\[ y''(x)=-\frac {y(x) (a v x-b)}{x^2 (a x+b)}-\frac {(2 a x+b) y'(x)}{x (a x+b)}+A x \] ✗ Mathematica : cpu = 300.001 (sec), leaf count = 0 , timed out
$Aborted
✓ Maple : cpu = 0.199 (sec), leaf count = 201
\[ \left \{ y \left ( x \right ) ={\frac {1}{{a}^{2} \left ( v+6 \right ) \left ( v+2 \right ) \left ( v+12 \right ) } \left ( {x}^{-{\frac {1}{2}}+{\frac {1}{2}\sqrt {1-4\,v}}}{a}^{2}{\it \_C2}\, \left ( v+6 \right ) \left ( v+2 \right ) \left ( v+12 \right ) {\mbox {$_2$F$_1$}({\frac {3}{2}}-{\frac {1}{2}\sqrt {1-4\,v}},-{\frac {1}{2}}-{\frac {1}{2}\sqrt {1-4\,v}};\,1-\sqrt {1-4\,v};\,-{\frac {b}{ax}})}+{a}^{2}{\it \_C1}\,{x}^{-{\frac {1}{2}}-{\frac {1}{2}\sqrt {1-4\,v}}} \left ( v+6 \right ) \left ( v+2 \right ) \left ( v+12 \right ) {\mbox {$_2$F$_1$}({\frac {3}{2}}+{\frac {1}{2}\sqrt {1-4\,v}},-{\frac {1}{2}}+{\frac {1}{2}\sqrt {1-4\,v}};\,1+\sqrt {1-4\,v};\,-{\frac {b}{ax}})}+A \left ( {x}^{2} \left ( v+6 \right ) \left ( v+2 \right ) {a}^{2}+bx \left ( v+4 \right ) \left ( v+2 \right ) a-3\,{b}^{2} \left ( v+4 \right ) \right ) x \right ) } \right \} \]