\[ y''(x)=-\frac {y'(x) \left (2 f(x) g(x) g'(x)^2-\left (g(x)^2-1\right ) \left (2 f'(x) g'(x)+f(x) g''(x)\right )\right )}{f(x) \left (g(x)^2-1\right ) g'(x)}-\frac {y(x) \left (\left (g(x)^2-1\right ) \left (f'(x) \left (2 f'(x) g'(x)+f(x) g''(x)\right )-f(x) f''(x) g'(x)\right )-f(x) g'(x)^2 \left (2 g(x) f'(x)+v (v+1) f(x) g'(x)\right )\right )}{f(x)^2 \left (g(x)^2-1\right ) g'(x)} \] ✓ Mathematica : cpu = 0.272891 (sec), leaf count = 24
\[\{\{y(x)\to c_1 f(x) P_v(g(x))+c_2 f(x) Q_v(g(x))\}\}\] ✓ Maple : cpu = 0.221 (sec), leaf count = 20
\[\{y \left (x \right ) = \left (c_{1} \LegendreP \left (v , g \left (x \right )\right )+c_{2} \LegendreQ \left (v , g \left (x \right )\right )\right ) f \left (x \right )\}\]