The choice of an appropriate strain energy function W is key to accurate modeling and computational finite element analysis of the mechanical behavior of unfilled non-crystalizing rubberlike materials. Despite the existing variety of models, finding a suitable model that can capture many deformation modes of a rubber specimen with a single set of parameter values and satisfy the a priori mathematical and structural requirements remains a formidable task. Previous work proposed a new generalized neo-Hookean W (I1) function (doi: 10.1016/j.ijnonlinmec.2020.103626), showing a promising fitting capability and enjoying a structural basis. In this work we use two extended forms of that model which include an I2 term adjunct, W(I1,I2), for application to various boundary value problems commonly encountered in rubber mechanics applications. Specifically, two functional forms of the I2 invariant are considered: a linear function and a logarithmic function. The boundary value problems of interest include the in-plane uniaxial, equi-biaxial, and pure shear deformations and simple shear, inflation, and nonhomogeneous deformations such as torsion. By simultaneous fitting of each model to various deformation modes of rubber specimens, it is demonstrated that a single set of model parameter values favorably captures the mechanical response for all the considered deformations of each specimen. It is further shown that the model with a logarithmic I2 function provides better fits than the linear function. Given the functional simplicity of the considered W (I1, I2) models, the low number of model parameters (three in total), the structurally motivated bases of the models, and their capability to capture the mechanical response for various deformations of rubber specimens, the considered models are recommended as a powerful tool for practical applications and analysis of rubber elasticity