\[ \boxed { {x}^{4}{\it d4y} \left ( x \right ) +6\,{x}^{3}{\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) + \left ( 4\,{x}^{4}+ \left ( -{\rho }^{2}-{\sigma }^{2}+7 \right ) {x}^{2} \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( 16\,{x}^{3}+ \left ( -{\rho }^{2}-{\sigma }^{2}+1 \right ) x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( {\rho }^{2}{\sigma }^{2}+8\,{x}^{2} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.518066 (sec), leaf count = 242 \[ \left \{\left \{y(x)\to c_1 x^{-\rho } \, _2F_3\left (\frac {1}{2}-\frac {\rho }{2},1-\frac {\rho }{2};1-\rho ,-\frac {\rho }{2}-\frac {\sigma }{2}+1,-\frac {\rho }{2}+\frac {\sigma }{2}+1;-x^2\right )+c_3 x^{-\sigma } \, _2F_3\left (\frac {1}{2}-\frac {\sigma }{2},1-\frac {\sigma }{2};1-\sigma ,-\frac {\rho }{2}-\frac {\sigma }{2}+1,\frac {\rho }{2}-\frac {\sigma }{2}+1;-x^2\right )+c_4 x^{\sigma } \, _2F_3\left (\frac {\sigma }{2}+\frac {1}{2},\frac {\sigma }{2}+1;-\frac {\rho }{2}+\frac {\sigma }{2}+1,\frac {\rho }{2}+\frac {\sigma }{2}+1,\sigma +1;-x^2\right )+c_2 x^{\rho } \, _2F_3\left (\frac {\rho }{2}+\frac {1}{2},\frac {\rho }{2}+1;\rho +1,\frac {\rho }{2}-\frac {\sigma }{2}+1,\frac {\rho }{2}+\frac {\sigma }{2}+1;-x^2\right )\right \}\right \} \]
Maple: cpu = 0.171 (sec), leaf count = 85 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\sl J}_{{\frac {\rho }{2}}+{ \frac {\sigma }{2}}}\left (x\right )}{{\sl J}_{{\frac {\rho }{2}}-{\frac { \sigma }{2}}}\left (x\right )}+{\it \_C2}\,{{\sl J}_{{\frac {\rho }{2}}+{ \frac {\sigma }{2}}}\left (x\right )}{{\sl Y}_{{\frac {\rho }{2}}-{\frac { \sigma }{2}}}\left (x\right )}+{\it \_C3}\,{{\sl Y}_{{\frac {\rho }{2}}+{ \frac {\sigma }{2}}}\left (x\right )}{{\sl J}_{{\frac {\rho }{2}}-{\frac { \sigma }{2}}}\left (x\right )}+{\it \_C4}\,{{\sl Y}_{{\frac {\rho }{2}}+{ \frac {\sigma }{2}}}\left (x\right )}{{\sl Y}_{{\frac {\rho }{2}}-{\frac { \sigma }{2}}}\left (x\right )} \right \} \]