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Given a rational homology 3-sphere M with the first integral homology of rankb and a link L inside M, colored by odd numbers, we construct a unifiedinvariant I_{M,L} belonging to a modification of the Habiro ring where b isinverted. 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 ofthe applications are the new Ohtsuki series (perturbative expansions of I_M atroots of unity) dominating all quantum SO(3) invariants.

A Unified Quantum SO(3) Invariant for Rational Homology 3-Spheres