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This paper presents a theoretical framework to analyze the performance of integrated unmanned aerial vehicle (UAV)-intelligent reflecting surface (IRS) relaying system in which IRS provides an additional degree of freedom combined with the flexible deployment of full-duplex UAV to enhance communication between ground nodes. Our framework considers three different transmission modes: {\bf (i)} UAV-only mode, {\bf (ii)} IRS-only mode, and {\bf (iii)} integrated UAV-IRS mode to achieve spectral and energy-efficient relaying. For the proposed modes, we provide exact and approximate expressions for the end-to-end outage probability, ergodic capacity, and energy efficiency (EE) in closed-form. We use the derived expressions to optimize key system parameters such as the UAV altitude and the number of elements on the IRS considering different modes. We formulate the problems in the form of fractional programming (e.g. single ratio, sum of multiple ratios or maximization-minimization of ratios) and devise optimal algorithms using quadratic transformations. Furthermore, we derive an analytic criterion to optimally select different transmission modes to maximize ergodic capacity and EE for a given number of IRS elements. Numerical results validate the derived expressions with Monte-Carlo simulations and the proposed optimization algorithms with the solutions obtained through exhaustive search. Insights are drawn related to the different communication modes, optimal number of IRS elements, and optimal UAV height.