Ionization Fraction of Helium in alternate, He dominated early universe
My answer is incomplete (but still possibly helpful, only posting because I don't have enough "points" to comment).

I did a bit of the math involved but did not substitute terms with n's and eta's. However, from the result at the bottom of that page, it looks to me like a Binomial Expansion (truncated Taylor Expansion). The link shows the highest order for your terms (which is okay since higher order terms can usually be regarded as negligible) with the same +/- pattern you expect in your answer.