New integrable models and analytical solutions in $f(R)$~cosmology with an ideal gas

In the context of $f\left(R\right)$-gravity with a spatially flat FLRW metric containing an ideal fluid, we use the method of invariant transformations to specify families of models which are integrable. We find three families of $f(R)$ theories for which new analytical solutions are given and closed-form solutions are provided.