\[ \frac {n x Q_n(x)-n Q_{n-1}(x)}{x^2-1}-n (n+1) y(x)+\left (x^2-1\right ) y''(x)=0 \] ✗ Mathematica : cpu = 299.998 (sec), leaf count = 0 , timed out
$Aborted
✓ Maple : cpu = 0.131 (sec), leaf count = 418
\[ \left \{ y \left ( x \right ) = \left ( -{x}^{2}+1 \right ) {\mbox {$_2$F$_1$}({\frac {n}{2}}+1,-{\frac {n}{2}}+{\frac {1}{2}};\,{\frac {1}{2}};\,{x}^{2})}{\it \_C2}+ \left ( -{x}^{3}+x \right ) {\mbox {$_2$F$_1$}(1-{\frac {n}{2}},{\frac {n}{2}}+{\frac {3}{2}};\,{\frac {3}{2}};\,{x}^{2})}{\it \_C1}-3\, \left ( n+1 \right ) \left ( x-1 \right ) \left ( 1+x \right ) \left ( -{\mbox {$_2$F$_1$}(1-n/2,n/2+3/2;\,3/2;\,{x}^{2})}\int \!1/3\,{\frac {{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})} \left ( x{\it LegendreQ} \left ( n,x \right ) -{\it LegendreQ} \left ( n+1,x \right ) \right ) }{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3} \left ( \left ( {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}+ \left ( {n}^{2}+n-2 \right ) {x}^{2}{\mbox {$_2$F$_1$}(2+n/2,3/2-n/2;\,3/2;\,{x}^{2})} \right ) {\mbox {$_2$F$_1$}(1-n/2,n/2+3/2;\,3/2;\,{x}^{2})}-1/3\,{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}{\mbox {$_2$F$_1$}(2-n/2,5/2+n/2;\,5/2;\,{x}^{2})}{x}^{2} \left ( n+3 \right ) \left ( n-2 \right ) \right ) }}\,{\rm d}xx+\int \!1/3\,{\frac {{\mbox {$_2$F$_1$}(1-n/2,n/2+3/2;\,3/2;\,{x}^{2})}x \left ( x{\it LegendreQ} \left ( n,x \right ) -{\it LegendreQ} \left ( n+1,x \right ) \right ) }{ \left ( 1+x \right ) ^{3} \left ( x-1 \right ) ^{3} \left ( \left ( {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}+ \left ( {n}^{2}+n-2 \right ) {x}^{2}{\mbox {$_2$F$_1$}(2+n/2,3/2-n/2;\,3/2;\,{x}^{2})} \right ) {\mbox {$_2$F$_1$}(1-n/2,n/2+3/2;\,3/2;\,{x}^{2})}-1/3\,{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}{\mbox {$_2$F$_1$}(2-n/2,5/2+n/2;\,5/2;\,{x}^{2})}{x}^{2} \left ( n+3 \right ) \left ( n-2 \right ) \right ) }}\,{\rm d}x{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})} \right ) \right \} \]