Evaluate $\int_{S}f$ where $f(x,y,z)=xz+y^2$ and $S$ is the surface parametrized by $\alpha(x,y)=(x^2,y^2/2,xy)\ \forall (x,y)\in[0,1]\times[0,2]$

Question

I have tried the problem in the following manner
By definition $\int_S f=\int_R f\circ \alpha \left\|D_1\alpha\times D_2\alpha\right\|\ldots(1)$
$D_1\alpha(x,y)=(2x,0,y)$ and $D_2\alpha(x,y)=(0,y,x)$. Simple calculation shows $D_1\alpha\times D_2\alpha=(-y^2,-2x^2,2xy)\implies\left\|D_1\alpha\times D_2\alpha\right\|=\sqrt{y^4+4x^4+4x^2y^2}$
From $(1)$, $\int_S f==\int\limits_{x=0}^{1}\int\limits_{y=0}^{2}(x^3y+y^4/4)\sqrt{y^4+4x^4+4x^2y^2}\ dy\ dx$
From this I can proceed further, the calculations look too much difficult. Can I use any change of variable to simplify the integration? Thanks for assistance in advance.