The finite cell method (FCM) belongs to the class of immersed boundary methods, and combines the fictitious domain approach with high-order approximation, adaptive integration and weak imposition of unfitted Dirichlet boundary conditions. Its main idea consists of the extension of the physical domain of interest beyond its potentially complex boundaries into a larger embedding domain of simple geometry, which can be meshed easily by a structured grid. We present an isogeometric design-through-analysis methodology based on the B-spline version of the finite cell method, which allows for the seamless integration of fully three-dimensional parameterizations of complex engineering parts described by T-spline surfaces into finite element analysis. The approach is demonstrated to achieve optimal rates of convergence and to yield accurate stress results not only within the domain of interest, but also directly on the immersed boundary. We also show that hierarchical refinement of B-splines considerably increases the flexibility of the immersed boundary approach in terms of adaptive resolution of local features in the geometry and the solution fields. At the same time, hierarchical refinement maintains the key advantage of fully automated mesh generation for complex geometries due to its simplicity and straightforward implementation. We illustrate the versatility of our methodology by two complex industrial examples of a ship propeller and an automobile wheel.