Fractional calculus yields numerous implementations in different fields such as biological materials, physical memory, oscillation, wave propagation, and viscoelastic dynamics. Due to the significant applications of fractional calculus, the current study deals with the fractional derivative base study of a Brinkman hybrid nanofluid with an inclined magnetic field. A three-parametric Prabhakar fractional derivative with the involvement of the Mittag-Leffler function is implemented. A vertical plate moving with exponential velocity is considered to be the source of the flow mechanism. Moreover, the effects of exponential heating are incorporated into the thermal analysis. An appropriate group of dimensionless ansatz is adopted to get the dimensionless setup of equations. The Prabhakar fractional operator is implemented in the dimensionless equations which are further tackled by an effectual Laplace transform technique. An inverse Stehfest method and Tzou's method are implemented to tackle the inversion of the Laplace transform. This study exhibits that the fractional constraints minimize both the fields of temperature and velocity. Moreover, the velocity distribution deteriorates corresponding to the improved Brinkman parameter. The Brinkman parameter and the fluid's viscosity are directly related to each other. With the improved Brinkman parameter, the viscosity of the fluid increases. As a result, the fluid motion decreases.