The increasing capacities of large language models (LLMs) present an unprecedented opportunity to scale up data analytics in the humanities and social sciences, augmenting and automating qualitative analytic tasks previously typically allocated to human labor. This contribution proposes a systematic mixed methods framework to harness qualitative analytic expertise, machine scalability, and rigorous quantification, with attention to transparency and replicability. 16 machine-assisted case studies are showcased as proof of concept. Tasks include linguistic and discourse analysis, lexical semantic change detection, interview analysis, historical event cause inference and text mining, detection of political stance, text and idea reuse, genre composition in literature and film; social network inference, automated lexicography, missing metadata augmentation, and multimodal visual cultural analytics. In contrast to the focus on English in the emerging LLM applicability literature, many examples here deal with scenarios involving smaller languages and historical texts prone to digitization distortions. In all but the most difficult tasks requiring expert knowledge, generative LLMs can demonstrably serve as viable research instruments. LLM (and human) annotations may contain errors and variation, but the agreement rate can and should be accounted for in subsequent statistical modeling; a bootstrapping approach is discussed. The replications among the case studies illustrate how tasks previously requiring potentially months of team effort and complex computational pipelines, can now be accomplished by an LLM-assisted scholar in a fraction of the time. Importantly, this approach is not intended to replace, but to augment researcher knowledge and skills. With these opportunities in sight, qualitative expertise and the ability to pose insightful questions have arguably never been more critical.