\[ y''(x)=\frac {y'(x) \left (2 (a-1) x^2-2 a+2 b c \left (x^2-1\right ) x^c\right )}{x \left (x^2-1\right )}-\frac {y(x) \left (b c (2 a-c-1) x^{c+2}-b c (2 a-c+1) x^c+x^2 ((a-1) a-v (v+1))-a (a+1)+b^2 c^2 \left (x^2-1\right ) x^{2 c}\right )}{x^2 \left (x^2-1\right )} \] ✓ Mathematica : cpu = 0.108569 (sec), leaf count = 42
DSolve[Derivative[2][y][x] == -(((-(a*(1 + a)) + ((-1 + a)*a - v*(1 + v))*x^2 - b*(1 + 2*a - c)*c*x^c + b*(-1 + 2*a - c)*c*x^(2 + c) + b^2*c^2*x^(2*c)*(-1 + x^2))*y[x])/(x^2*(-1 + x^2))) + ((-2*a + 2*(-1 + a)*x^2 + 2*b*c*x^c*(-1 + x^2))*Derivative[1][y][x])/(x*(-1 + x^2)),y[x],x]
\[\left \{\left \{y(x)\to c_1 P_v(x) e^{a \log (x)+b x^c}+c_2 Q_v(x) e^{a \log (x)+b x^c}\right \}\right \}\] ✓ Maple : cpu = 0.1 (sec), leaf count = 25
dsolve(diff(diff(y(x),x),x) = 1/x*(2*b*c*x^c*(x^2-1)+2*(a-1)*x^2-2*a)/(x^2-1)*diff(y(x),x)-(b^2*c^2*x^(2*c)*(x^2-1)+b*c*x^(c+2)*(2*a-c-1)-b*c*x^c*(2*a-c+1)+x^2*(a*(a-1)-v*(v+1))-a*(a+1))/x^2/(x^2-1)*y(x),y(x))
\[y \left (x \right ) = {\mathrm e}^{b \,x^{c}} x^{a} \left (c_{1} \LegendreP \left (v , x\right )+c_{2} \LegendreQ \left (v , x\right )\right )\]