Multiple definitions of casus irreducibilis In the case of cubic equations,

Casus irreducibilis occurs when none of the roots is rational and when all three roots are distinct and real (...)
  —Wikipedia's Casus irreducibilis article

So, $x^3-3x+1=0$ is definitely an example of casus irreducibilis.
Cardano's formula can express a rational root in terms of non-real radicals (yet it is unnecessary), as in this example: $x^3-15x-4=0$. Some (Working with casus irreducibilis) call this equation a casus irreducibilis, but this disagrees with the (supposed) Wikipedia definition (which is described below), as it has a rational solution, namely $x=\sqrt[3]{2+11i}+\sqrt[3]{2-11i}=4$.
Does the question in the link just involve a misinterpretation of casus irreducibilis, or are there any trustworthy books or other sources which support the fact that equations like $x^3-15x-4=0$ (which yield a rational root through Cardano's formula, though unnecessarily, using roots of complex numbers) are casus irreducibilis?
I suppose that the Wikipedia definition should read

Casus irreducibilis occurs if and only if none of the roots is rational and if and only if all three roots are distinct and real (...)

instead, as this defines casus irreducibilis precisely.
 A: The "irreducibilis" part of casus irreducibilis is irreducibility over the rationals. Hence $x^3-15x-4=0$ is not casus irreducibilis.
The linked question, however, has not really misused the term.

My question is, using Cardano's method for casus irreducibilis...

implies that it is about a situation where the rational root test isn't used beforehand, and the equation is assumed casus irreducibilis; we want to tell if the root obtained from Cardano's formula is really a rational in disguise.
A: Historically casus irreducibilis simply meant the case where the discriminant is negative regardless of the existence of rational roots.
For instance, Lagrange's lectures on elementary mathematics discusses the irreducible case in connection with Bombelli's example $x^3 = 15x + 4$, which was chosen because it has an integer solution and so can be used to test the cubic formula.
The idea that "irreducible case" means the cubic polynomial has to be irreducible seems to be a modern idea from field and Galois theory, projected backward.  Before Galois the concern was with generic solution formulas where the coefficients of the polynomial are parameters.   Solvability and unsolvability of particular equations not in a parametric family was not really subject to a general theory until Galois placed it in the context of field extensions, Galois group actions and so on.
In short, the Wikipedia is probably wrong, but for the subsequent modern analysis of whether casus irreducibilis can be circumvented, of course the first thing to require is that there be no rational solution.
