\[ \boxed { 3\,x \left ( {x}^{2}-1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +x \left ( y \left ( x \right ) \right ) ^{2}- \left ( {x}^{2}+1 \right ) y \left ( x \right ) -3\,x=0} \]
Mathematica: cpu = 1.566699 (sec), leaf count = 2816 \[ \left \{\left \{y(x)\to \frac {3 \left (x^2-1\right ) \left (\frac {e^{\int _1^x \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} \text {Root}\left [125 x^8-164 x^6+70 x^4-20 x^2+\left (1296 x^{12}-5184 x^{10}+7776 x^8-5184 x^6+1296 x^4\right ) \text {$\#$1}^4+\left (-3456 x^{11}+12096 x^9-15552 x^7+8640 x^5-1728 x^3\right ) \text {$\#$1}^3+\left (3240 x^{10}-9504 x^8+9936 x^6-4320 x^4+648 x^2\right ) \text {$\#$1}^2+\left (-1200 x^9+2736 x^7-2160 x^5+720 x^3-96 x\right ) \text {$\#$1}+5\& ,1\right ] \int _1^x e^{-2 \int _1^{K[2]} \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} \, dK[2]}{\sqrt [6]{x} \sqrt [3]{1-x^2}}-\frac {e^{\int _1^x \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} \int _1^x e^{-2 \int _1^{K[2]} \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} \, dK[2]}{6 x^{7/6} \sqrt [3]{1-x^2}}+\frac {2 e^{\int _1^x \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} x^{5/6} \int _1^x e^{-2 \int _1^{K[2]} \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} \, dK[2]}{3 \left (1-x^2\right )^{4/3}}+c_1 \left (\frac {e^{\int _1^x \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} \text {Root}\left [125 x^8-164 x^6+70 x^4-20 x^2+\left (1296 x^{12}-5184 x^{10}+7776 x^8-5184 x^6+1296 x^4\right ) \text {$\#$1}^4+\left (-3456 x^{11}+12096 x^9-15552 x^7+8640 x^5-1728 x^3\right ) \text {$\#$1}^3+\left (3240 x^{10}-9504 x^8+9936 x^6-4320 x^4+648 x^2\right ) \text {$\#$1}^2+\left (-1200 x^9+2736 x^7-2160 x^5+720 x^3-96 x\right ) \text {$\#$1}+5\& ,1\right ]}{\sqrt [6]{x} \sqrt [3]{1-x^2}}-\frac {e^{\int _1^x \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]}}{6 x^{7/6} \sqrt [3]{1-x^2}}+\frac {2 e^{\int _1^x \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} x^{5/6}}{3 \left (1-x^2\right )^{4/3}}\right )+\frac {e^{-\int _1^x \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]}}{\sqrt [6]{x} \sqrt [3]{1-x^2}}\right )}{\frac {e^{\int _1^x \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} c_1}{\sqrt [6]{x} \sqrt [3]{1-x^2}}+\frac {e^{\int _1^x \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} \int _1^x e^{-2 \int _1^{K[2]} \text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {$\#$1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {$\#$1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {$\#$1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {$\#$1}+5\& ,1\right ] \, dK[1]} \, dK[2]}{\sqrt [6]{x} \sqrt [3]{1-x^2}}}\right \}\right \} \]
Maple: cpu = 0.109 (sec), leaf count = 145 \[ \left \{ y \left ( x \right ) ={\frac {35\,{\it \_C1}\,{x}^{4}-35\,{\it \_C1}\,{x}^{2}}{8} {\mbox {$_2$F$_1$}({\frac {11}{6}},{\frac {13}{6}};\,{\frac {7}{3}};\,{x}^{2})} {\frac {1}{\sqrt [3]{x}}} \left ( {x}^{{\frac {2}{3}}} {\mbox {$_2$F$_1$}({\frac {5}{6}},{\frac {7}{6}};\,{\frac {4}{3}};\,{x}^{2})} {\it \_C1}+ {\mbox {$_2$F$_1$}({\frac {1}{2}},{\frac {5}{6}};\,{\frac {2}{3}};\,{x}^{2})} \right ) ^{-1}}+{\frac {1}{8} \left ( \left ( 40\,{\it \_C1}\,{x}^{2}- 16\,{\it \_C1} \right ) {\mbox {$_2$F$_1$}({\frac {5}{6}},{\frac {7}{6}};\,{\frac {4}{3}};\,{x}^{2})} + \left ( 30\,{x}^{10/3}-30\,{x}^{4/3} \right ) {\mbox {$_2$F$_1$}({\frac {3}{2}},{\frac {11}{6}};\,{\frac {5}{3}};\,{x}^{2})} +24\,{\mbox {$_2$F$_1$}(1/2,5/6;\,2/3;\,{x}^{2})}{x}^{4/3} \right ) { \frac {1}{\sqrt [3]{x}}} \left ( {x}^{{\frac {2}{3}}} {\mbox {$_2$F$_1$}({\frac {5}{6}},{\frac {7}{6}};\,{\frac {4}{3}};\,{x}^{2})} {\it \_C1}+ {\mbox {$_2$F$_1$}({\frac {1}{2}},{\frac {5}{6}};\,{\frac {2}{3}};\,{x}^{2})} \right ) ^{-1}} \right \} \]