\[ y''(x)=-\frac {y(x) \left (x \left (a^2-b^2\right )+c^2\right )}{4 (x-1) x^2}-\frac {((a+1) x-1) y'(x)}{(x-1) x} \] ✓ Mathematica : cpu = 0.147618 (sec), leaf count = 114
DSolve[Derivative[2][y][x] == -1/4*((c^2 + (a^2 - b^2)*x)*y[x])/((-1 + x)*x^2) - ((-1 + (1 + a)*x)*Derivative[1][y][x])/((-1 + x)*x),y[x],x]
\[\left \{\left \{y(x)\to i^{-c} c_1 x^{-c/2} \, _2F_1\left (\frac {a}{2}-\frac {b}{2}-\frac {c}{2},\frac {a}{2}+\frac {b}{2}-\frac {c}{2};1-c;x\right )+i^c c_2 x^{c/2} \, _2F_1\left (\frac {a}{2}-\frac {b}{2}+\frac {c}{2},\frac {a}{2}+\frac {b}{2}+\frac {c}{2};c+1;x\right )\right \}\right \}\] ✓ Maple : cpu = 0.072 (sec), leaf count = 89
dsolve(diff(diff(y(x),x),x) = -((a+1)*x-1)/x/(x-1)*diff(y(x),x)-1/4*((a^2-b^2)*x+c^2)/x^2/(x-1)*y(x),y(x))
\[y \left (x \right ) = \left (x -1\right )^{1-a} \left (x^{-\frac {c}{2}} \hypergeom \left (\left [-\frac {a}{2}-\frac {b}{2}-\frac {c}{2}+1, -\frac {a}{2}+\frac {b}{2}-\frac {c}{2}+1\right ], \left [-c +1\right ], x\right ) c_{2}+x^{\frac {c}{2}} \hypergeom \left (\left [-\frac {a}{2}+\frac {b}{2}+\frac {c}{2}+1, -\frac {a}{2}-\frac {b}{2}+\frac {c}{2}+1\right ], \left [c +1\right ], x\right ) c_{1}\right )\]