In this paper, we first present a simple seven-term 4D hyperchaotic system based on the classical Sprott-C 3D chaotic system. This novel system is inspired by the simple 4D hyperchaotic system based on Sprott-B proposed by A. T. Sheet (2022). We discuss the phenomenon of premature divergence brought about by the improper choice of coupling parameters in that paper and describe the basic properties of the new system with phase diagrams, Lyapunov exponential spectra and bifurcation diagrams. Then, we find that the dynamical behaviors of the system suffer from the limitation of the control parameters and cannot represent the process of motion in detail. To improve the system, we expand the dimensionality and add the control parameters and memristors. A 5D memristive hyperchaotic system with hidden attractors is proposed, and the basic dynamical properties of the system, such as its dissipation, equilibrium point, stability, Lyapunov exponential spectra and bifurcation diagram, are analyzed. Finally, the hardware circuits of the 4D Sprott-C system and the 5D memristive hyperchaotic system were realized by a field programmable gate array (FPGA) and verified by an experiment. The experimental results are consistent with the numerical simulation results obtained in MATLAB, which demonstrates the feasibility and potential of the system.