Cannot solve $ y = xy'+x^3(y')^2 $

This equation has four solutions all roughly as awful as this one (produced by Mathematica 11.3): $$ y(x) = -\frac{\sqrt{3} x \sqrt{-\frac{c_1^2 x^3-\sqrt[3]{-c_1^6 x^9+2160 c_1^8 x^7+93312 c_1^{10} x^5+48 \sqrt{3} \sqrt{c_1^{14} \left(-x^{10}\right) \left(x^2-108 c_1^2\right){}^3}}-\frac{c_1^4 x^6}{\sqrt[3]{-c_1^6 x^9+2160 c_1^8 x^7+93312 c_1^{10} x^5+48 \sqrt{3} \sqrt{c_1^{14} \left(-x^{10}\right) \left(x^2-108 c_1^2\right){}^3}}}-\frac{864 c_1^6 x^4}{\sqrt[3]{-c_1^6 x^9+2160 c_1^8 x^7+93312 c_1^{10} x^5+48 \sqrt{3} \sqrt{c_1^{14} \left(-x^{10}\right) \left(x^2-108 c_1^2\right){}^3}}}-48 c_1^4 x}{c_1^4 x^3}}+\sqrt{3} x \sqrt{-\frac{x \left(864 c_1^2+x^2\right)}{\sqrt[3]{-c_1^6 x^9+2160 c_1^8 x^7+93312 c_1^{10} x^5+48 \sqrt{3} \sqrt{c_1^{14} \left(-x^{10}\right) \left(x^2-108 c_1^2\right){}^3}}}-\frac{\sqrt[3]{-c_1^6 x^9+2160 c_1^8 x^7+93312 c_1^{10} x^5+48 \sqrt{3} \sqrt{c_1^{14} \left(-x^{10}\right) \left(x^2-108 c_1^2\right){}^3}}}{c_1^4 x^3}+\frac{96 \sqrt{3} \left(4 c_1^2+x^2\right)}{c_1^2 x^3 \sqrt{-\frac{c_1^2 x^3-\sqrt[3]{-c_1^6 x^9+2160 c_1^8 x^7+93312 c_1^{10} x^5+48 \sqrt{3} \sqrt{c_1^{14} \left(-x^{10}\right) \left(x^2-108 c_1^2\right){}^3}}-\frac{c_1^4 x^6}{\sqrt[3]{-c_1^6 x^9+2160 c_1^8 x^7+93312 c_1^{10} x^5+48 \sqrt{3} \sqrt{c_1^{14} \left(-x^{10}\right) \left(x^2-108 c_1^2\right){}^3}}}-\frac{864 c_1^6 x^4}{\sqrt[3]{-c_1^6 x^9+2160 c_1^8 x^7+93312 c_1^{10} x^5+48 \sqrt{3} \sqrt{c_1^{14} \left(-x^{10}\right) \left(x^2-108 c_1^2\right){}^3}}}-48 c_1^4 x}{c_1^4 x^3}}}-\frac{2}{c_1^2}+\frac{96}{x^2}}-36}{24 x} \text{,}$$ where $c_1$ is the constant of integration.

(It is likely that this cannot be rendered intelligibly by the MathJax interperter. However, the point is that this solution has no hope of being elementary, which seems to be adequately indicated.)