  The static dipole polarizability for the hydrogen molecular ions H$_2^+$,
HD$^+$, and D$_2^+$ are calculated. These new data for polarizability takes
into account the leading order relativistic corrections to the wave function of
the three-body system resulted from the Breit-Pauli Hamiltonian of $m\alpha^4$
order. Our study covers a wide range of rotational ($L=0-5$) and vibrational
($v=0-10$) states, which are of practical interest for precision spectroscopy
of the hydrogen molecular ions.
