\[ y''(x)=-\frac {y'(x) \left (a p x^b+q\right )}{x \left (a x^b-1\right )}-\frac {y(x) \left (a r x^b+s\right )}{x^2 \left (a x^b-1\right )} \] ✓ Mathematica : cpu = 0.0865379 (sec), leaf count = 481
\[\left \{\left \{y(x)\to c_1 i^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{b}} a^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \left (x^b\right )^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \, _2F_1\left (\frac {p}{2 b}+\frac {q}{2 b}-\frac {\sqrt {p^2-2 p-4 r+1}}{2 b}-\frac {\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {p}{2 b}+\frac {q}{2 b}+\frac {\sqrt {p^2-2 p-4 r+1}}{2 b}-\frac {\sqrt {q^2+2 q+4 s+1}}{2 b};1-\frac {\sqrt {q^2+2 q+4 s+1}}{b};a x^b\right )+c_2 i^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{b}} a^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \left (x^b\right )^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \, _2F_1\left (\frac {p}{2 b}+\frac {q}{2 b}-\frac {\sqrt {p^2-2 p-4 r+1}}{2 b}+\frac {\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {p}{2 b}+\frac {q}{2 b}+\frac {\sqrt {p^2-2 p-4 r+1}}{2 b}+\frac {\sqrt {q^2+2 q+4 s+1}}{2 b};\frac {\sqrt {q^2+2 q+4 s+1}}{b}+1;a x^b\right )\right \}\right \}\] ✓ Maple : cpu = 0.239 (sec), leaf count = 253
\[y \relax (x ) = c_{1} \hypergeom \left (\left [\frac {p +q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q +\sqrt {q^{2}+2 q +4 s +1}-\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [\frac {b +\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right ) x^{\frac {q}{2}+\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}+\frac {1}{2}}+c_{2} \hypergeom \left (\left [-\frac {-p -q +\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}, \frac {p +q -\sqrt {q^{2}+2 q +4 s +1}+\sqrt {p^{2}-2 p -4 r +1}}{2 b}\right ], \left [\frac {b -\sqrt {q^{2}+2 q +4 s +1}}{b}\right ], a \,x^{b}\right ) x^{\frac {q}{2}-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}+\frac {1}{2}}\]