\[ \boxed { \left ( {x}^{2}-1 \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -n \left ( n+1 \right ) y \left ( x \right ) +{\frac {\partial }{\partial x}}{\it LegendreQ} \left ( n,x \right ) =0} \]
Mathematica: cpu = 0 (sec), leaf count = 0 \[ \text {Hanged} \]
Maple: cpu = 0.109 (sec), leaf count = 418 \[ \left \{ y \left ( x \right ) = \left ( -{x}^{2}+1 \right ) {\mbox {$_2$F$_1$}({\frac {n}{2}}+1,-{\frac {n}{2}}+{\frac {1}{2}};\,{\frac {1}{2}};\,{x}^{2})} {\it \_C2}+ \left ( -{x}^{3}+x \right ) {\mbox {$_2$F$_1$}(-{\frac {n}{2}}+1,{\frac {n}{2}}+{\frac {3}{2}};\,{\frac {3}{2}};\,{x}^{2})} {\it \_C1}-3\, \left ( n+1 \right ) \left ( x-1 \right ) \left ( 1+x \right ) \left ( -\int \!1/3\,{\frac { {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})} \left ( x{\it LegendreQ} \left ( n,x \right ) -{\it LegendreQ} \left ( n+1,x \right ) \right ) }{ \left ( \left ( {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}+ \left ( {n}^{2}+n-2 \right ) {x}^{2}{\mbox {$_2$F$_1$}(n/2+2,3/2-n/2;\,3/2;\,{x}^{2})} \right ) {\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})}-1/3\, {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})} {\mbox {$_2$F$_1$}(-n/2+2,n/2+5/2;\,5/2;\,{x}^{2})}{x}^{2} \left ( n+3 \right ) \left ( n-2 \right ) \right ) \left ( x-1 \right ) ^{3} \left ( 1+x \right ) ^{3}}}\,{\rm d}x {\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})}x+\int \!1/3\,{ \frac {{\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})}x \left ( x{ \it LegendreQ} \left ( n,x \right ) -{\it LegendreQ} \left ( n+1,x \right ) \right ) }{ \left ( \left ( {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}+ \left ( {n}^{2}+n-2 \right ) {x}^{2}{\mbox {$_2$F$_1$}(n/2+2,3/2-n/2;\,3/2;\,{x}^{2})} \right ) {\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})}-1/3\, {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})} {\mbox {$_2$F$_1$}(-n/2+2,n/2+5/2;\,5/2;\,{x}^{2})}{x}^{2} \left ( n+3 \right ) \left ( n-2 \right ) \right ) \left ( x-1 \right ) ^{3} \left ( 1+x \right ) ^{3}}}\,{\rm d}x {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})} \right ) \right \} \]