We know that, the "greater" the degree of a polynomial equation, the greater the "complexity" of the roots in general.
For example, the overall complexity of the roots of a general cubic equation is obviously more complicated than the general quadratic. However, the overall complexity of the roots of a general quartic equation is "greater" than any of them. We understand this when we look at the general formulas.
Motivation:
I'm wondering how much "more" the complexity of the roots of a solvable quintic equation can be than the general quartic.
But, here I would like to observe this "event" on the precise example quintic, rather than just Galois theory.
For example, I would like to "compare" a solvable "most complex" quintic with the complexity of the roots of the equation $$2x^4+5x^3+7x^2+13x+3=0:$$
Then, I found the this solvable quintic:
$$x^5-5x+12=0$$
and the exact real root equals to:
However, I do not have information that this is included in the examples I am looking for.
In short, I am looking for a precise example for the solvable quintic whose roots are the most complicated.
Of course, the example this type of quintic itself can be interesting. But, I would like to see its roots expressed in radicals.
At least I know that the equation I'm looking for will be irreducible.