  Given a rational homology 3-sphere M with the first integral homology of rank
b and a link L inside M, colored by odd numbers, we construct a unified
invariant I_{M,L} belonging to a modification of the Habiro ring where b is
inverted. Our unified invariant dominates the whole set of the SO(3)
Witten-Reshetikhin-Turaev invariants of the pair (M,L). If b=1 and L is empty,
I_M coincides with Habiro's invariant of integral homology 3-spheres. For b>1,
the unified invariant defined by the third author is determined by I_M. One of
the applications are the new Ohtsuki series (perturbative expansions of I_M at
roots of unity) dominating all quantum SO(3) invariants.
