The nonlinear vibrations of a pipe conveying fluid with simple supports are investigated in this paper. The equation governing transverse motion of the pipe with partial-differential nonlinearity is derived from the Newton′s second law. Under the quasi-static stretch assumption, another governing equation with integro-partial-differential nonlinearity can be obtained, which is widely adopted vibrations of fluid-conveying pipes. The multiple scale method is applied directly to the two types of governing equations with different nonlinear terms. The natural frequencies of different velocities are presented for the two types of nonlinearity. It can be concluded that the integro-partial-differential nonlinearity is much weaker than the partial-differential one.