  The deviation from geodesic motion of the world line of an extended body
endowed with multipolar structure up to the mass quadrupole moment is studied
in the Kerr background according to the Mathisson-Papapetrou-Dixon model. The
properties of the quadrupole tensor are clarified by identifying the relevant
components which enter the equations of motion, leading to the definition of an
effective quadrupole tensor sharing its own algebraic symmetries, but also
obeying those implied by the Mathisson-Papapetrou-Dixon model itself. The
equations of motion are then solved analytically in the limit of small values
of the characteristic length scales associated with the spin and quadrupole
variables in comparison with the one associated with the background curvature
and under special assumptions on body's structure and motion. The resulting
quasi-circular orbit is parametrized in a Keplerian-like form, so that
temporal, radial and azimuthal eccentricities as well as semi-major axis,
period and periastron advance are explicitly computed and expressed in terms of
gauge-invariant variables in the weak field and slow motion limit. A companion
numerical study of the equations of motion is performed too.
