  We present an integral representation for the tensor product $L$-function of
a pair of automorphic cuspidal representations, one of a classical group, the
other of a general linear group. Our construction is uniform over all classical
groups, and is applicable to all cuspidal representations; it does not require
genericity. The main new ideas of the construction are the use of generalized
Speh representations as inducing data for the Eisenstein series and the
introduction of a new (global and local) model, which generalizes the Whittaker
model.
  This is the first in a series of papers, treating symplectic and even
orthogonal groups. Subsequent papers (in preparation) will treat odd orthogonal
and general spin groups, the metaplectic covering version of these integrals,
and applications to functoriality coming from combining this work with the
converse theorem (and independent of the trace formula).
