How to calculate this kind of double definite integral directly

Let $ D=\left\{(x, y) \mid x^{2}+y^{2} \leq \sqrt{2}, x \geq 0, y \geq 0\right\}$ , $ \left[1+x^{2}+y^{2}\right]$ represents the largest integer not greater than $ 1+x^{2}+y^{2}$ , now I want to calculate this double integral $ \iint_{D} x y\left[1+x^{2}+y^{2}\right] d x d y$ .

reg = ImplicitRegion[x^2 + y^2 <= Sqrt[2] && x >= 0 && y >= 0, {x, y}]; Integrate[x*y*Round[1 + x^2 + y^2], {x, y} ∈ reg] 

But the result I calculated using the above method is not correct, the answer is $ \frac{3}{8}$ , what should I do to directly calculate this double integral (without using the technique of turning double integral into iterated integral)?