  In this article we perform a second order perturbation analysis of the
gravitational metric theory of gravity $ f(\chi) = \chi^{3/2} $ developed by
Bernal et al. (2011). We show that the theory accounts in detail for two
observational facts: (1) the phenomenology of flattened rotation curves
associated to the Tully-Fisher relation observed in spiral galaxies, and (2)
the details of observations of gravitational lensing in galaxies and groups of
galaxies, without the need of any dark matter. We show how all dynamical
observations on flat rotation curves and gravitational lensing can be
synthesised in terms of the empirically required metric coefficients of any
metric theory of gravity. We construct the corresponding metric components for
the theory presented at second order in perturbation, which are shown to be
perfectly compatible with the empirically derived ones. It is also shown that
under the theory being presented, in order to obtain a complete full agreement
with the observational results, a specific signature of Riemann's tensor has to
be chosen. This signature corresponds to the one most widely used nowadays in
relativity theory. Also, a computational program, the MEXICAS (Metric
EXtended-gravity Incorporated through a Computer Algebraic System) code,
developed for its usage in the Computer Algebraic System (CAS) Maxima for
working out perturbations on a metric theory of gravity, is presented and made
publicly available.
