Pyramid 461
Preview $$U_{461}(x, y) = \frac{\left(y + 1\right)^{4} - \sqrt{4 x \left(y + 1\right)^{2} + \left(y + 1\right)^{8}}}{2 \left(y + 1\right)^{2}}$$
Pyramid 462
Preview $$U_{462}(x, y) = \frac{\left(y + 1\right)^{4}}{9 \sqrt[3]{\frac{x}{2 \left(y + 1\right)^{2}} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(y + 1\right)^{8}\right)}}{18 \left(y + 1\right)^{2}} + \frac{\left(y + 1\right)^{6}}{27}}} + \frac{\left(y + 1\right)^{2}}{3} + \sqrt[3]{\frac{x}{2 \left(y + 1\right)^{2}} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(y + 1\right)^{8}\right)}}{18 \left(y + 1\right)^{2}} + \frac{\left(y + 1\right)^{6}}{27}}$$
Pyramid 463
Preview $$U_{463}(x, y) = \frac{\left(1 - y\right) \left(\sqrt{\frac{4 x}{1 - y} + \frac{1}{\left(1 - y\right)^{4}}} + \frac{1}{\left(1 - y\right)^{2}}\right)}{2}$$
Pyramid 464
Preview $$U_{464}(x, y) = \frac{\left(1 - y\right) \left(- \sqrt{\frac{4 x}{1 - y} + \frac{1}{\left(1 - y\right)^{4}}} + \frac{1}{\left(1 - y\right)^{2}}\right)}{2}$$
Pyramid 465
Preview $$U_{465}(x, y) = \sqrt[3]{\frac{x \left(1 - y\right)}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{4}}\right)} \left(1 - y\right)}{18} + \frac{1}{27 \left(1 - y\right)^{3}}} + \frac{1}{3 - 3 y} + \frac{1}{9 \left(1 - y\right)^{2} \sqrt[3]{\frac{x \left(1 - y\right)}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{4}}\right)} \left(1 - y\right)}{18} + \frac{1}{27 \left(1 - y\right)^{3}}}}$$
Pyramid 466
Preview $$U_{466}(x, y) = \frac{1 - \sqrt{- 4 x y^{4} - 16 x y^{3} - 24 x y^{2} - 16 x y - 4 x + 1}}{2 x y + 2 x}$$
Pyramid 467
Preview $$U_{467}(x, y) = \sqrt[3]{\frac{x \left(1 - y\right)^{2}}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{8}}\right)} \left(1 - y\right)^{2}}{18} + \frac{1}{27 \left(1 - y\right)^{6}}} + \frac{1}{3 \left(1 - y\right)^{2}} + \frac{1}{9 \left(1 - y\right)^{4} \sqrt[3]{\frac{x \left(1 - y\right)^{2}}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{8}}\right)} \left(1 - y\right)^{2}}{18} + \frac{1}{27 \left(1 - y\right)^{6}}}}$$
Pyramid 468
Preview $$U_{468}(x, y) = \frac{\sqrt{3} \sqrt{y \left(16 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- x}}{2} \right)}}{3} \right)} - 12\right) + 3} - 3}{y \left(8 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- x}}{2} \right)}}{3} \right)} - 6\right)}$$
Pyramid 469
Preview $$U_{469}(x, y) = y \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- x}}{2} \right)}}{3} \right)}}{3}\right) + 1$$
Pyramid 470
Preview $$U_{470}(x, y) = \frac{\sqrt{y \left(4 - \frac{16 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- x}}{2} \right)}}{3} \right)}}{3}\right) + 1}}{2} + \frac{1}{2}$$
Pyramid 471
Preview $$U_{471}(x, y) = \frac{\sqrt{4 y + 1}}{6} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{36 \sqrt[3]{\frac{x}{\sqrt{4 y + 1} + 1} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{4}}{4}\right)}}{9 \sqrt{4 y + 1} + 9} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{216}}} + \sqrt[3]{\frac{x}{\sqrt{4 y + 1} + 1} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{4}}{4}\right)}}{9 \sqrt{4 y + 1} + 9} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{216}} + \frac{1}{6}$$
Pyramid 472
Preview $$U_{472}(x, y) = \frac{\left(1 - \sqrt{1 - 4 x}\right) \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)}{2 x}$$
Pyramid 473
Preview $$U_{473}(x, y) = - \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{2}}{x + \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3} - 1}$$
Pyramid 474
Preview $$U_{474}(x, y) = \frac{x + \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{2}}{1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}}$$
Pyramid 475
Preview $$U_{475}(x, y) = \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{2} + \sqrt{4 x \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right) + \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{4}}}{2 - \frac{8 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}}$$
Pyramid 476
Preview $$U_{476}(x, y) = \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{2} - \sqrt{4 x \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right) + \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{4}}}{2 - \frac{8 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}}$$
Pyramid 477
Preview $$U_{477}(x, y) = \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{2}}{9 \sqrt[3]{\frac{x}{2 - \frac{8 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{4}\right)}}{18 \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)} + \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{3}}{27}}} + \sqrt[3]{\frac{x}{2 - \frac{8 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}} + \frac{\sqrt{3} \sqrt{x \left(27 x + 4 \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{4}\right)}}{18 \left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)} + \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{3}\right)^{3}}{27}} - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- y}}{2} \right)}}{3} \right)}}{9} + \frac{1}{3}$$
Pyramid 478
Preview $$U_{478}(x, y) = \frac{\sqrt{3} \sqrt{y} \left(\sqrt{\frac{8 \sqrt{3} x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}} + \frac{16 \sin^{4}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 y^{2}}} + \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 y}\right)}{4 \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}$$
Pyramid 479
Preview $$U_{479}(x, y) = \frac{\sqrt{3} \sqrt{y} \left(- \sqrt{\frac{8 \sqrt{3} x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}} + \frac{16 \sin^{4}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 y^{2}}} + \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 y}\right)}{4 \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}$$
Pyramid 480
Preview $$U_{480}(x, y) = \frac{\left(y + 1\right)^{2} \left(- 2 x - \sqrt{1 - 4 x} + 1\right)}{2 x^{2}}$$
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