\[ \boxed { {\it d4y} \left ( x \right ) - \left ( 12\,{k}^{2} \left ( {\it JacobiSN} \left ( z,x \right ) \right ) ^{2}+a \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +b{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( \alpha \, \left ( {\it JacobiSN} \left ( z,x \right ) \right ) ^{2}+\beta \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.100013 (sec), leaf count = 47 \[ \text {DSolve}\left [-y''(x) \left (a+12 k^2 \text {sn}(z|x)^2\right )+y(x) \left (\alpha \text {sn}(z|x)^2+\beta \right )+b y'(x)+y^{(4)}(x)=0,y(x),x\right ] \]
Maple: cpu = 5.319 (sec), leaf count = 59 \[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac { {\rm d}^{4}}{{\rm d}{x}^{4}}}{\it \_Y} \left ( x \right ) + \left ( -12\, {k}^{2} \left ( {\it JacobiSN} \left ( z,x \right ) \right ) ^{2}-a \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +b{\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) + \left ( \alpha \, \left ( {\it JacobiSN} \left ( z,x \right ) \right ) ^{2 }+\beta \right ) {\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]