\[ y''(x)=-\frac {y(x) \left (\left (x^2-1\right ) \left (a x^2+b x+c\right )-k^2\right )}{\left (x^2-1\right )^2}-\frac {2 x y'(x)}{x^2-1} \] ✓ Mathematica : cpu = 0.305321 (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 = 0.286 (sec), leaf count = 110
\[\left \{y \left (x \right ) = \left (c_{2} \sqrt {2 x -2}\, \left (x +1\right )^{-\frac {k}{2}} \left (x -1\right )^{\frac {k}{2}-\frac {1}{2}} \HeunC \left (4 \sqrt {-a}, -k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {x}{2}+\frac {1}{2}\right )+c_{1} \left (x^{2}-1\right )^{\frac {k}{2}} \HeunC \left (4 \sqrt {-a}, k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {x}{2}+\frac {1}{2}\right )\right ) {\mathrm e}^{\sqrt {-a}\, x}\right \}\]