\[ y''(x)=-\frac {y(x) (a v x-b)}{x^2 (a x+b)}-\frac {(2 a x+b) y'(x)}{x (a x+b)}+A x \] ✓ Mathematica : cpu = 61.5978 (sec), leaf count = 2924
DSolve[Derivative[2][y][x] == A*x - ((-b + a*v*x)*y[x])/(x^2*(b + a*x)) - ((b + 2*a*x)*Derivative[1][y][x])/(x*(b + a*x)),y[x],x]
\[\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 = 0.15 (sec), leaf count = 201
dsolve(diff(diff(y(x),x),x) = -1/x*(2*a*x+b)/(a*x+b)*diff(y(x),x)-(a*v*x-b)/(a*x+b)/x^2*y(x)+A*x,y(x))
\[y \left (x \right ) = \frac {x^{-\frac {1}{2}+\frac {\sqrt {1-4 v}}{2}} a^{2} c_{2} \left (v +6\right ) \left (v +2\right ) \left (v +12\right ) \hypergeom \left (\left [\frac {3}{2}-\frac {\sqrt {1-4 v}}{2}, -\frac {1}{2}-\frac {\sqrt {1-4 v}}{2}\right ], \left [1-\sqrt {1-4 v}\right ], -\frac {b}{x a}\right )+a^{2} c_{1} x^{-\frac {1}{2}-\frac {\sqrt {1-4 v}}{2}} \left (v +6\right ) \left (v +2\right ) \left (v +12\right ) \hypergeom \left (\left [\frac {3}{2}+\frac {\sqrt {1-4 v}}{2}, -\frac {1}{2}+\frac {\sqrt {1-4 v}}{2}\right ], \left [1+\sqrt {1-4 v}\right ], -\frac {b}{x a}\right )+\left (x^{2} \left (v +6\right ) \left (v +2\right ) a^{2}+b x \left (v +4\right ) \left (v +2\right ) a -3 b^{2} \left (v +4\right )\right ) A x}{a^{2} \left (v +6\right ) \left (v +2\right ) \left (v +12\right )}\]