\[ \boxed { {x}^{2} \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +a \left ( y \left ( x \right ) \right ) ^{2} \right ) +b{x}^{\alpha }+c=0} \]
Mathematica: cpu = 0.168521 (sec), leaf count = 1712 \[ \left \{\left \{y(x)\to \frac {a^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {i \sqrt {4 a c-1}}{2 \alpha }} \alpha ^{-\frac {i \sqrt {4 a c-1} \alpha +\alpha }{\alpha ^2}+\frac {i \sqrt {4 a c-1}}{\alpha }+1} b^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {i \sqrt {4 a c-1}}{2 \alpha }} \left (\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {i \sqrt {4 a c-1}}{2 \alpha }\right ) \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {i \sqrt {4 a c-1}}{2 \alpha }-1} J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (\frac {\sqrt {1-4 a c}}{\alpha }+1\right ) x^{\alpha -1}+\frac {1}{2} a^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {i \sqrt {4 a c-1}}{2 \alpha }+\frac {1}{2}} \alpha ^{\frac {i \sqrt {4 a c-1}}{\alpha }-\frac {i \sqrt {4 a c-1} \alpha +\alpha }{\alpha ^2}} b^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {i \sqrt {4 a c-1}}{2 \alpha }+\frac {1}{2}} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {i \sqrt {4 a c-1}}{2 \alpha }-\frac {1}{2}} \left (J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )-J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}+1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )\right ) \Gamma \left (\frac {\sqrt {1-4 a c}}{\alpha }+1\right ) x^{\alpha -1}+c_1 \left (a^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {i \sqrt {4 a c-1}}{2 \alpha }} \alpha ^{-\frac {\alpha -i \alpha \sqrt {4 a c-1}}{\alpha ^2}-\frac {i \sqrt {4 a c-1}}{\alpha }+1} b^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {i \sqrt {4 a c-1}}{2 \alpha }} \left (\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {i \sqrt {4 a c-1}}{2 \alpha }\right ) \left (x^{\alpha }\right )^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {i \sqrt {4 a c-1}}{2 \alpha }-1} J_{-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) x^{\alpha -1}+\frac {1}{2} a^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {i \sqrt {4 a c-1}}{2 \alpha }+\frac {1}{2}} \alpha ^{-\frac {\alpha -i \alpha \sqrt {4 a c-1}}{\alpha ^2}-\frac {i \sqrt {4 a c-1}}{\alpha }} b^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {i \sqrt {4 a c-1}}{2 \alpha }+\frac {1}{2}} \left (x^{\alpha }\right )^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {i \sqrt {4 a c-1}}{2 \alpha }-\frac {1}{2}} \left (J_{-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )-J_{1-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )\right ) \Gamma \left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) x^{\alpha -1}\right )}{a \left (a^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {i \sqrt {4 a c-1}}{2 \alpha }} b^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {i \sqrt {4 a c-1}}{2 \alpha }} \left (x^{\alpha }\right )^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {i \sqrt {4 a c-1}}{2 \alpha }} J_{-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) c_1 \Gamma \left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) \alpha ^{-\frac {\alpha -i \alpha \sqrt {4 a c-1}}{\alpha ^2}-\frac {i \sqrt {4 a c-1}}{\alpha }}+a^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {i \sqrt {4 a c-1}}{2 \alpha }} b^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {i \sqrt {4 a c-1}}{2 \alpha }} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {i \sqrt {4 a c-1}}{2 \alpha }} J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (\frac {\sqrt {1-4 a c}}{\alpha }+1\right ) \alpha ^{\frac {i \sqrt {4 a c-1}}{\alpha }-\frac {i \sqrt {4 a c-1} \alpha +\alpha }{\alpha ^2}}\right )}\right \}\right \} \]
Maple: cpu = 0.078 (sec), leaf count = 244 \[ \left \{ y \left ( x \right ) =-{\frac {1}{2\,ax} \left ( \left ( -\sqrt {-4\,ac+1}{\it \_C1}-{\it \_C1} \right ) {{\sl Y}_{{\frac {1}{\alpha } \sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }} \right )}+2\,{x}^{\alpha /2}{{\sl Y}_{{\frac {\sqrt {-4\,ac+1}+\alpha }{ \alpha }}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )} \sqrt {ab}{\it \_C1}+ \left ( -\sqrt {-4\,ac+1}-1 \right ) {{\sl J}_{{ \frac {1}{\alpha }\sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{ \alpha /2}}{\alpha }}\right )}+2\,{{\sl J}_{{\frac {\sqrt {-4\,ac+1}+ \alpha }{\alpha }}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }} \right )}\sqrt {ab}{x}^{\alpha /2} \right ) \left ( {{\sl Y}_{{\frac {1}{ \alpha }\sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{ \alpha }}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{\alpha }\sqrt {-4\,ac+1 }}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )} \right ) ^{-1}} \right \} \]