Simplify the polynomial $xy(x+y)+(x+y)+(x+y)^2=13xy$ to the form $y^2=4x^3-g_{2}x-g_{3}$ 
I try to simplify a polynomial to the form: $y^2=4x^3-g_{2}x-g_{3}$, which is the elliptic curves. And the polynomial is $xy(x+y)+(x+y)+(x+y)^2=13xy$. 

I try to let the $u=x+y$ and $v=x-y$, then I get the $u^3-uv^2+4u-9u^2+13v^2=0$. But how to get ahead?

You made a good start but there was a complication you did not anticipate.
The homogeneous version of your equation is:
$\, 0 = -W X Y Z + (X+Y)(X+Z)(Y+Z), \,$ where $\, W=13 \,$ is a constant. Now substitute $\, X = 1 + c_3 x + \sqrt{c_1} y, $
$\, Y = 1 + c_3 x - \sqrt{c_1} y, $ $\, Z = 2 x + c_2, \,$
where $\,c_1,c_2,c_3 \,$ depend on $\,W.\,$ After the substitutions, we eliminate the $\, x y^2 \,$ term with $\, c_3 = W. \,$
We eliminate the $\,x^2\,$ term with
$\, c_2 = (2W^2 +16W + 8)/(W^3 - 4W^2 - 8W). \,$ Now the coefficients of $\,y^2\,$ and of $\,x^3\,$ needs adjustment to get the final form. Let $\, c_1 = (-W^3 + 4W^2 + 8W)/12 .\,$
The equation now is
$\, 0 = 2W(1+W)( -y^2 + 4x^3 - g_2x - g_3), \,$ where
$$\, g_2 = \frac{12(W^4 - 8W^3 + 16W + 16)}{W^2(W^2 - 4W - 8)^2},
  \quad g_3 = \frac{8(W^4 - 8W^3 - 8W - 8)}{W^3(W^2 - 4W - 8)^2}. $$
In our case of $\, W=13 \,$ these invariants become
$\, g_2 = 134508/2007889, \, g_3 = 86984/26102557. \,$
There is a $2$-torsion point $\, (-1/13,0). \,$ There is a generator point
$\, (-97/1417, 168 \sqrt{-3}/1417^{3/2}). \,$
After noticing the $1417$ in the denominators, we can scale $\,x,y\,$ to simplify the equation. After scaling it is
$\, 0 = -y^2 + 4x^3 - 134508x - 9481256, \,$ with generator point
$\, (-97, 168 \sqrt{-3}). \,$

This is the elliptic curve with LMFDB label 8190.bw4 which has no rational generator.
