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• The axisymmetric restricted five-body problem within the frame of variable mass has been studied. • The positions of libration points are illustrated. • The basins of convergence are unveiled. • The basin entropy is presented. The present manuscript reveals the existence and stability of the libration points (LPs) in the axisymmetric restricted five-body problem (AR5BP) in which mass of the fifth particle varies with respect to time as per Jeans' law. It has been assumed that the four bodies P i with masses m i , i = 1 , 2 , ... , 4 (when m 3 = m 4 = m ˜) form an axisymmetric convex configuration. The system of differential equations which govern the motion of the test particle (whose mass is variable) moving under the gravitational influence of the four primaries, have been presented, indeed, these equations are different from those of the test particle with constant mass in the AR5BP. We have determined the in-plane and out-of-plane LPs along with their linear stability. Furthermore, it is observed that the existence of these LPs depends not only on the angle parameters α and β but also on the parameters occur due to variation in mass namely γ (0 < γ ≤ 1) and σ (0 ≤ σ ≤ 2.2 , a proportionality constant occurs in Jeans' law). Moreover, we have investigated the evolution of the regions of possible motion as function of variable mass parameters where the fifth body can move freely. The topology of the basins of convergence (BoC) linked with the LPs as function of σ is unveiled by deploying the bivariate version of Newton-Raphson (NR) iterative method, and, in order to measure the uncertainty of the basins, the basin entropy is also evaluated. The co-relations between the domains of basins of convergence (DoBoC) and the required number of iterations to achieve predefined accuracy, and also with the corresponding probability distributions are illustrated. [ABSTRACT FROM AUTHOR]