\[ a y(x)^2-b x^{2 \nu }-c x^{\nu -1}+y'(x)=0 \] ✓ Mathematica : cpu = 0.229701 (sec), leaf count = 618
\[\left \{\left \{y(x)\to -\frac {x^{\nu } \left (\sqrt {b} c_1 (\nu +1) \sqrt {(\nu +1)^2} U\left (\frac {\sqrt {b} \sqrt {(\nu +1)^2} \nu +\sqrt {a} c (\nu +1)}{2 \sqrt {b} (\nu +1) \sqrt {(\nu +1)^2}},\frac {\nu }{\nu +1},\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )+c_1 \left (\sqrt {a} c (\nu +1)+\sqrt {b} \sqrt {(\nu +1)^2} \nu \right ) U\left (\frac {\sqrt {a} c (\nu +1)+\sqrt {b} \sqrt {(\nu +1)^2} (3 \nu +2)}{2 \sqrt {b} (\nu +1) \sqrt {(\nu +1)^2}},\frac {\nu }{\nu +1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )+\sqrt {b} (\nu +1) \sqrt {(\nu +1)^2} \left (L_{\frac {-\sqrt {a} c (\nu +1)-\sqrt {b} \sqrt {(\nu +1)^2} \nu }{2 \sqrt {b} (\nu +1) \sqrt {(\nu +1)^2}}}^{-\frac {1}{\nu +1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )+2 L_{\frac {-\sqrt {a} c (\nu +1)-\sqrt {b} \sqrt {(\nu +1)^2} (3 \nu +2)}{2 \sqrt {b} (\nu +1) \sqrt {(\nu +1)^2}}}^{\frac {\nu }{\nu +1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )\right )\right )}{\sqrt {a} (\nu +1)^2 \left (c_1 U\left (\frac {\sqrt {b} \sqrt {(\nu +1)^2} \nu +\sqrt {a} c (\nu +1)}{2 \sqrt {b} (\nu +1) \sqrt {(\nu +1)^2}},\frac {\nu }{\nu +1},\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )+L_{\frac {-\sqrt {a} c (\nu +1)-\sqrt {b} \sqrt {(\nu +1)^2} \nu }{2 \sqrt {b} (\nu +1) \sqrt {(\nu +1)^2}}}^{-\frac {1}{\nu +1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )\right )}\right \}\right \}\]
✓ Maple : cpu = 0.309 (sec), leaf count = 348
\[ \left \{ y \left ( x \right ) =-{\frac {1}{2\,ax} \left ( \left ( \left ( -\nu -2 \right ) {b}^{{\frac {3}{2}}}+\sqrt {a}bc \right ) {{\sl M}_{-{\frac {1}{2\,\nu +2} \left ( \left ( -2\,\nu -2 \right ) \sqrt {b}+\sqrt {a}c \right ) {\frac {1}{\sqrt {b}}}},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )}+2\,{b}^{3/2}{\it \_C1}\, \left ( \nu +1 \right ) {{\sl W}_{-{\frac { \left ( -2\,\nu -2 \right ) \sqrt {b}+\sqrt {a}c}{\sqrt {b} \left ( 2\,\nu +2 \right ) }},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )}+ \left ( {{\sl W}_{-{\frac {c}{2\,\nu +2}\sqrt {a}{\frac {1}{\sqrt {b}}}},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )}{\it \_C1}+{{\sl M}_{-{\frac {c}{2\,\nu +2}\sqrt {a}{\frac {1}{\sqrt {b}}}},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )} \right ) \left ( {b}^{{\frac {3}{2}}}\nu -2\,\sqrt {a}b \left ( {x}^{\nu +1}b+c/2 \right ) \right ) \right ) {b}^{-{\frac {3}{2}}} \left ( {{\sl W}_{-{\frac {c}{2\,\nu +2}\sqrt {a}{\frac {1}{\sqrt {b}}}},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )}{\it \_C1}+{{\sl M}_{-{\frac {c}{2\,\nu +2}\sqrt {a}{\frac {1}{\sqrt {b}}}},\, \left ( 2\,\nu +2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}\sqrt {b}{x}^{\nu +1}}{\nu +1}}\right )} \right ) ^{-1}} \right \} \]
\begin {align} y^{\prime }+ay^{2}-bx^{2v}-cx^{v-1} & =0\nonumber \\ y^{\prime } & =bx^{v}+cx^{v-1}-ay^{2}\tag {1}\\ & =P\left ( x\right ) +Q\left ( x\right ) y+R\left ( x\right ) y^{2}\nonumber \end {align}
This is Riccati first order non-linear ODE with \(P\left ( x\right ) =bx^{v}+cx^{v-1},Q\left ( x\right ) =0,R\left ( x\right ) =-a\).
Need to do this later.