  We investigate the radial behavior of galactic rotation curves by a Fourth
Order Gravity adding also the Dark Matter component. The Fourth Order Gravity
is a Lagrangian containing the Ricci scalar, the Ricci and Riemann tensor, but
the rotation curves are depending only on two free parameters. A systematic
analysis of rotation curves, in the Newtonian Limit of theory, induced by all
galactic sub-structures of ordinary matter is shown. This analysis is presented
for Fourth Order Gravity with and without Dark Matter. The outcomes are
compared with respect to classical outcomes of General Relativity. The
gravitational potential of point-like mass is the usual potential corrected by
two Yukawa terms. The rotation curve is higher or also lower than curve of
General Relativity if in the Lagrangian the Ricci scalar square is dominant or
not with respect to the contribution of the Ricci tensor square. The curves are
compared with the experimental data for the Milky Way and the galaxy NGC 3198.
Although the Fourth Order Gravity gives more rotational contributions, in the
limit of large distances the Keplerian behavior is present, and it is missing
if we add a Dark Matter component. By modifying the theory of Gravitation
consequently also the spatial description of Dark Matter could undergo a
modification. At last we compare the gravitational potential by Fourth Order
Gravity with respect to more used potential induced by power law of Ricci
scalar.
