Integer programming feasibility is NP-what 
What is the complexity class of the general problem of integer programming feasibility?

The sources I've looked at are, in my opinion, very confusing. Some say NP-hard, some say NP-complete. Some do not distinguish between the general problem and the binary case. Some do not distinguish between the optimization problem and the feasibility problem. As far as I can tell, there appears to be no correlation between these three factors. I have only seen an actual proof of any claim in the binary feasibility case, which is NP-complete.
If I understand correctly, NP-completeness is a stronger condition than NP-hardness, since NP-hard problems do not need to be NP (or decision problems, but I can ignore that because I'm only interested in feasibility). So I am not surprised that the two are used interchangeably around for the binary case.
A reference would also be greatly appreciated.
 A: Edit 29 March 2015:  As D.W. points out in a comment, Sahni's "P-hard" and "P-complete" do not mean what one might expect from modern usage.  His terms are (roughly) conditional on P=NP.  His section 1.1, Definition 4 is a clear example: "A problem $L$ is P-Hard iff a polynomial algorithm for $L$ implies P = NP."  As D.W. asserts, this eliminates the discrepancy between Sahni and G&J:  the problem is NP-hard/-complete, depending on the signs of the constraint coefficients.
P-hard/-complete, depending on the signs of the constraint coefficients.  See Sahni, S., "Computationally Related Problems" SIAM J. Computing vol. 3, pp. 262-279, 1974.  Also here (Section 2.5, I1 and I2).
Referenced as "A6 MP1" at Garey and Johnson, "Computers and Intractability:  A Guide to the Theory of NP-Completeness", 1979.  However, there must be an error in this reference,  G&J assert NP-hard/-complete, but Sahni asserts P-Hard (positive or negative constraint coefficients) and P-complete (positive constraints).
