3 Mathematica. Time used: 0.07 (sec). Leaf size: 34

ode=(1+2*x[t])+(4-t^2)*D[x[t],t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]

\begin{align*} x(t)&\to -\frac {1}{2}-\frac {c_1 (t-2)}{\sqrt {4-t^2}}\\ x(t)&\to -\frac {1}{2} \end{align*}