Find $ \int_0^2 \int_0^2\sqrt{5x^2+5y^2+8xy+1}\hspace{1mm}dy\hspace{1mm}dx$

Question

I need the approximation to four decimals

Not sure how to start or if a closed form solution exists

All Ideas are appreciated

Answer 1

In Maple environment:

[> s:= Int(sqrt(5*x^2+8*x*y+5*y^2+1), x = 0 .. 2);
           /2                          (1/2)   
          |   /   2              2    \        
     s:=  |   \5 x  + 8 x y + 5 y  + 1/      dx
          |
          |                                    
         /0                                    

[> int(s, y = 0 .. 2, numeric);

                                  17.71654322

Answer 2

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.