\[ x^2 \left (a y(x)^2+y'(x)\right )+b x^{\alpha }+c=0 \] ✓ Mathematica : cpu = 0.181345 (sec), leaf count = 1588
\[\left \{\left \{y(x)\to \frac {a^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}} \alpha ^{\frac {2 i \sqrt {4 a c-1}}{\alpha }+1} \left (x^{\alpha }\right )^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+\frac {1}{2}} J_{-\frac {\sqrt {\alpha ^2 (1-4 a 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 ) b^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}}+a^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}} \alpha ^{\frac {2 i \sqrt {4 a c-1}}{\alpha }} \sqrt {\alpha ^2 (1-4 a c)} \left (x^{\alpha }\right )^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+\frac {1}{2}} J_{-\frac {\sqrt {\alpha ^2 (1-4 a 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 ) b^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}}-i a^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}} \alpha ^{\frac {2 i \sqrt {4 a c-1}}{\alpha }+1} \sqrt {4 a c-1} \left (x^{\alpha }\right )^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+\frac {1}{2}} J_{-\frac {\sqrt {\alpha ^2 (1-4 a 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 ) b^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}}-a^{\frac {i \sqrt {4 a c-1}}{\alpha }} \alpha ^{\frac {2 \sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}} \left (\alpha \left (-i \sqrt {4 a c-1}-1\right )+\sqrt {\alpha ^2 (1-4 a c)}\right ) \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}} J_{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (\frac {\alpha +\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {i \sqrt {4 a c-1}}{\alpha }}+a^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+\frac {1}{2}} \alpha ^{\frac {2 i \sqrt {4 a c-1}}{\alpha }+1} \left (x^{\alpha }\right )^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+1} J_{-\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}-1}\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 ) b^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+\frac {1}{2}}-a^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+\frac {1}{2}} \alpha ^{\frac {2 i \sqrt {4 a c-1}}{\alpha }+1} \left (x^{\alpha }\right )^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+1} J_{1-\frac {\sqrt {\alpha ^2 (1-4 a 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 ) b^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+\frac {1}{2}}+a^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}} \alpha ^{\frac {2 \sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+1} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1}}{\alpha }+1} J_{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (\frac {\alpha +\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}}-a^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}} \alpha ^{\frac {2 \sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+1} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1}}{\alpha }+1} J_{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}+1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (\frac {\alpha +\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}}}{2 a \alpha x \sqrt {x^{\alpha }} \left (a^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}} \alpha ^{\frac {2 i \sqrt {4 a c-1}}{\alpha }} b^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}} J_{-\frac {\sqrt {\alpha ^2 (1-4 a 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 ) \left (x^{\alpha }\right )^{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}}+a^{\frac {i \sqrt {4 a c-1}}{\alpha }} \alpha ^{\frac {2 \sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}} b^{\frac {i \sqrt {4 a c-1}}{\alpha }} J_{\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (\frac {\alpha +\sqrt {1-4 a c}}{\alpha }\right ) \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1}}{\alpha }}\right )}\right \}\right \}\]
✓ Maple : cpu = 0.126 (sec), leaf count = 219
\[ \left \{ y \left ( x \right ) ={\frac {1}{2\,ax} \left ( -2\, \left ( {{\sl Y}_{{\frac {\sqrt {-4\,ac+1}+\alpha }{\alpha }}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )}{\it \_C1}+{{\sl J}_{{\frac {\sqrt {-4\,ac+1}+\alpha }{\alpha }}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )} \right ) \sqrt {ab}{x}^{\alpha /2}+ \left ( \sqrt {-4\,ac+1}+1 \right ) \left ( {{\sl Y}_{{\frac {1}{\alpha }\sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{\alpha }\sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )} \right ) \right ) \left ( {{\sl Y}_{{\frac {1}{\alpha }\sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{\alpha }\sqrt {-4\,ac+1}}}\left (2\,{\frac {\sqrt {ab}{x}^{\alpha /2}}{\alpha }}\right )} \right ) ^{-1}} \right \} \]