\[ y''(x)=-\frac {y'(x) \left (a p x^b+q\right )}{x \left (a x^b-1\right )}-\frac {y(x) \left (a r x^b+s\right )}{x^2 \left (a x^b-1\right )} \] ✓ Mathematica : cpu = 0.145655 (sec), leaf count = 405
\[\left \{\left \{y(x)\to c_1 i^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{b}} a^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \left (x^b\right )^{\frac {-\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \, _2F_1\left (\frac {p+q-\sqrt {p^2-2 p-4 r+1}-\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {p+q+\sqrt {p^2-2 p-4 r+1}-\sqrt {q^2+2 q+4 s+1}}{2 b};1-\frac {\sqrt {q^2+2 q+4 s+1}}{b};a x^b\right )+c_2 i^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{b}} a^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \left (x^b\right )^{\frac {\sqrt {q^2+2 q+4 s+1}+q+1}{2 b}} \, _2F_1\left (\frac {p+q-\sqrt {p^2-2 p-4 r+1}+\sqrt {q^2+2 q+4 s+1}}{2 b},\frac {p+q+\sqrt {p^2-2 p-4 r+1}+\sqrt {q^2+2 q+4 s+1}}{2 b};\frac {b+\sqrt {q^2+2 q+4 s+1}}{b};a x^b\right )\right \}\right \}\]
✓ Maple : cpu = 0.464 (sec), leaf count = 253
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}({\frac {1}{2\,b} \left ( p+q+\sqrt {{q}^{2}+2\,q+4\,s+1}-\sqrt {{p}^{2}-2\,p-4\,r+1} \right ) },{\frac {1}{2\,b} \left ( p+q+\sqrt {{q}^{2}+2\,q+4\,s+1}+\sqrt {{p}^{2}-2\,p-4\,r+1} \right ) };\,{\frac {1}{b} \left ( b+\sqrt {{q}^{2}+2\,q+4\,s+1} \right ) };\,a{x}^{b})}{x}^{{\frac {q}{2}}+{\frac {1}{2}\sqrt {{q}^{2}+2\,q+4\,s+1}}+{\frac {1}{2}}}+{\it \_C2}\,{\mbox {$_2$F$_1$}(-{\frac {1}{2\,b} \left ( -p-q+\sqrt {{q}^{2}+2\,q+4\,s+1}+\sqrt {{p}^{2}-2\,p-4\,r+1} \right ) },{\frac {1}{2\,b} \left ( p+q-\sqrt {{q}^{2}+2\,q+4\,s+1}+\sqrt {{p}^{2}-2\,p-4\,r+1} \right ) };\,{\frac {1}{b} \left ( b-\sqrt {{q}^{2}+2\,q+4\,s+1} \right ) };\,a{x}^{b})}{x}^{{\frac {q}{2}}-{\frac {1}{2}\sqrt {{q}^{2}+2\,q+4\,s+1}}+{\frac {1}{2}}} \right \} \]