\[ \boxed { {\it d4y} \left ( x \right ) + \left ( a{x}^{2}+b\lambda +c \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( a{x}^{2}+\beta \,\lambda +\gamma \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 80.218186 (sec), leaf count = 73 \[ \left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (a \unicode {f817}^2+\beta \lambda +\gamma \right ) \unicode {f818}(\unicode {f817})+\left (a \unicode {f817}^2+c+b \lambda \right ) \unicode {f818}''(\unicode {f817})+\unicode {f818}^{(4)}(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2,\unicode {f818}''(0)=c_3,\unicode {f818}^{(3)}(0)=c_4\right \}\right )(x)\right \}\right \} \]
Maple: cpu = 0.062 (sec), leaf count = 50 \[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ \left ( a{x}^ {2}+\beta \,\lambda +\gamma \right ) {\it \_Y} \left ( x \right ) + \left ( a{x}^{2}+b\lambda +c \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +{\frac {{\rm d}^{4}}{{\rm d}{x}^{4}}}{\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]