\[ y''(x)=-\frac {\left (a x^2+a-1\right ) y'(x)}{x \left (x^2+1\right )}-\frac {y(x) \left (b x^2+c\right )}{x^2 \left (x^2+1\right )} \] ✓ Mathematica : cpu = 0.482 (sec), leaf count = 288
DSolve[Derivative[2][y][x] == -(((c + b*x^2)*y[x])/(x^2*(1 + x^2))) - ((-1 + a + a*x^2)*Derivative[1][y][x])/(x*(1 + x^2)),y[x],x]
\[\left \{\left \{y(x)\to c_1 x^{\frac {1}{2} \left (-\sqrt {a^2-4 a-4 c+4}-a+2\right )} \, _2F_1\left (-\frac {1}{4} \sqrt {a^2-2 a-4 b+1}-\frac {1}{4} \sqrt {a^2-4 a-4 c+4}+\frac {1}{4},\frac {1}{4} \sqrt {a^2-2 a-4 b+1}-\frac {1}{4} \sqrt {a^2-4 a-4 c+4}+\frac {1}{4};1-\frac {1}{2} \sqrt {a^2-4 a-4 c+4};-x^2\right )+c_2 x^{\frac {1}{2} \left (\sqrt {a^2-4 a-4 c+4}-a+2\right )} \, _2F_1\left (-\frac {1}{4} \sqrt {a^2-2 a-4 b+1}+\frac {1}{4} \sqrt {a^2-4 a-4 c+4}+\frac {1}{4},\frac {1}{4} \sqrt {a^2-2 a-4 b+1}+\frac {1}{4} \sqrt {a^2-4 a-4 c+4}+\frac {1}{4};\frac {1}{2} \sqrt {a^2-4 a-4 c+4}+1;-x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.102 (sec), leaf count = 97
dsolve(diff(diff(y(x),x),x) = -1/x*(a*x^2+a-1)/(x^2+1)*diff(y(x),x)-(b*x^2+c)/x^2/(x^2+1)*y(x),y(x))
\[y \left (x \right ) = x^{1-\frac {a}{2}} \left (\LegendreP \left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right ) c_{1}+\LegendreQ \left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right ) c_{2}\right )\]