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.

14 pages, 10 figures