Applying the Extended Hamilton's Principle for nonmaterial volumes, a nonlinear reduced-order planar model of a cantilevered pipe conveying fluid is developed, consistently considering the effects of axial extensibility and conservation of mass associated to the internal flow. Unlike the corresponding inextensible pipe models, in which the term of transport of kinetic energy in the Extended Hamilton's Principle cancels out, in the present model such a term is not identically zero since the velocity of the flow along the pipe length is a function both of the generalized velocities and coordinates of the problem. The system dynamics is then investigated, assessing how extensibility and mass conservation affect dynamic bifurcations, by comparing root locus diagrams, and by simulating the resulting nonlinear model in some selected scenarios.