There is a method for bounding the cyclicity of nondegenerate monodromic singularities of polynomial planar families of vector fields Xλ which can work even in case that the Poincar´e first return map has associated a non-radical Bautin ideal B. The method is based on the stabilization of the integral closures of an ascending chain of polynomial ideals in the ring of polynomials in the parameters λ of the family that stabilizes at B. In this work we use computational algebra methods to provide an explicit example in which the classical procedure to find the Bautin depth of B fails but the new approach is successful.