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This paper introduces a new analytical framework to evaluate the capacity of intelligent reconfigurable surface (IRS)-aided wireless networks in the presence of a direct link (DL). The obtained analysis is used to characterize the signal-to-noise ratio (SNR) at the user equipment (UE) while using adaptive power and rate transmission. In particular, we consider the channel inversion with a fixed rate, optimum power and rate adaptation, and the truncated channel inversion with a fixed rate. The obtained expressions are derived in a unified closed-form. All the single-hop channel gains are modeled as independent and identically distributed Nakagami-m fading channels. Consequently, the channels’ gains at the receiver become independent and non-identically distributed. The moment generating function (MGF) is used to derive an accurate approximation of the probability density and cumulative distribution functions of the instantaneous SNR, which are used to evaluate the channel capacity at low and high SNRs to quantify the achievable multiplexing gain. The obtained analytical and simulation results indicated that a strong DL may significantly enhance the channel capacity gain obtained using the IRS. In particular scenarios, the capacity improved by about 30% for a large number of IRS elements when the DL Nakagami fading parameter m is increased from 2 to 6.