Solve x^3+x^2y+xy^2+y^3=81

$$x ^ { 3 } + x ^ { 2 } y + x y ^ { 2 } + y ^ { 3 } = 81$$

Solve for x

$$x=-\frac{\sqrt[3]{2}\left(1+\sqrt{3}i\right)\left(3\sqrt{3\left(16y^{6}-3240y^{3}+177147\right)}-20y^{3}+2187\right)^{-\frac{1}{3}}\left(2^{\frac{2}{3}}\left(-\sqrt{3}i+1\right)y\sqrt[3]{3\sqrt{3\left(16y^{6}-3240y^{3}+177147\right)}-20y^{3}+2187}+\sqrt[3]{2}\left(-\sqrt{3}i-1\right)\left(3\sqrt{3\left(16y^{6}-3240y^{3}+177147\right)}-20y^{3}+2187\right)^{\frac{2}{3}}-8y^{2}\right)}{24}$$

$$x=\frac{2^{\frac{2}{3}}\left(3\sqrt{3\left(16y^{6}-3240y^{3}+177147\right)}-20y^{3}+2187\right)^{-\frac{1}{3}}\left(-y\sqrt[3]{2\left(3\sqrt{3\left(16y^{6}-3240y^{3}+177147\right)}-20y^{3}+2187\right)}+\left(3\sqrt{3\left(16y^{6}-3240y^{3}+177147\right)}-20y^{3}+2187\right)^{\frac{2}{3}}-2\times 2^{\frac{2}{3}}y^{2}\right)}{6}$$

$$x=-\frac{\sqrt[3]{2}\left(-\sqrt{3}i+1\right)\left(3\sqrt{3\left(16y^{6}-3240y^{3}+177147\right)}-20y^{3}+2187\right)^{-\frac{1}{3}}\left(2^{\frac{2}{3}}\left(1+\sqrt{3}i\right)y\sqrt[3]{3\sqrt{3\left(16y^{6}-3240y^{3}+177147\right)}-20y^{3}+2187}+\sqrt[3]{2}\left(-1+\sqrt{3}i\right)\left(3\sqrt{3\left(16y^{6}-3240y^{3}+177147\right)}-20y^{3}+2187\right)^{\frac{2}{3}}-8y^{2}\right)}{24}$$

Solve for y

$$y=-\frac{\sqrt[3]{2}\left(1+\sqrt{3}i\right)\left(3\sqrt{3\left(16x^{6}-3240x^{3}+177147\right)}-20x^{3}+2187\right)^{-\frac{1}{3}}\left(2^{\frac{2}{3}}\left(-\sqrt{3}i+1\right)x\sqrt[3]{3\sqrt{3\left(16x^{6}-3240x^{3}+177147\right)}-20x^{3}+2187}+\sqrt[3]{2}\left(-\sqrt{3}i-1\right)\left(3\sqrt{3\left(16x^{6}-3240x^{3}+177147\right)}-20x^{3}+2187\right)^{\frac{2}{3}}-8x^{2}\right)}{24}$$

$$y=\frac{2^{\frac{2}{3}}\left(3\sqrt{3\left(16x^{6}-3240x^{3}+177147\right)}-20x^{3}+2187\right)^{-\frac{1}{3}}\left(-x\sqrt[3]{2\left(3\sqrt{3\left(16x^{6}-3240x^{3}+177147\right)}-20x^{3}+2187\right)}+\left(3\sqrt{3\left(16x^{6}-3240x^{3}+177147\right)}-20x^{3}+2187\right)^{\frac{2}{3}}-2\times 2^{\frac{2}{3}}x^{2}\right)}{6}$$

$$y=-\frac{\sqrt[3]{2}\left(-\sqrt{3}i+1\right)\left(3\sqrt{3\left(16x^{6}-3240x^{3}+177147\right)}-20x^{3}+2187\right)^{-\frac{1}{3}}\left(2^{\frac{2}{3}}\left(1+\sqrt{3}i\right)x\sqrt[3]{3\sqrt{3\left(16x^{6}-3240x^{3}+177147\right)}-20x^{3}+2187}+\sqrt[3]{2}\left(-1+\sqrt{3}i\right)\left(3\sqrt{3\left(16x^{6}-3240x^{3}+177147\right)}-20x^{3}+2187\right)^{\frac{2}{3}}-8x^{2}\right)}{24}$$