\[ \text {A0} y(x) (a x+b)+\text {A1} (a x+b) y'(x)+\text {A2} (a x+b)^2 y''(x)=0 \] ✓ Mathematica : cpu = 0.0593741 (sec), leaf count = 243
DSolve[A0*(b + a*x)*y[x] + A1*(b + a*x)*Derivative[1][y][x] + A2*(b + a*x)^2*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (\frac {2 b}{a}+2 x\right )^{\frac {\text {A1}}{2 a \text {A2}}} (2 a \text {A2} x+2 \text {A2} b)^{-\frac {\text {A1}}{2 a \text {A2}}} \left (-\frac {\text {A0} \left (\frac {b}{a}+x\right )}{a \text {A2}}\right )^{\frac {1}{2}-\frac {\text {A1}}{2 a \text {A2}}} I_{\frac {\text {A1}}{a \text {A2}}-1}\left (2 \sqrt {-\frac {\text {A0} \left (\frac {b}{a}+x\right )}{a \text {A2}}}\right )+c_2 (-1)^{1-\frac {\text {A1}}{a \text {A2}}} \left (\frac {2 b}{a}+2 x\right )^{\frac {\text {A1}}{2 a \text {A2}}} (2 a \text {A2} x+2 \text {A2} b)^{-\frac {\text {A1}}{2 a \text {A2}}} \left (-\frac {\text {A0} \left (\frac {b}{a}+x\right )}{a \text {A2}}\right )^{\frac {1}{2}-\frac {\text {A1}}{2 a \text {A2}}} K_{\frac {\text {A1}}{a \text {A2}}-1}\left (2 \sqrt {-\frac {\text {A0} \left (\frac {b}{a}+x\right )}{a \text {A2}}}\right )\right \}\right \}\] ✓ Maple : cpu = 0.056 (sec), leaf count = 98
dsolve(A2*(a*x+b)^2*diff(diff(y(x),x),x)+A1*(a*x+b)*diff(y(x),x)+A0*(a*x+b)*y(x)=0,y(x))
\[y \left (x \right ) = \left (a x +b \right )^{-\frac {-a \mathit {A2} +\mathit {A1}}{2 a \mathit {A2}}} \left (\BesselY \left (\frac {a \mathit {A2} -\mathit {A1}}{a \mathit {A2}}, 2 \sqrt {\mathit {A0}}\, \sqrt {\frac {a x +b}{a^{2} \mathit {A2}}}\right ) c_{2}+\BesselJ \left (\frac {a \mathit {A2} -\mathit {A1}}{a \mathit {A2}}, 2 \sqrt {\mathit {A0}}\, \sqrt {\frac {a x +b}{a^{2} \mathit {A2}}}\right ) c_{1}\right )\]