  Large-scale structure surveys in the coming years will measure the
redshift-space power spectrum to unprecedented accuracy, allowing for powerful
new tests of the LambdaCDM picture as well as measurements of particle physics
parameters such as the neutrino masses. We extend the Time-RG perturbative
framework to redshift space, computing the power spectrum P_s(k,mu) in massive
neutrino cosmologies with time-dependent dark energy equations of state w(z).
Time-RG is uniquely capable of incorporating scale-dependent growth into the
P_s(k,mu) computation, which is important for massive neutrinos as well as
modified gravity models. Although changes to w(z) and the neutrino mass
fraction both affect the late-time scale-dependence of the non-linear power
spectrum, we find that the two effects depend differently on the line-of-sight
angle mu. Finally, we use the HACC N-body code to quantify errors in the
perturbative calculations. For a LambdaCDM model at redshift z=1, our procedure
predicts the monopole~(quadrupole) to 1% accuracy up to a wave number 0.19h/Mpc
(0.28h/Mpc), compared to 0.08h/Mpc (0.07h/Mpc) for the Kaiser approximation and
0.19h/Mpc (0.16h/Mpc) for the current state-of-the-art perturbation scheme. Our
calculation agrees with the simulated redshift-space power spectrum even for
neutrino masses above the current bound, and for rapidly-evolving dark energy
equations of state, |dw/dz| ~ 1. Along with this article, we make our
redshift-space Time-RG implementation publicly available as the code redTime.
