Try:

 eq = y''[x] + y'[x] == x^2*Sqrt[x];
 sol = DSolve[eq, y, x]
 (* {{y -> Function[{x}, (2 x^(7/2))/7 - E^-x C[1] + C[2] + (
      E^-x Sqrt[-x] Gamma[7/2, -x])/Sqrt[x]]}} *)
 eq /. sol // FullSimplify(*Solution is Correct*)
 (* {True}*)

Using FunctionExpand to get Erfi[x] function:

 sol2 = DSolveValue[eq, y[x], x] // FunctionExpand // Simplify
(*-((15 Sqrt[x])/4) + (5 x^(3/2))/2 - x^(5/2) + (2 x^(7/2))/7 + 
 E^-x ((15 Sqrt[\[Pi]] Sqrt[-x])/(8 Sqrt[x]) - C[1]) + C[2] + 
 15/8 E^-x Sqrt[\[Pi]] Erfi[Sqrt[x]] *)

eq /. {{y -> Function[{x}, Evaluate@sol2]}} // FullSimplify(*Solution is correct*)
  (* {True}*)

Check if yours solution are correct:

eq /. {{y -> 
      Function[{x}, 
       C[2] + 1/
          56 E^-x (2 (E^x Sqrt[x] (-105 + 70 x - 28 x^2 + 8 x^3) - 
              28 C[1]) + 105 Sqrt[Pi]*Erfi[Sqrt[x]])]}} // FullSimplify
(* {True}*)

Regards M.I.