Find mass,  starting from density:

rho = 4*z + x*y + x

mass = Integrate[ rho, {z,0,1}, {x,-z,z}, {y,-z,z} ]

Now find the moments:

Myz = Integrate[  (rho)*x, {z,0,1}, {x,-z,z}, {y,-z,z} ]

Mxy = Integrate[  (rho)*z, {z,0,1}, {x,-z,z}, {y,-z,z} ]

Mxz = Integrate[  (rho)*y, {z,0,1}, {x,-z,z}, {y,-z,z} ]

Use the moments to calculate coordinates of the center of mass.

xBar = Myz / mass

yBar = Mxz / mass

zBar = Mxy / mass