Community search is a problem that seeks cohesive and connected subgraphs in a graph that satisfy certain topology constraints, e.g., degree constraints. The majority of existing works focus exclusively on the topology and ignore the nodes' influence in the communities. To tackle this deficiency, influential community search is further proposed to include the node's influence. Each node has a weight, namely influence value, in the influential community search problem to represent its network influence. The influence value of a community is produced by an aggregated function, e.g., max, min, avg, and sum, over the influence values of the nodes in the same community. The objective of the influential community search problem is to locate the top-r communities with the highest influence values while satisfying the topology constraints. Existing studies on influential community search have several limitations: (i) they focus exclusively on simple aggregation functions such as min, which may fall short of certain requirements in many real-world scenarios, and (ii) they impose no limitation on the size of the community, whereas most real-world scenarios do. This motivates us to conduct a new study to fill this gap. We consider the problem of identifying the top-r influential communities with/without size constraints while using more complicated aggregation functions such as sum or avg. We give a theoretical analysis demonstrating the hardness of the problems and propose efficient and effective heuristic solutions for our topr influential community search problems. Extensive experiments on real large graphs demonstrate that our proposed solution is significantly more efficient than baseline solutions.