$$U_{1177}(x, y) = \frac{\sqrt{4 x + 1} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{2 y^{2}}$$

$$Tsqrt(n, k) = \begin{cases}\binom{m}{n} 4^{n}&\text{if k even, where m =} \frac{k} {2},\\\frac {{(-1)}^{n-m} \binom{n}{m} \binom{2n}{n}} {\binom{2n}{2m}} &\text{if k odd, n > m, where m =} \frac{k+1} {2},\\\frac {\binom{2m}{2n} \binom{2n}{n}} {\binom{m}{n}}&\text{if k odd, n} \le \text{m, where m =} \frac{k+1} {2},\\\end{cases} $$$$T_{1177}(n, m, k) = \frac{k \operatorname{Tsqrt}{\left(n,k \right)} {\binom{2 k + 2 m}{m}}}{k + m}$$