\[ \boxed { {x}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +ax{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( b{x}^{m}+c \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.069509 (sec), leaf count = 445 \[ \left \{\left \{y(x)\to c_1 m^{-\frac {-\sqrt {a^2-2 a-4 c+1}-a+1}{m}-\frac {\sqrt {a^2-2 a-4 c+1}}{m}} b^{\frac {-\sqrt {a^2-2 a-4 c+1}-a+1}{2 m}+\frac {\sqrt {a^2-2 a-4 c+1}}{2 m}} \left (x^m\right )^{\frac {-\sqrt {a^2-2 a-4 c+1}-a+1}{2 m}+\frac {\sqrt {a^2-2 a-4 c+1}}{2 m}} \Gamma \left (1-\frac {\sqrt {a^2-2 a-4 c+1}}{m}\right ) J_{-\frac {\sqrt {a^2-2 a-4 c+1}}{m}}\left (\frac {2 \sqrt {b} \sqrt {x^m}}{m}\right )+c_2 m^{\frac {\sqrt {a^2-2 a-4 c+1}}{m}-\frac {\sqrt {a^2-2 a-4 c+1}-a+1}{m}} b^{\frac {\sqrt {a^2-2 a-4 c+1}-a+1}{2 m}-\frac {\sqrt {a^2-2 a-4 c+1}}{2 m}} \left (x^m\right )^{\frac {\sqrt {a^2-2 a-4 c+1}-a+1}{2 m}-\frac {\sqrt {a^2-2 a-4 c+1}}{2 m}} \Gamma \left (\frac {\sqrt {a^2-2 a-4 c+1}}{m}+1\right ) J_{\frac {\sqrt {a^2-2 a-4 c+1}}{m}}\left (\frac {2 \sqrt {b} \sqrt {x^m}}{m}\right )\right \}\right \} \]
Maple: cpu = 0.031 (sec), leaf count = 85 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{-{\frac {a}{2}}+{\frac { 1}{2}}}{{\sl J}_{{\frac {1}{m}\sqrt {{a}^{2}-2\,a-4\,c+1}}}\left (2\,{ \frac {\sqrt {b}{x}^{m/2}}{m}}\right )}+{\it \_C2}\,{x}^{-{\frac {a}{2} }+{\frac {1}{2}}}{{\sl Y}_{{\frac {1}{m}\sqrt {{a}^{2}-2\,a-4\,c+1}} }\left (2\,{\frac {\sqrt {b}{x}^{m/2}}{m}}\right )} \right \} \]