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Your $f$ can be simplified to:

v*(u + v)*(-x^2 - 6*(x + y)*z - 5*z^2 + 6*(x + y + z)^2 + (u + 2*v)*(x + 2*y + z)) + 
 (x + y)*(x + y + z)*(u^2 + x*(x + 2*y + z) + (u + 2*v)*(2*x + 2*y + z))

Is it what you wanted?

Your $f$ can be simplified to:

v*(u + v)*(-x^2 - 6*(x + y)*z - 5*z^2 + 6*(x v)*(6*(x + y + z)^2 - (x^2 + (u 6*(x + 2*v)*(x y)*z + 5*z^2) + (x + 2*y + z)) z)*(u +  2*v)) + (x + y)*(x + y + z)*(u^2 + x*(x + 2*y + z) + (u + 2*v)*(2*x + 2*y + z))

Is it what you wanted?

Your $f$ can be simplified to:

v*(u + v)*(6*(x + y + z)^2 - (x^2 + 6*(x + y)*z + 5*z^2) + (x + 2*y + z)*(u + 2*v)) +  (x + y)*(x + y + z)*(u^2 + x*(x + 2*y + z) + (u + 2*v)*(2*x + 2*y + z))

Is it what you wanted?

Your $f$ can be simplified to:

v*(u + v)*(6*(x + y + z)^2 - (x^2 + 6*(x + y)*z + 5*z^2) + (x + 2*y + z)*(u + 2*v)) +  (x + y)*(x + y + z)*(u^2 + x*(x + 2*y + z) + (u + 2*v)*(2*x + 2*y + z))

Is it what you wanted?

Your $f$ can be simplified to:

v*(u + v)*(6*(x + y + z)^2 - (x^2 + 6*(x + y)*z + 5*z^2) + (x + 2*y + z)*(u + 2*v)) + (x + y)*(x + y + z)*(u^2 + x*(x + 2*y + z) + (u + 2*v)*(2*x + 2*y + z))

or if you prefer it without minus $-$ sign:

v*(u + v)*(5*x^2 + 12*x*y + 6*y^2 + 6*x*z + 6*y*z + z^2 + (x + 2*y + z)*(u + 2*v)) + (x + y)*(x + y + z)*(u^2 + x*(x + 2*y + z) + (u + 2*v)*(2*x + 2*y + z))

Is it what you wanted?

Your $f$ can be simplified to:

v*(u + v)*(6*(x + y + z)^2 - (x^2 + 6*(x + y)*z + 5*z^2) + (x + 2*y + z)*(u + 2*v)) + (x + y)*(x + y + z)*(u^2 + x*(x + 2*y + z) + (u + 2*v)*(2*x + 2*y + z))

or if you prefer it without minus $-$ sign:

v*(u + v)*(5*x^2 v)*((x + 12*x*y z)*(5*x + 6*y^2 z) + 6*x*z 6*y*(2*x + 6*y*z y + z^2 z) + (x + 2*y + z)*(u + 2*v)) + (x + y)*(x + y + z)*(u^2 + x*(x + 2*y + z) + (u + 2*v)*(2*x + 2*y + z))

Is it what you wanted?