This work considers the design and performance analysis of multirate non-orthogonal multiple access (MR-NOMA) with arbitrary rates and various modulation orders, where closed-form bit error rate (BER) expressions are derived for the joint-multiuser maximum likelihood sequence detector (JMLSD) and low-complexity optimal successive interference cancellation (SIC) receiver. The derived expressions are used to optimize the power assignment (PA) at the base station to minimize the BER while strictly satisfying certain BER requirements. The presented results show that MR-NOMA can increase the trade-off freedom between spectral efficiency and robustness to errors due to its ability to control the energy by varying the power and symbol duration, simultaneously, while single-rate non-orthogonal multiple access (SR-NOMA) can only control the power only. The mathematical derivations are validated via Monte Carlo simulation.