\[ \boxed { {\it a2}\,{x}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( {\it a1}\,{x}^{2}+{\it b1}\,x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( {\it a0}\,{x}^{2}+{\it b0}\,x+{\it c0} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.571573 (sec), leaf count = 356 \[ \left \{\left \{y(x)\to c_1 U\left (-\frac {2 \text {b0} \text {a2}-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} \text {a2}-\text {a1} \text {b1}-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} \sqrt {\text {a2}^2-2 \text {b1} \text {a2}-4 \text {c0} \text {a2}+\text {b1}^2}}{2 \text {a2} \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}},\frac {\sqrt {\text {a2}^2-2 \text {b1} \text {a2}-4 \text {c0} \text {a2}+\text {b1}^2}}{\text {a2}}+1,\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right ) \exp \left (\frac {\log (x) \left (\sqrt {\text {a2}^2-2 \text {a2} (\text {b1}+2 \text {c0})+\text {b1}^2}+\text {a2}-\text {b1}\right )-x \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}\right )}{2 \text {a2}}\right )+c_2 L_{\frac {-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} \sqrt {\text {a2}^2-2 \text {a2} \text {b1}-4 \text {a2} \text {c0}+\text {b1}^2}-\text {a2} \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {a1} \text {b1}+2 \text {a2} \text {b0}}{2 \text {a2} \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}}^{\frac {\sqrt {\text {a2}^2-2 \text {a2} \text {b1}-4 \text {a2} \text {c0}+\text {b1}^2}}{\text {a2}}}\left (\frac {x \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}{\text {a2}}\right ) \exp \left (\frac {\log (x) \left (\sqrt {\text {a2}^2-2 \text {a2} (\text {b1}+2 \text {c0})+\text {b1}^2}+\text {a2}-\text {b1}\right )-x \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}\right )}{2 \text {a2}}\right )\right \}\right \} \]
Maple: cpu = 0.218 (sec), leaf count = 165 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{-{\frac {{\it b1}}{2\,{ \it a2}}}}{{\rm e}^{-{\frac {{\it a1}\,x}{2\,{\it a2}}}}}{{\sl M}_{-{ \frac {{\it a1}\,{\it b1}-2\,{\it a2}\,{\it b0}}{2\,{\it a2}}{\frac {1 }{\sqrt {-4\,{\it a0}\,{\it a2}+{{\it a1}}^{2}}}}},\,{\frac {1}{2\,{ \it a2}}\sqrt {{{\it a2}}^{2}+ \left ( -2\,{\it b1}-4\,{\it c0} \right ) {\it a2}+{{\it b1}}^{2}}}}\left ({\frac {x}{{\it a2}}\sqrt {-4 \,{\it a0}\,{\it a2}+{{\it a1}}^{2}}}\right )}+{\it \_C2}\,{x}^{-{ \frac {{\it b1}}{2\,{\it a2}}}}{{\rm e}^{-{\frac {{\it a1}\,x}{2\,{ \it a2}}}}}{{\sl W}_{-{\frac {{\it a1}\,{\it b1}-2\,{\it a2}\,{\it b0} }{2\,{\it a2}}{\frac {1}{\sqrt {-4\,{\it a0}\,{\it a2}+{{\it a1}}^{2}} }}},\,{\frac {1}{2\,{\it a2}}\sqrt {{{\it a2}}^{2}+ \left ( -2\,{\it b1 }-4\,{\it c0} \right ) {\it a2}+{{\it b1}}^{2}}}}\left ({\frac {x}{{\it a2}}\sqrt {-4\,{\it a0}\,{\it a2}+{{\it a1}}^{2}}}\right )} \right \} \]