The full title of the book is Multivariable model building: A pragmatic approach to regression analysis based on fractional polynomials for modelling continuous variables . It’s a good book. It clearly presents its rationale for using a restricted set of fractional polynomials instead of either linear terms or splines.
I don’t agree with all of it. I think there is too much emphasis on statistical significance rather than practical importance; if a complicated fractional polynomial does a tiny bit better than a linear model, but the difference is significant because of large sample size, then which should we choose?
The authors also do not deal with missing data, merely noting that it makes things much more complicated. Indeed it does. And methods that work with linear regression may not work well with fractional polynomials.
Nevertheless, I may wind up using fractional polynomials in some future work and I think the book merits reading and could be a source of further research.