  The computational technique of $N$-fold Mellin-Barnes (MB) integrals,
presented in a companion paper by the same authors, is used to derive sets of
series representations of the massive one-loop conformal 3-point Feynman
integral in various configurations. This shows the great simplicity and
efficiency of the method in nonresonant cases (generic propagator powers) as
well as some of its subtleties in the resonant ones (for unit propagator
powers). We confirm certain results in the physics and mathematics literature
and provide many new results, some of them dealing with the more general
massive one-loop conformal $n$-point case. In particular, we prove two recent
conjectures that give the massive one-loop conformal $n$-point integral (for
generic propagator powers) in terms of multiple hypergeometric series. We show
how these conjectures, that were deduced from a Yangian bootstrap analysis, are
related by a tower of new quadratic transformations in Hypergeometric Functions
Theory. Finally, we also use our MB method to identify spurious contributions
that can arise in the Yangian approach.
