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We compute 1-loop corrections to the redshift space galaxy power spectrum in cosmologies containing additional scales, and hence kernels different from Einstein-de Sitter (EdS). Specifically, our method is tailored for cosmologies in the presence of massive neutrinos and some modified gravity models; in this article we concentrate on the former case. The perturbative kernels have contributions that we notice appear either from the logarithmic growth factor $f(k,t)$, which is scale-dependent because of the neutrino free-streaming, or from the failure of the commonly used approximation $f^2=Ω_m$. The latter contributions make the computation of loop corrections quite slow, precluding full-shape analyses for parameter estimation. However, we identify that the dominant pieces of the kernels come from the growth factor, allowing us to simplify the kernels but retaining the characteristic free-streaming scale introduced by the neutrinos' mass. Moreover, with this simplification one can exploit FFTLog methods to speed up the computations even more. We validate our analytical modeling and numerical method with halo catalogs extracted from the Quijote simulations finding good agreement with the, a priori, known cosmological parameters. We make public our Python code FOLPS$ν$ to compute the redshift space power spectrum in a fraction of second. Code available at https://github.com/henoriega/FOLPS-nu.