax^2+bx+d=0. Removing the x^2 term will requires solving a quintic, so you end up having to apply a series to an infinite series: First for the bring radical and then for the sextic trinomial. Likewise, trying to invert the sextic x^6+ax^2+bx+d=0 will require a double summation. One can check that the Lagrange inversion theorem will not generate a single series for a general 4-term expression.