The contribution presents anextensionandapplicationof a recently proposed finite element formulation for quasi-inextensible and quasi-incompressible finite hyperelasticity to fibrous soft biological tissues and touches in particular upon computational aspects thereof. In line with theoretical framework presented by Dal (Int J Numer Methods Eng 117:118-140, 2019), the mixed variational formulation is extended to two families of fibers as often encountered while dealing with fibrous tissues. Apart from that, the purelyEuleriansetting features the additive decomposition of the free energy function into volumetric, isotropic and anisotropic parts. The multiplicative split of the deformation gradient and all the outcomes thereof, e.g., unimodular invariants, are simply dispensed with in the three element formulations investigated, namelyQ1,Q1P0and the proposedQ1P0F0. For the quasi-incompressible response, theQ1P0element formulation is briefly outlined where the pressure-type Lagrange multiplier and its conjugate enter the variational formulation as an extended set of variables. Using the similar argumentation, an extended Hu-Washizu-type mixed variational potential is introduced where the volume averaged squares of fiber stretches and associated fiber stresses are additional field variables. The resulting finite element formulation calledQ1P0F0is very attractive as it is based on mean values of the additional field variables at element level through integration over the element domain in a preprocessing step, earning the model vast utilization areas. The proposed approach is examined through representative boundary value problems pertaining to fibrous biological tissues. For all the cases studied, the proposedQ1P0F0formulation elicits the most compliant mechanical response, thereby outperforming the standardQ1andQ1P0element formulations through mesh-refinement analyses. Results prompt further experimental investigations as to true deformation fields under biologically relevant loading conditions which would make the assessment ofQ1P0andQ1P0F0more based on physical grounds.