Pyramid 401
Preview $$U_{401}(x, y) = \sqrt[3]{\frac{x \left(1 - \sqrt{1 - 4 y}\right)}{4 y} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(1 - \sqrt{1 - 4 y}\right)^{2}}{y^{2}}\right)} \left(1 - \sqrt{1 - 4 y}\right)}{36 y} + \frac{\left(1 - \sqrt{1 - 4 y}\right)^{3}}{216 y^{3}}} + \frac{1 - \sqrt{1 - 4 y}}{6 y} + \frac{\left(1 - \sqrt{1 - 4 y}\right)^{2}}{36 y^{2} \sqrt[3]{\frac{x \left(1 - \sqrt{1 - 4 y}\right)}{4 y} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(1 - \sqrt{1 - 4 y}\right)^{2}}{y^{2}}\right)} \left(1 - \sqrt{1 - 4 y}\right)}{36 y} + \frac{\left(1 - \sqrt{1 - 4 y}\right)^{3}}{216 y^{3}}}}$$
Pyramid 402
Preview $$U_{402}(x, y) = \frac{\left(x + 1\right) \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{2 y^{2}}$$
Pyramid 403
Preview $$U_{403}(x, y) = \frac{- 2 y - \sqrt{1 - 4 y} + 1}{x \left(2 y + \sqrt{1 - 4 y} - 1\right) + 2 y^{2}}$$
Pyramid 404
Preview $$U_{404}(x, y) = \frac{- y^{2} + \sqrt{x \sqrt{1 - 4 y} \left(2 - 4 y\right) + x \left(- 4 y^{2} + 8 y - 2\right) + y^{4}}}{x \sqrt{1 - 4 y} + x \left(2 y - 1\right)}$$
Pyramid 405
Preview $$U_{405}(x, y) = \frac{\sqrt{\frac{2 x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{y^{2}} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{4 y^{4}}}}{2} + \frac{- 2 y - \sqrt{1 - 4 y} + 1}{4 y^{2}}$$
Pyramid 406
Preview $$U_{406}(x, y) = - \frac{\sqrt{\frac{2 x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{y^{2}} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{4 y^{4}}}}{2} + \frac{- 2 y - \sqrt{1 - 4 y} + 1}{4 y^{2}}$$
Pyramid 407
Preview $$U_{407}(x, y) = \sqrt[3]{\frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{4 y^{2}} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{y^{4}}\right)} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{36 y^{2}} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{216 y^{6}}} + \frac{- 2 y - \sqrt{1 - 4 y} + 1}{6 y^{2}} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{36 y^{4} \sqrt[3]{\frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{4 y^{2}} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{y^{4}}\right)} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{36 y^{2}} + \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{216 y^{6}}}}$$
Pyramid 408
Preview $$U_{408}(x, y) = - \frac{\sqrt{\frac{8 \sqrt{3} x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}} + \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 y}}}{2} + \frac{\sqrt{3} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}}$$
Pyramid 409
Preview $$U_{409}(x, y) = \sqrt[3]{\frac{\sqrt{3} x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}} + \frac{\sqrt{x \left(27 x + \frac{16 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 y}\right)} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 \sqrt{y}} + \frac{8 \sqrt{3} \sin^{3}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{243 y^{\frac{3}{2}}}} + \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{27 y \sqrt[3]{\frac{\sqrt{3} x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}} + \frac{\sqrt{x \left(27 x + \frac{16 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 y}\right)} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 \sqrt{y}} + \frac{8 \sqrt{3} \sin^{3}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{243 y^{\frac{3}{2}}}}} + \frac{2 \sqrt{3} \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 \sqrt{y}}$$
Pyramid 410
Preview $$U_{410}(x, y) = \frac{\sqrt{4 y + 1}}{4} + \frac{\sqrt{2 x \left(\sqrt{4 y + 1} + 1\right) + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4}}}{2} + \frac{1}{4}$$
Pyramid 411
Preview $$U_{411}(x, y) = \frac{\sqrt{4 y + 1}}{4} - \frac{\sqrt{2 x \left(\sqrt{4 y + 1} + 1\right) + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4}}}{2} + \frac{1}{4}$$
Pyramid 412
Preview $$U_{412}(x, y) = \frac{\sqrt{4 y + 1}}{6} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{36 \sqrt[3]{\frac{x \left(\sqrt{4 y + 1} + 1\right)}{4} + \frac{\sqrt{3} \sqrt{x \left(27 x + \left(\sqrt{4 y + 1} + 1\right)^{2}\right)} \left(\sqrt{4 y + 1} + 1\right)}{36} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{216}}} + \sqrt[3]{\frac{x \left(\sqrt{4 y + 1} + 1\right)}{4} + \frac{\sqrt{3} \sqrt{x \left(27 x + \left(\sqrt{4 y + 1} + 1\right)^{2}\right)} \left(\sqrt{4 y + 1} + 1\right)}{36} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{216}} + \frac{1}{6}$$
Pyramid 413
Preview $$U_{413}(x, y) = x + \frac{\sqrt{4 y + 1}}{2} + \frac{1}{2}$$
Pyramid 414
Preview $$U_{414}(x, y) = \frac{\sqrt{4 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4}}}{2} + \frac{\sqrt{4 y + 1}}{4} + \frac{1}{4}$$
Pyramid 415
Preview $$U_{415}(x, y) = - \frac{\sqrt{4 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4}}}{2} + \frac{\sqrt{4 y + 1}}{4} + \frac{1}{4}$$
Pyramid 416
Preview $$U_{416}(x, y) = \frac{\sqrt{4 y + 1}}{6} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{36 \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{2}\right)}}{18} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{216}}} + \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{2}\right)}}{18} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{3}}{216}} + \frac{1}{6}$$
Pyramid 417
Preview $$U_{417}(x, y) = \frac{1}{\left(1 - x\right) \left(1 - y\right)^{2}}$$
Pyramid 418
Preview $$U_{418}(x, y) = \frac{y + \sqrt{- 4 x + y^{2} - 2 y + 1} - 1}{2 x y - 2 x}$$
Pyramid 419
Preview $$U_{419}(x, y) = \frac{- y \left(2 - 2 x\right) - \sqrt{y \left(4 - 4 x\right) + 1} - 1}{2 x - 2}$$
Pyramid 420
Preview $$U_{420}(x, y) = \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{6}}\right)}}{18} + \frac{1}{27 \left(1 - y\right)^{6}}} + \frac{1}{3 \left(1 - y\right)^{2}} + \frac{1}{9 \left(1 - y\right)^{4} \sqrt[3]{\frac{x}{2} + \frac{\sqrt{3} \sqrt{x \left(27 x + \frac{4}{\left(1 - y\right)^{6}}\right)}}{18} + \frac{1}{27 \left(1 - y\right)^{6}}}}$$
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