\[ y''(x)=-\frac {y(x) \left (\left (x^2-1\right ) \left (a x^2+b x+c\right )-k^2\right )}{\left (x^2-1\right )^2}-\frac {2 x y'(x)}{x^2-1} \] ✓ Mathematica : cpu = 0.264252 (sec), leaf count = 202
\[\left \{\left \{y(x)\to c_1 e^{\sqrt {-a} x} \left (x^2-1\right )^{k/2} \text {HeunC}\left [(k+1) \left (2 \sqrt {-a}-k\right )-a+b-c,2 \left (2 \sqrt {-a} (k+1)+b\right ),k+1,k+1,4 \sqrt {-a},\frac {x}{2}+\frac {1}{2}\right ]+c_2 \sqrt {2 x-2} e^{\sqrt {-a} x} (x+1)^{-k/2} (x-1)^{\frac {k}{2}-\frac {1}{2}} \text {HeunC}\left [-2 \sqrt {-a} (k-1)-a+b-c,2 \left (2 \sqrt {-a}+b\right ),1-k,k+1,4 \sqrt {-a},\frac {x}{2}+\frac {1}{2}\right ]\right \}\right \}\] ✓ Maple : cpu = 2.259 (sec), leaf count = 110
\[ \left \{ y \left ( x \right ) ={{\rm e}^{\sqrt {-a}x}} \left ( {\it HeunC} \left ( 4\,\sqrt {-a},-k,k,2\,b,{\frac {{k}^{2}}{2}}+a-b+c,{\frac {1}{2}}+{\frac {x}{2}} \right ) \sqrt {2\,x-2} \left ( 1+x \right ) ^{-{\frac {k}{2}}} \left ( x-1 \right ) ^{{\frac {k}{2}}-{\frac {1}{2}}}{\it \_C2}+{\it HeunC} \left ( 4\,\sqrt {-a},k,k,2\,b,{\frac {{k}^{2}}{2}}+a-b+c,{\frac {1}{2}}+{\frac {x}{2}} \right ) \left ( {x}^{2}-1 \right ) ^{{\frac {k}{2}}}{\it \_C1} \right ) \right \} \]