Given a rational homology 3-sphere M with |H 1 (M,ℤ)|=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=∅,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 . Important applications are the new Ohtsuki series (perturbative expansions of I M ) dominating quantum SO(3) invariants at roots of unity whose order is not a power of a prime. These series are not known to be determined by the LMO invariant.