\[ y''(x)=-\frac {y(x) (a v x-b)}{x^2 (a x+b)}-\frac {(2 a x+b) y'(x)}{x (a x+b)}+A x \] ✓ Mathematica : cpu = 62.0389 (sec), leaf count = 2924
\[\left \{\left \{y(x)\to \frac {a x c_1 \, _2F_1\left (\frac {3}{2}-\frac {1}{2} \sqrt {1-4 v},\frac {1}{2} \sqrt {1-4 v}+\frac {3}{2};3;-\frac {a x}{b}\right )}{b}+c_2 G_{2,2}^{2,0}\left (-\frac {a x}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+\frac {a x \, _2F_1\left (\frac {3}{2}-\frac {1}{2} \sqrt {1-4 v},\frac {1}{2} \sqrt {1-4 v}+\frac {3}{2};3;-\frac {a x}{b}\right ) \int _1^x\left (-\frac {9 A \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[1]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right ){}^2 b^3}{a^2 \left (v \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[1]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+2 \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[1]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+3 \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[1]}{b}\right ) G_{3,3}^{3,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\\end {array}\right )\right ) \left (-3 b \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[1]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+2 a \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[1]}{b}\right ) K[1] G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+a v \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[1]}{b}\right ) K[1] G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+3 a \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[1]}{b}\right ) K[1] G_{3,3}^{3,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\\end {array}\right )\right )}-\frac {3 A G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right ) b^2}{a^2 \left (v \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[1]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+2 \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[1]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+3 \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[1]}{b}\right ) G_{3,3}^{3,0}\left (-\frac {a K[1]}{b}|\begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\\end {array}\right )\right )}\right )dK[1]+b G_{2,2}^{2,0}\left (-\frac {a x}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right ) \int _1^x\left (\frac {27 A \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[2]}{b}\right ){}^2 \, _2F_1\left (\frac {3}{2}-\frac {1}{2} \sqrt {1-4 v},\frac {1}{2} \sqrt {1-4 v}+\frac {3}{2};3;-\frac {a K[2]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right ){}^2 b^3}{a^2 \left (v \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[2]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+2 \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[2]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+3 \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[2]}{b}\right ) G_{3,3}^{3,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\\end {array}\right )\right ){}^2 \left (-3 b \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[2]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+2 a \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[2]}{b}\right ) K[2] G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+a v \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[2]}{b}\right ) K[2] G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+3 a \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[2]}{b}\right ) K[2] G_{3,3}^{3,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\\end {array}\right )\right )}+\frac {9 A \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[2]}{b}\right ) \, _2F_1\left (\frac {3}{2}-\frac {1}{2} \sqrt {1-4 v},\frac {1}{2} \sqrt {1-4 v}+\frac {3}{2};3;-\frac {a K[2]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right ) b^2}{a^2 \left (v \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[2]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+2 \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[2]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+3 \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[2]}{b}\right ) G_{3,3}^{3,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\\end {array}\right )\right ){}^2}+\frac {3 A \, _2F_1\left (\frac {3}{2}-\frac {1}{2} \sqrt {1-4 v},\frac {1}{2} \sqrt {1-4 v}+\frac {3}{2};3;-\frac {a K[2]}{b}\right ) K[2] b}{a \left (v \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[2]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+2 \, _2F_1\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right );4;-\frac {a K[2]}{b}\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\\end {array}\right )+3 \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right );3;-\frac {a K[2]}{b}\right ) G_{3,3}^{3,0}\left (-\frac {a K[2]}{b}|\begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\\end {array}\right )\right )}\right )dK[2]}{b}\right \}\right \}\] ✓ Maple : cpu = 0.858 (sec), leaf count = 201
\[ \left \{ y \left ( x \right ) ={\frac {1}{{a}^{2} \left ( v+6 \right ) \left ( v+2 \right ) \left ( v+12 \right ) } \left ( {x}^{-{\frac {1}{2}}+{\frac {1}{2}\sqrt {1-4\,v}}}{a}^{2}{\it \_C2}\, \left ( v+6 \right ) \left ( v+2 \right ) \left ( v+12 \right ) {\mbox {$_2$F$_1$}({\frac {3}{2}}-{\frac {1}{2}\sqrt {1-4\,v}},-{\frac {1}{2}}-{\frac {1}{2}\sqrt {1-4\,v}};\,1-\sqrt {1-4\,v};\,-{\frac {b}{ax}})}+{a}^{2}{\it \_C1}\,{x}^{-{\frac {1}{2}}-{\frac {1}{2}\sqrt {1-4\,v}}} \left ( v+6 \right ) \left ( v+2 \right ) \left ( v+12 \right ) {\mbox {$_2$F$_1$}({\frac {3}{2}}+{\frac {1}{2}\sqrt {1-4\,v}},-{\frac {1}{2}}+{\frac {1}{2}\sqrt {1-4\,v}};\,1+\sqrt {1-4\,v};\,-{\frac {b}{ax}})}+A \left ( {x}^{2} \left ( v+6 \right ) \left ( v+2 \right ) {a}^{2}+bx \left ( v+4 \right ) \left ( v+2 \right ) a-3\,{b}^{2} \left ( v+4 \right ) \right ) x \right ) } \right \} \]