\[ (a x+b) y'(x)+y(x) \left (\text {a1} x^2+\text {b1} x+\text {c1}\right )+y''(x)=0 \] ✓ Mathematica : cpu = 0.117508 (sec), leaf count = 421
DSolve[(c1 + b1*x + a1*x^2)*y[x] + (b + a*x)*Derivative[1][y][x] + Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \exp \left (\frac {-b x \sqrt {a^2-4 \text {a1}}-\frac {1}{2} a x^2 \sqrt {a^2-4 \text {a1}}-\frac {1}{2} a^2 x^2-a b x+2 \text {a1} x^2+2 \text {b1} x}{2 \sqrt {a^2-4 \text {a1}}}\right ) H_{\frac {-a^3+2 \text {c1} a^2-\sqrt {a^2-4 \text {a1}} a^2+4 \text {a1} a-2 b \text {b1} a+2 \text {a1} b^2+2 \text {b1}^2+4 \sqrt {a^2-4 \text {a1}} \text {a1}-8 \text {a1} \text {c1}}{2 \left (a^2-4 \text {a1}\right )^{3/2}}}\left (\frac {a b-2 \text {b1}}{\sqrt {2} \left (a^2-4 \text {a1}\right )^{3/4}}+\frac {\sqrt [4]{a^2-4 \text {a1}} x}{\sqrt {2}}\right )+c_2 \exp \left (\frac {-b x \sqrt {a^2-4 \text {a1}}-\frac {1}{2} a x^2 \sqrt {a^2-4 \text {a1}}-\frac {1}{2} a^2 x^2-a b x+2 \text {a1} x^2+2 \text {b1} x}{2 \sqrt {a^2-4 \text {a1}}}\right ) \, _1F_1\left (-\frac {-a^3+2 \text {c1} a^2-\sqrt {a^2-4 \text {a1}} a^2+4 \text {a1} a-2 b \text {b1} a+2 \text {a1} b^2+2 \text {b1}^2+4 \sqrt {a^2-4 \text {a1}} \text {a1}-8 \text {a1} \text {c1}}{4 \left (a^2-4 \text {a1}\right )^{3/2}};\frac {1}{2};\left (\frac {a b-2 \text {b1}}{\sqrt {2} \left (a^2-4 \text {a1}\right )^{3/4}}+\frac {\sqrt [4]{a^2-4 \text {a1}} x}{\sqrt {2}}\right )^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.248 (sec), leaf count = 262
dsolve(diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+(a1*x^2+b1*x+c1)*y(x)=0,y(x))
\[y \left (x \right ) = {\mathrm e}^{-\frac {x \left (\left (a x +2 b \right ) \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}+\left (a^{2}-4 \mathit {a1} \right ) \left (a^{2} x +2 a b -4 \mathit {a1} x -4 \mathit {b1} \right )\right )}{4 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}} \left (c_{2} \left (a^{2} x +a b -4 \mathit {a1} x -2 \mathit {b1} \right ) \hypergeom \left (\left [\frac {3 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}+a^{3}-2 a^{2} \mathit {c1} +\left (2 \mathit {b1} b -4 \mathit {a1} \right ) a +\left (-2 b^{2}+8 \mathit {c1} \right ) \mathit {a1} -2 \mathit {b1}^{2}}{4 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (a^{2} x +a b -4 \mathit {a1} x -2 \mathit {b1} \right )^{2}}{2 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}\right )+c_{1} \hypergeom \left (\left [\frac {\left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}+a^{3}-2 a^{2} \mathit {c1} +\left (2 \mathit {b1} b -4 \mathit {a1} \right ) a +\left (-2 b^{2}+8 \mathit {c1} \right ) \mathit {a1} -2 \mathit {b1}^{2}}{4 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (a^{2} x +a b -4 \mathit {a1} x -2 \mathit {b1} \right )^{2}}{2 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}\right )\right )\]