The initial aim of the talk is to present the description of an effective method for computing d-invariants of infinite families of Brieskorn spheres. Associating our work with the recent results in involutive Floer homology, these spheres will be new 3-manifolds generating infinite rank summands in the integral homology cobordism group. The next intention of the talk is to bring your attention to new infinite families of Brieskorn spheres bounding rational homology balls but not integral homology balls. They will be useful in understanding the homomorphism between homology cobordism groups. The first part is a joint work with Çağrı Karakurt.