2.1248 ODE No. 1248
\[ a x y'(x)+y(x) \left (b x^2+c x+d\right )+\left (x^2-1\right ) y''(x)=0 \]
✓ Mathematica : cpu = 0.176943 (sec), leaf count = 238
DSolve[(d + c*x + b*x^2)*y[x] + a*x*Derivative[1][y][x] + (-1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_2 \left (\frac {x}{2}-\frac {1}{2}\right )^{a/4} \left (x^2-1\right )^{-a/4} \left (\frac {x}{2}+\frac {1}{2}\right )^{1-\frac {a}{4}} e^{\sqrt {-b} x} \text {HeunC}\left [\frac {1}{4} a \left (a-4 \sqrt {-b}-2\right )-b+4 \sqrt {-b}+c-d,2 \left (2 \sqrt {-b}+c\right ),2-\frac {a}{2},\frac {a}{2},4 \sqrt {-b},\frac {x}{2}+\frac {1}{2}\right ]+c_1 ((x-1) (x+1))^{a/4} \left (x^2-1\right )^{-a/4} e^{\sqrt {-b} x} \text {HeunC}\left [a \sqrt {-b}-b+c-d,2 \left (a \sqrt {-b}+c\right ),\frac {a}{2},\frac {a}{2},4 \sqrt {-b},\frac {x}{2}+\frac {1}{2}\right ]\right \}\right \}\]
✓ Maple : cpu = 0.318 (sec), leaf count = 134
dsolve((x^2-1)*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(b*x^2+c*x+d)*y(x)=0,y(x))
\[y \left (x \right ) = {\mathrm e}^{\sqrt {-b}\, x} \left (x^{2}-1\right )^{-\frac {a}{4}} \left (\left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {a}{4}+1} \operatorname {HeunC}\left (4 \sqrt {-b}, 1-\frac {a}{2}, \frac {a}{2}-1, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {1}{2}+\frac {x}{2}\right ) \left (-\frac {1}{2}+\frac {x}{2}\right )^{\frac {a}{4}} c_{2} +\operatorname {HeunC}\left (4 \sqrt {-b}, \frac {a}{2}-1, \frac {a}{2}-1, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {1}{2}+\frac {x}{2}\right ) \left (\left (x -1\right ) \left (1+x \right )\right )^{\frac {a}{4}} c_{1} \right )\]