We consider the representations of the group SO3(R) that contain the standard representation R3 as an irreducible component. This is a large class of representations containing the vector spaces of ternary forms of any odd degree, the space of Piezoelectric tensors and the classical product space R3 × . . . × R3 which appears in multiple physical and engineering situations. For any such representation we contruct a set of algebraically independent rational invariants that separate orbits outside of an identified hypersurface. As such they generate the field of rational invariants. Our key ingredient is a constructive use of Seshadri slice lemma. This latter establishes an isomor- phism between the SO3(R)-invariant field on the full representation R3 ⊕ H with the invariant field of O2(R) acting on a subspace R ⊕ H. The O2(R)-invariants are given explicitly as polynomials. We pre- dict the denominators of the related SO3(R)-invariants and show how to evaluate these latter thanks to Wigner matrices. The results are then extended to O3(R) actions and several cases of interest to applications are detailed as examples.