Current article investigates the non-Newtonian MHD flow and heat transfer of copper-alumina/water hybrid nanofluid due to permeable stretching/shrinking surface with full slip model. The system of nonlinear partial differential equations was transformed to a system of ordinary differential equations via proper similarity variables. On obtaining exact solutions in terms of incomplete gamma function, the impact of various physical parameters was discussed for velocity and temperature distributions. There is an establishment that, the temperature distribution and thermal boundary layer increase as the one of the parameters viz, magnetic parameter, Biot number, radiation parameter is increasing. Additionally, transverse and axial velocities of the nanofluid are more than that of hybrid nanofluid in stretching case and reverse in shrinking case. However, the temperature for hybrid nanofluid will be more than that of nanofluid in stretching case and reverse in shrinking case. Further, present analysis reveals that stronger mass transpiration is needed for steady flow of MHD Casson liquid. Moreover, the skin friction will enhance with the mass suction and MHD in case of stretching sheet, where there is declines in rate of heat transfer. There is an increase in the skin friction as the intensity of suction parameter increases, accordingly which progress the enhancement of the heat transfer and deteriorate the temperature of the hybrid nanofluid. The consequence has possible industrial and technological applications in fluid based systems involving shrinkable/stretchable materials, MHD flow meters and pumps, plastic films drawing. Hybrid nanoparticles increase the rate of cooling, so become significant in machining and manufacturing.