\[ \boxed { {x}^{2} \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2} \right ) +a{x}^{k}-b \left ( b-1 \right ) =0} \]
Mathematica: cpu = 0.131017 (sec), leaf count = 821 \[ \left \{\left \{y(x)\to -\frac {a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )} \left (\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )\right ) x^{k-1} \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )-1} J_{\frac {2 b-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) k^{1-\frac {1}{k}}+\frac {1}{2} a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )+\frac {1}{2}} x^{k-1} \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )-\frac {1}{2}} \left (J_{\frac {2 b-1}{k}-1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-J_{\frac {2 b-1}{k}+1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right ) \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) k^{-1/k}+c_1 \left (a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )} \left (\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )\right ) k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}+1} x^{k-1} J_{\frac {1-2 b}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )-1}+\frac {1}{2} a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )+\frac {1}{2}} k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}} x^{k-1} \left (J_{\frac {1-2 b}{k}-1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-J_{\frac {1-2 b}{k}+1}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right ) \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )-\frac {1}{2}}\right )}{-a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )} k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}} J_{\frac {1-2 b}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) c_1 \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )}-a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )} k^{-1/k} J_{\frac {2 b-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right ) \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )}}\right \}\right \} \]
Maple: cpu = 0.094 (sec), leaf count = 296 \[ \left \{ y \left ( x \right ) =-{\frac {{\it \_C1}}{x}\sqrt {a}{x}^{{ \frac {k}{2}}}{{\sl Y}_{{\frac {1}{k} \left ( \sqrt { \left ( -1+2\,b \right ) ^{2}}+k \right ) }}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k}} \right )} \left ( {{\sl Y}_{{\frac {1}{k}\sqrt { \left ( -1+2\,b \right ) ^{2}}}}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k}}\right )}{\it \_C1}+{ {\sl J}_{{\frac {1}{k}\sqrt { \left ( -1+2\,b \right ) ^{2}}}}\left (2\,{ \frac {\sqrt {a}{x}^{k/2}}{k}}\right )} \right ) ^{-1}}+{\frac {1}{2\,x} \left ( \left ( {\it csgn} \left ( -1+2\,b \right ) \left ( -1+2\,b \right ) {\it \_C1}+{\it \_C1} \right ) {{\sl Y}_{{\frac {1}{k}\sqrt { \left ( -1+2\,b \right ) ^{2}}}}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k} }\right )}-2\,{{\sl J}_{{\frac {\sqrt { \left ( -1+2\,b \right ) ^{2}}+k }{k}}}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k}}\right )}\sqrt {a}{x}^{k/ 2}+ \left ( {\it csgn} \left ( -1+2\,b \right ) \left ( -1+2\,b \right ) + 1 \right ) {{\sl J}_{{\frac {1}{k}\sqrt { \left ( -1+2\,b \right ) ^{2}}} }\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k}}\right )} \right ) \left ( { {\sl Y}_{{\frac {1}{k}\sqrt { \left ( -1+2\,b \right ) ^{2}}}}\left (2\,{ \frac {\sqrt {a}{x}^{k/2}}{k}}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{ k}\sqrt { \left ( -1+2\,b \right ) ^{2}}}}\left (2\,{\frac {\sqrt {a}{x}^ {k/2}}{k}}\right )} \right ) ^{-1}} \right \} \]