Higher-order gravity models: corrections up to cubic curvature invariants and inflation

We construct a higher-order gravity model including all corrections up to mass dimension six. Starting from the Jordan frame, we derive the field equations and specialize to the FLRW background, where the dynamics take the form of a four-dimensional autonomous system. Focusing on the $R+R^{2}+RR_{\mu\nu}R^{\mu\nu}$ case, we obtain linearized equations in the parameter $\gamma_{0}$ and analyze the resulting phase space. The model exhibits the main desirable features of an inflationary regime, with a slow-roll attractor and a stable critical point corresponding to the end of inflation. Analytical expressions for the scalar spectral index $n_{s}$ and the tensor-to-scalar ratio $r$ show that the model is consistent with Planck, BICEP/Keck, and BAO data if $|\gamma_{0}|\lesssim 10^{-3}$. Moreover, negative values of $\gamma_{0}$ restore compatibility with recent ACT, Planck, and DESI results, suggesting that higher-order corrections may be relevant in refining inflationary cosmology.