  Based on the minimal length concept, inspired by Heisenberg algebra, a closed
analytical formula is derived for the energy spectrum of the prolate
{\gamma}-rigid Bohr-Mottelson Hamiltonian of nuclei, within a quantum
perturbation method (QPM), by considering a scaled Davidson potential in
\b{eta} shape variable. In the resulting solution, called X(3)-D-ML, the ground
state and the first \b{eta}-band are all studied as a function of the free
parameters. The fact of introducing the minimal length concept with a QPM makes
the model very flexible and a powerful approach to describe nuclear collective
excitations of a variety of vibrational-like nuclei. The introduction of
scaling parameters in the Davidson potential enables us to get a physical
minimum of this latter in comparison with previous works. The analysis of the
corrected wave function, as well as the probability density distribution, shows
that the minimal length parameter has a physical upper bound limit.
