\[ a y(x)^2-b y(x)-c x^{\beta }+x y'(x)=0 \] ✓ Mathematica : cpu = 0.180257 (sec), leaf count = 244
\[\left \{\left \{y(x)\to -\frac {\sqrt {-a} \sqrt {c} x^{\beta /2} \left (-2 J_{\frac {b}{\beta }-1}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+c_1 J_{1-\frac {b}{\beta }}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )-c_1 J_{-\frac {b+\beta }{\beta }}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )\right )-b c_1 J_{-\frac {b}{\beta }}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )}{2 a \left (J_{\frac {b}{\beta }}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+c_1 J_{-\frac {b}{\beta }}\left (\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )\right )}\right \}\right \}\] ✓ Maple : cpu = 0.102 (sec), leaf count = 171
\[y \relax (x ) = \frac {-\left (\BesselY \left (\frac {b +\beta }{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1}+\BesselJ \left (\frac {b +\beta }{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right ) \sqrt {-a c}\, x^{\frac {\beta }{2}}+b \left (\BesselY \left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1}+\BesselJ \left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right )}{a \left (\BesselY \left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1}+\BesselJ \left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right )}\]