$$U_{1037}(x, y) = \left(y + 1\right)^{7} \sqrt{4 x \left(y + 1\right)^{3} + 1}$$

$$Tsqrt(n, k) = \begin{cases}\binom{m}{n} 4^{n}&\text{if k even, where m =} \frac{k} {2},\\\frac {{(-1)}^{n-m} \binom{n}{m} \binom{2n}{n}} {\binom{2n}{2m}} &\text{if k odd, n > m, where m =} \frac{k+1} {2},\\\frac {\binom{2m}{2n} \binom{2n}{n}} {\binom{m}{n}}&\text{if k odd, n} \le \text{m, where m =} \frac{k+1} {2},\\\end{cases} $$$$T_{1037}(n, m, k) = \operatorname{Tsqrt}{\left(n,k \right)} {\binom{3 k + 3 n}{m}}$$