We present a new analytical solution to compute the full-tensor gravity gradient due to a body mass of uniform density with arbitrary geometry. The solution is an extension of an existing analytical computation of gravitational anomalies of a polyhedron source, based on a transition of the general expressions from surface to line integrals. These developments enable the computation of the gravity gradient tensor using the same simple procedures as the gravitational field. The method is validated by comparing with a closed analytical solution, including on/in the near field of the body surface. The algorithm is implemented in the freely available MATLAB-based software called Gal Eötvös Earth Calculator. It is tested successfully for various measurement distances and body mass sizes, enabling applications from local geophysical prospecting to global topographic effect for satellite data. Due to its flexibility, the new solution, and the associated software, is particularly well suited for joint analyses of all types of gravity measurements regardless of the extent, altitude and irregularity of their spatial distribution.