Find $ \int_0^2 \int_0^2\sqrt{5x^2+5y^2+8xy+1}\hspace{1mm}dy\hspace{1mm}dx$ I need the approximation to four decimals
Not sure how to start or if a closed form solution exists
All Ideas are appreciated

Possible hint
It is not the most pleasant integral. However, using a CAS, the following result was obtained $$\int\sqrt{5x^2+5y^2+8xy+1}\hspace{1mm}dy=\sqrt{5 x^2+8 x y+5 y^2+1} \left(\frac{2 x}{5}+\frac{y}{2}\right)+\frac{\left(9
   x^2+5\right) \log \left(\sqrt{5} \sqrt{5 x^2+8 x y+5 y^2+1}+4 x+5 y\right)}{10
   \sqrt{5}}$$ So $$\int_0^2\sqrt{5x^2+5y^2+8xy+1}\hspace{1mm}dy=\frac{1}{50} \left(-20 \sqrt{5 x^2+1} x-\sqrt{5} \left(9 x^2+5\right) \log
   \left(\sqrt{25 x^2+5}+4 x\right)+\sqrt{5} \left(9 x^2+5\right) \log \left(4
   x+\sqrt{5} \sqrt{x (5 x+16)+21}+10\right)+10 (2 x+5) \sqrt{x (5 x+16)+21}\right)$$ Looking at this result over the range $0\leq x \leq 2$ reveals a function which looks very  linear. So I hope that a simple quadrature method would lead to the required result.
