\[ \boxed { x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +a \left ( y \left ( x \right ) \right ) ^{2}-by \left ( x \right ) -c{x}^{\beta }=0} \]
Mathematica: cpu = 0.017002 (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 = 0.078 (sec), leaf count = 237 \[ \left \{ y \left ( x \right ) =-{\frac {{\it \_C1}}{a}\sqrt {-ac}{x}^{{ \frac {\beta }{2}}}{{\sl Y}_{{\frac {b+\beta }{\beta }}}\left (2\,{\frac { \sqrt {-ac}{x}^{\beta /2}}{\beta }}\right )} \left ( {{\sl Y}_{{\frac {b}{ \beta }}}\left (2\,{\frac {\sqrt {-ac}{x}^{\beta /2}}{\beta }}\right )}{ \it \_C1}+{{\sl J}_{{\frac {b}{\beta }}}\left (2\,{\frac {\sqrt {-ac}{x} ^{\beta /2}}{\beta }}\right )} \right ) ^{-1}}-{\frac {1}{a} \left ( { {\sl J}_{{\frac {b+\beta }{\beta }}}\left (2\,{\frac {\sqrt {-ac}{x}^{ \beta /2}}{\beta }}\right )}\sqrt {-ac}{x}^{{\frac {\beta }{2}}}-{{\sl Y} _{{\frac {b}{\beta }}}\left (2\,{\frac {\sqrt {-ac}{x}^{\beta /2}}{\beta } }\right )}{\it \_C1}\,b-b{{\sl J}_{{\frac {b}{\beta }}}\left (2\,{\frac { \sqrt {-ac}{x}^{\beta /2}}{\beta }}\right )} \right ) \left ( {{\sl Y}_{{ \frac {b}{\beta }}}\left (2\,{\frac {\sqrt {-ac}{x}^{\beta /2}}{\beta }} \right )}{\it \_C1}+{{\sl J}_{{\frac {b}{\beta }}}\left (2\,{\frac { \sqrt {-ac}{x}^{\beta /2}}{\beta }}\right )} \right ) ^{-1}} \right \} \]