Evaluating the disruptive nature of academic ideas is a new area of research evaluation that moves beyond standard citation-based metrics by taking into account the broader citation context of publications or patents. The "$CD$ index" and a number of related indicators have been proposed in order to characterise mathematically the disruptiveness of scientific publications or patents. This research area has generated a lot of attention in recent years, yet there is no general consensus on the significance and reliability of disruption indices. More experimentation and evaluation would be desirable, however is hampered by the fact that these indicators are expensive and time-consuming to calculate, especially if done at scale on large citation networks. We present a novel method to calculate disruption indices that leverages the Dimensions cloud-based research infrastructure and reduces the computational time taken to produce such indices by an order of magnitude, as well as making available such functionalities within an online environment that requires no set-up efforts. We explain the novel algorithm and describe how its results align with preexisting implementations of disruption indicators. This method will enable researchers to develop, validate and improve mathematical disruption models more quickly and with more precision, thus contributing to the development of this new research area.