Why Can't `DSolve` Find a Solution for this ODE?

When DSolve runs a long time, it's probably because either Integrate or Solve is chewing over a tough problem. Here's a way to make Integrate give up quicker, so that you can see if that is the problem.

Note/warning: The code returns Inactive[Integrate][...] instead of an unevaluated Integrate[...]. The latter seems preferable, but I was unable to figure out how to do it. Luckily, the inactive version seems to work.

Base code: The function withTimedIntegrate runs code with Integrate under a time constraint of tc seconds.

ClearAll[withTimedIntegrate];
SetAttributes[withTimedIntegrate, HoldFirst];
withTimedIntegrate[code_, tc_] := Module[{$in},
   Internal`InheritedBlock[{Integrate},
    Unprotect[Integrate];
    i : Integrate[___] /; ! TrueQ[$in] :=
     Block[{$in = True},
      TimeConstrained[i, tc, Inactivate[i, Integrate]]
      ];
    Protect[Integrate];
    code
    ]
   ];

OP's example:

withTimedIntegrate[{dsol} = DSolve[ode == 0, y, x], 1]; // AbsoluteTiming
dsol

OP's example with an IVP:

withTimedIntegrate[
  {dsolIVP} = DSolve[{ode == 0, y[0] == 1, y'[0] == 0}, y, x], 
  1] // AbsoluteTiming
y[1] /. dsolIVP // Activate // N

Check:

NDSolveValue[{ode == 0, y[0] == 1, y'[0] == 0}, y[1], {x, 0, 1}]
(*  1.83793  *)

With

$Version
(* 10.4.1 for Microsoft Windows (64-bit) (April 11, 2016) *)

the code in the question produces the desired answer, although slowly,

(* {{y -> Function[{x}, E^x*C[1] + E^x*C[2]*Integrate[E^(-ArcTan[K[1]] - 2*K[1]), 
     {K[1], 1, x}]]}} *)

Computation time, as measured by AbsoluteTiming, is about 40 minutes on my PC.

Addendum

As is so often the case, DSolve performs much better when given some help. Begin with the substitution, y[x] -> Exp[x] z[x].

Unevaluated[D[y[x], {x, 2}] + D[y[x], {x, 1}]/(1 + x^2) - 
    y[x] (1 + 1/(1 + x^2))] /. y[x] -> Exp[x] z[x];
Simplify[% Exp[-x]] // Apart
(* ((3 + 2*x^2)*Derivative[1][z][x])/(1 + x^2) + Derivative[2][z][x] *)

One might think that DSolve could solve this greatly simplified ode in seconds, but in fact it takes 36 minutes! (Perhaps, DSolve is searching for a solution that does not involve Integrate.)

DSolve[% == 0, z[x], x]
(* {{z[x] -> C[2] + Integrate[E^(-ArcTan[K[1]] - 2*K[1])*C[1], {K[1], 1, x}]}} *)

The obvious substitution z'[x] -> w[x] finally allows DSolve to proceed quickly.

%% /. {z''[x] -> w'[x], z'[x] -> w[x]};
DSolve[% == 0, w[x], x]
(* {{w[x] -> E^(-2 x - ArcTan[x]) C[1]}} *)

Back substitution and an additional integration then yield the desired result.