9.4.22 Curvilinear integrals

Problem: Having checked that the integrand is a total differential, calculate the integral: \[ \int_{(1,1,1)}^{(2,3,4)}\left(2 x y+y^{2}+y z^{2}\right) d x+\left(x^{2}+2 x y+x z^{2}\right) d y+2 x y z d z \]