\[ \boxed { x{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +a{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +b{x}^{{\it a1}}y \left ( x \right ) =0} \]
Mathematica: cpu = 0.054507 (sec), leaf count = 447 \[ \left \{\left \{y(x)\to c_1 \left (\frac {1}{\text {a1}}+1\right )^{\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}} \text {a1}^{\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}} b^{\frac {1}{2} \left (\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}\right )} \left (x^{\text {a1}}\right )^{\left (\frac {1}{2 \text {a1}}+\frac {1}{2}\right ) \left (\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}\right )} \Gamma \left (\frac {a}{\text {a1}+1}+\frac {\text {a1}}{\text {a1}+1}\right ) J_{\frac {a-1}{\text {a1}+1}}\left (\frac {2 \sqrt {b} \left (x^{\text {a1}}\right )^{\frac {1}{2} \left (1+\frac {1}{\text {a1}}\right )}}{\left (1+\frac {1}{\text {a1}}\right ) \text {a1}}\right )+c_2 \left (\frac {1}{\text {a1}}+1\right )^{-\frac {2 (1-a)}{\text {a1}+1}-\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}+\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}} \text {a1}^{-\frac {2 (1-a)}{\text {a1}+1}-\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}+\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}} b^{\frac {1-a}{\text {a1}+1}+\frac {1}{2} \left (\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}\right )} \left (x^{\text {a1}}\right )^{\frac {(1-a) \left (\frac {1}{\text {a1}}+1\right )}{\text {a1}+1}+\left (\frac {1}{2 \text {a1}}+\frac {1}{2}\right ) \left (\frac {a}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}-\frac {1}{\left (\frac {1}{\text {a1}}+1\right ) \text {a1}}\right )} \Gamma \left (-\frac {a}{\text {a1}+1}+\frac {\text {a1}}{\text {a1}+1}+\frac {2}{\text {a1}+1}\right ) J_{\frac {1-a}{\text {a1}+1}}\left (\frac {2 \sqrt {b} \left (x^{\text {a1}}\right )^{\frac {1}{2} \left (1+\frac {1}{\text {a1}}\right )}}{\left (1+\frac {1}{\text {a1}}\right ) \text {a1}}\right )\right \}\right \} \]
Maple: cpu = 0.062 (sec), leaf count = 77 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{-{\frac {a}{2}}+{\frac { 1}{2}}}{{\sl J}_{{\frac {a-1}{{\it a1}+1}}}\left (2\,{\frac {\sqrt {b}{ x}^{{\it a1}/2+1/2}}{{\it a1}+1}}\right )}+{\it \_C2}\,{x}^{-{\frac {a }{2}}+{\frac {1}{2}}}{{\sl Y}_{{\frac {a-1}{{\it a1}+1}}}\left (2\,{ \frac {\sqrt {b}{x}^{{\it a1}/2+1/2}}{{\it a1}+1}}\right )} \right \} \]