\[ \boxed { {x}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( a{x}^{k}-b \left ( b-1 \right ) \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.052507 (sec), leaf count = 225 \[ \left \{\left \{y(x)\to c_1 k^{-\frac {2 (1-b)}{k}-\frac {2 b}{k}+\frac {1}{k}} a^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )} \left (x^k\right )^{\frac {1-b}{k}+\frac {1}{2} \left (\frac {2 b}{k}-\frac {1}{k}\right )} \Gamma \left (-\frac {2 b}{k}+\frac {1}{k}+1\right ) J_{\frac {1-2 b}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+c_2 k^{-1/k} a^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )} \left (x^k\right )^{\frac {b}{k}+\frac {1}{2} \left (\frac {1}{k}-\frac {2 b}{k}\right )} \Gamma \left (\frac {2 b}{k}-\frac {1}{k}+1\right ) J_{\frac {2 b-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right \}\right \} \]
Maple: cpu = 0.031 (sec), leaf count = 69 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sqrt {x}{{\sl J}_{{\frac {1 }{k}\sqrt { \left ( 2\,b-1 \right ) ^{2}}}}\left (2\,{\frac {\sqrt {a}{x} ^{k/2}}{k}}\right )}+{\it \_C2}\,\sqrt {x}{{\sl Y}_{{\frac {1}{k}\sqrt { \left ( 2\,b-1 \right ) ^{2}}}}\left (2\,{\frac {\sqrt {a}{x}^{k/2}}{k} }\right )} \right \} \]