\[ a x y'(x)+y(x) \left (b x^m+c\right )+x^2 y''(x)=0 \] ✓ Mathematica : cpu = 0.0516641 (sec), leaf count = 445
DSolve[(c + b*x^m)*y[x] + a*x*Derivative[1][y][x] + x^2*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 m^{-\frac {-\sqrt {a^2-2 a-4 c+1}-a+1}{m}-\frac {\sqrt {a^2-2 a-4 c+1}}{m}} b^{\frac {-\sqrt {a^2-2 a-4 c+1}-a+1}{2 m}+\frac {\sqrt {a^2-2 a-4 c+1}}{2 m}} \left (x^m\right )^{\frac {-\sqrt {a^2-2 a-4 c+1}-a+1}{2 m}+\frac {\sqrt {a^2-2 a-4 c+1}}{2 m}} \Gamma \left (1-\frac {\sqrt {a^2-2 a-4 c+1}}{m}\right ) J_{-\frac {\sqrt {a^2-2 a-4 c+1}}{m}}\left (\frac {2 \sqrt {b} \sqrt {x^m}}{m}\right )+c_2 m^{\frac {\sqrt {a^2-2 a-4 c+1}}{m}-\frac {\sqrt {a^2-2 a-4 c+1}-a+1}{m}} b^{\frac {\sqrt {a^2-2 a-4 c+1}-a+1}{2 m}-\frac {\sqrt {a^2-2 a-4 c+1}}{2 m}} \left (x^m\right )^{\frac {\sqrt {a^2-2 a-4 c+1}-a+1}{2 m}-\frac {\sqrt {a^2-2 a-4 c+1}}{2 m}} \Gamma \left (\frac {\sqrt {a^2-2 a-4 c+1}}{m}+1\right ) J_{\frac {\sqrt {a^2-2 a-4 c+1}}{m}}\left (\frac {2 \sqrt {b} \sqrt {x^m}}{m}\right )\right \}\right \}\] ✓ Maple : cpu = 0.034 (sec), leaf count = 79
dsolve(x^2*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(b*x^m+c)*y(x)=0,y(x))
\[y \left (x \right ) = x^{-\frac {a}{2}+\frac {1}{2}} \left (\BesselY \left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right ) c_{2}+\BesselJ \left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right ) c_{1}\right )\]