\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-2\,{\frac {x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{{x}^{2}-1}}-{\frac { \left ( \left ( {x}^{2}-1 \right ) \left ( a{x}^{2}+bx+c \right ) -{k}^{2} \right ) y \left ( x \right ) }{ \left ( {x}^{2}-1 \right ) ^{2}}}=0} \]
Mathematica: cpu = 4.812611 (sec), leaf count = 93 \[ \left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (a \unicode {f817}^4+b \unicode {f817}^3-a \unicode {f817}^2+c \unicode {f817}^2-b \unicode {f817}-k^2-c\right ) \unicode {f818}(\unicode {f817})+\left (2 \unicode {f817}^3-2 \unicode {f817}\right ) \unicode {f818}'(\unicode {f817})+\left (\unicode {f817}^4-2 \unicode {f817}^2+1\right ) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \} \]
Maple: cpu = 0.187 (sec), leaf count = 120 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{\sqrt {-a}x}}{\it HeunC} \left ( 4\,\sqrt {-a},k,k,2\,b,{\frac {{k}^{2}}{2}}+a-b+c,{ \frac {1}{2}}+{\frac {x}{2}} \right ) \left ( {x}^{2}-1 \right ) ^{{ \frac {k}{2}}}+{{\it \_C2}{{\rm e}^{\sqrt {-a}x}}{\it HeunC} \left ( 4 \,\sqrt {-a},-k,k,2\,b,{\frac {{k}^{2}}{2}}+a-b+c,{\frac {1}{2}}+{ \frac {x}{2}} \right ) \sqrt {2\,x-2} \left ( 1+x \right ) ^{-{\frac {k}{ 2}}} \left ( x-1 \right ) ^{{\frac {k}{2}}}{\frac {1}{\sqrt {x-1}}}} \right \} \]