  We implement the effective field theory for gravitating spinning objects in
the post-Newtonian scheme at the next-to-next-to-leading order level to derive
the gravitational spin-orbit interaction potential at the third and a half
post-Newtonian order for rapidly rotating compact objects. From the
next-to-next-to-leading order interaction potential, which we obtain here in a
Lagrangian form for the first time, we derive straightforwardly the
corresponding Hamiltonian. The spin-orbit sector constitutes the most elaborate
spin dependent sector at each order, and accordingly we encounter a
proliferation of the relevant Feynman diagrams, and a significant increase of
the computational complexity. We present the evaluation of the interaction
potential, going over contributing Feynman diagrams. The computation is carried
out in terms of the nonrelativistic gravitational fields, together with the
various gauge choices included in the effective field theory for gravitating
spinning objects, which optimize the calculation. In addition, we automatize
the effective field theory computations, and carry out the automated
computations in parallel. Such automated effective field theory computations
would be most useful to obtain higher order post-Newtonian corrections. We
compare our Hamiltonian to the ADM Hamiltonian, and arrive at a complete
agreement between the ADM and effective field theory results. We provide
complete gauge invariant relations among the binding energy, angular momentum,
and orbital frequency of an inspiralling binary with generic compact spinning
components to third and a half post-Newtonian order. The derivation presented
here is essential to obtain further higher order post-Newtonian corrections,
and to reach the accuracy level required for the successful detection of
gravitational radiation.
