\[ y''(x)=-\frac {y(x) \left (\left (x^2-1\right ) \left (a x^2+b x+c\right )-k^2\right )}{\left (x^2-1\right )^2}-\frac {2 x y'(x)}{x^2-1} \] ✓ Mathematica : cpu = 0.242671 (sec), leaf count = 202
DSolve[Derivative[2][y][x] == -(((-k^2 + (-1 + x^2)*(c + b*x + a*x^2))*y[x])/(-1 + x^2)^2) - (2*x*Derivative[1][y][x])/(-1 + x^2),y[x],x]
\[\left \{\left \{y(x)\to c_1 e^{\sqrt {-a} x} \left (x^2-1\right )^{k/2} \text {HeunC}\left [(k+1) \left (2 \sqrt {-a}-k\right )-a+b-c,2 \left (2 \sqrt {-a} (k+1)+b\right ),k+1,k+1,4 \sqrt {-a},\frac {x}{2}+\frac {1}{2}\right ]+c_2 \sqrt {2 x-2} e^{\sqrt {-a} x} (x+1)^{-k/2} (x-1)^{\frac {k}{2}-\frac {1}{2}} \text {HeunC}\left [-2 \sqrt {-a} (k-1)-a+b-c,2 \left (2 \sqrt {-a}+b\right ),1-k,k+1,4 \sqrt {-a},\frac {x}{2}+\frac {1}{2}\right ]\right \}\right \}\] ✓ Maple : cpu = 0.239 (sec), leaf count = 110
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-((x^2-1)*(a*x^2+b*x+c)-k^2)/(x^2-1)^2*y(x),y(x))
\[y \left (x \right ) = {\mathrm e}^{\sqrt {-a}\, x} \left (\HeunC \left (4 \sqrt {-a}, -k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {1}{2}+\frac {x}{2}\right ) \sqrt {2 x -2}\, \left (1+x \right )^{-\frac {k}{2}} \left (x -1\right )^{\frac {k}{2}-\frac {1}{2}} c_{2}+\HeunC \left (4 \sqrt {-a}, k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {1}{2}+\frac {x}{2}\right ) \left (x^{2}-1\right )^{\frac {k}{2}} c_{1}\right )\]