In this short communication, we deal with an integrability analysis of nonlinear three-dimensional differential systems. Right-hand sides of these systems are linear in one variable, which enables one to find explicitly a particular solution and to calculate variational equations along this solution. The conditions for the complete integrability with two functionally independent rational first integrals for B-integrability and the partial integrability are obtained from an analysis of properties of the differential Galois group of variational equations. They have a very simple form of numbers, which is necessary to check whether they are appropriate integers. An application of the obtained conditions to some exemplary nonlinear three-dimensional differential systems is shown.