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.
