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Continuous integration (CI) is a well-established technique in commercial and open-source software projects, although not routinely used in scientific publishing. In the scientific software context, CI can serve two functions to increase reproducibility of scientific results: providing an established platform for testing the reproducibility of these results, and demonstrating to other scientists how the code and data generate the published results. We explore scientific software testing and CI strategies using two articles published in the areas of applied mathematics and computational physics. We discuss lessons learned from reproducing these articles as well as examine and discuss existing tests. We introduce the notion of a scientific test as one that produces computational results from a published article. We then consider full result reproduction within a CI environment. If authors find their work too time or resource intensive to easily adapt to a CI context, we recommend the inclusion of results from reduced versions of their work (e.g., run at lower resolution, with shorter time scales, with smaller data sets) alongside their primary results within their article. While these smaller versions may be less interesting scientifically, they can serve to verify that published code and data are working properly. We demonstrate such reduction tests on the two articles studied.