This paper presents an efficient approximation algorithm to solve the task scheduling problem on heterogeneous platform for the particular case of the linear chain of tasks. The objective is to minimize both the total execution time (makespan) and the total energy consumed by the system. For this purpose, we introduce a constraint on the energy consumption during execution. Our goal is to provides an algorithm with a performance guarantee. Two algorithms have been proposed; the first provides an optimal solution for preemptive scheduling. This solution is then used in the second algorithm to provide an approximate solution for non-preemptive scheduling. Numerical evaluations demonstrate that the proposed algorithm achieves a close-to-optimal performance compared to exact solution obtained by CPLEX for small instances. For large instances, CPLEX is struggling to provide a feasible solution, whereas our approach takes less than a second to produce a solution for an instance of 10000 tasks.