\[ y''(x)=-\frac {y(x) (a v x-b)}{x^2 (a x+b)}-\frac {(2 a x+b) y'(x)}{x (a x+b)}+A x \] ✗ Mathematica : cpu = 3599.98 (sec), leaf count = 0 , timed out
$Aborted
✓ Maple : cpu = 0.178 (sec), leaf count = 195
\[ \left \{ y \left ( x \right ) ={\mbox {$_2$F$_1$}({\frac {3}{2}}-{\frac {1}{2}\sqrt {1-4\,v}},-{\frac {1}{2}}-{\frac {1}{2}\sqrt {1-4\,v}};\,1-\sqrt {1-4\,v};\,-{\frac {b}{ax}})}{x}^{-{\frac {1}{2}}+{\frac {1}{2}\sqrt {1-4\,v}}}{\it \_C2}+{\mbox {$_2$F$_1$}({\frac {3}{2}}+{\frac {1}{2}\sqrt {1-4\,v}},-{\frac {1}{2}}+{\frac {1}{2}\sqrt {1-4\,v}};\,1+\sqrt {1-4\,v};\,-{\frac {b}{ax}})}{x}^{-{\frac {1}{2}}-{\frac {1}{2}\sqrt {1-4\,v}}}{\it \_C1}+{\frac {Ax \left ( ax \left ( ax+b \right ) {v}^{2}+ \left ( 8\,{a}^{2}{x}^{2}+6\,abx-3\,{b}^{2} \right ) v+12\,{a}^{2}{x}^{2}+8\,abx-12\,{b}^{2} \right ) }{{a}^{2} \left ( v+6 \right ) \left ( v+2 \right ) \left ( v+12 \right ) }} \right \} \]