\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( a{{\rm e}^{2\,x}}+b{{\rm e}^{x}}+c \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.680586 (sec), leaf count = 180 \[ \left \{\left \{y(x)\to c_1 e^{i \left (\sqrt {c} \log \left (e^x\right )-\sqrt {a} e^x\right )} U\left (\frac {i \left (b-i \sqrt {a}+2 \sqrt {a} \sqrt {c}\right )}{2 \sqrt {a}},2 i \sqrt {c}+1,2 i \sqrt {a} e^x\right )+c_2 e^{i \left (\sqrt {c} \log \left (e^x\right )-\sqrt {a} e^x\right )} L_{-\frac {i \left (2 \sqrt {a} \sqrt {c}-i \sqrt {a}+b\right )}{2 \sqrt {a}}}^{2 i \sqrt {c}}\left (2 i \sqrt {a} e^x\right )\right \}\right \} \]
Maple: cpu = 0.141 (sec), leaf count = 61 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{-{\frac {x}{2}}}}{ {\sl M}_{{-{\frac {i}{2}}b{\frac {1}{\sqrt {a}}}},\,i\sqrt {c}}\left (2 \,i\sqrt {a}{{\rm e}^{x}}\right )}+{\it \_C2}\,{{\rm e}^{-{\frac {x}{2} }}}{{\sl W}_{{-{\frac {i}{2}}b{\frac {1}{\sqrt {a}}}},\,i\sqrt {c} }\left (2\,i\sqrt {a}{{\rm e}^{x}}\right )} \right \} \]