InverseFourierTransform takes forever

Apart function helps:

$Version
(*12.1.0 for Microsoft Windows (64-bit) (March 14, 2020)*)

FUNC = InverseFourierSinTransform[Apart[1/(w^6 (b + w^2)), w], w, r, 
Assumptions -> {r > 0, b > 0}, FourierParameters -> {1, -1}]

(*(r (-7200 - 2200 b r^2 - 137 b^2 r^4 + 
60 EulerGamma (120 + 20 b r^2 + b^2 r^4) + 
60 (120 + 20 b r^2 + b^2 r^4) Log[r]))/(7200 b^3 \[Pi]) + 
MeijerG[{{1/2}, {}}, {{1/2, 1/2}, {0}}, (b r^2)/4]/(
2 b^(7/2) Sqrt[\[Pi]])*)

FUNC // Simplify // Expand

(*-(r/(b^3 \[Pi])) + (EulerGamma r)/(b^3 \[Pi]) - (11 r^3)/(
36 b^2 \[Pi]) + (EulerGamma r^3)/(6 b^2 \[Pi]) - (137 r^5)/(
7200 b \[Pi]) + (EulerGamma r^5)/(120 b \[Pi]) + (r Log[r])/(
b^3 \[Pi]) + (r^3 Log[r])/(6 b^2 \[Pi]) + (r^5 Log[r])/(
120 b \[Pi]) + MeijerG[{{1/2}, {}}, {{1/2, 1/2}, {0}}, (b r^2)/4]/(
2 b^(7/2) Sqrt[\[Pi]]) *)

$$\text{InverseFourierSinTransform}\left[\frac{1}{w^6 \left(b+w^2\right)},w,r\right]=\\-\frac{r}{b^3 \pi }+\frac{\gamma r}{b^3 \pi }-\frac{11 r^3}{36 b^2 \pi }+\frac{\gamma r^3}{6 b^2 \pi }-\frac{137 r^5}{7200 b \pi }+\frac{\gamma r^5}{120 b \pi }+\frac{r \log (r)}{b^3 \pi }+\frac{r^3 \log (r)}{6 b^2 \pi }+\frac{r^5 \log (r)}{120 b \pi }+\frac{G_{1,3}^{2,1}\left(\frac{b r^2}{4}| \begin{array}{c} \frac{1}{2} \\ \frac{1}{2},\frac{1}{2},0 \\ \end{array} \right)}{2 b^{7/2} \sqrt{\pi }}$$

Amplifying on Mariusz Iwaniuk's answer

$Version

(* "12.1.0 for Mac OS X x86 (64-bit) (March 14, 2020)" *)

Clear["Global`*"]

ift = InverseFourierSinTransform[1/(w^6 (w^2 + b)) // Apart, w, r, 
  Assumptions -> {b > 0}, FourierParameters -> {1, -1}]

(* (1/(7200 b^3 π))
 r (-7200 - 2200 b r^2 - 137 b^2 r^4 + 
    60 EulerGamma (120 + 20 b r^2 + b^2 r^4) + 
    60 (120 + 20 b r^2 + b^2 r^4) Log[r]) + 
 MeijerG[{{1/2}, {}}, {{1/2, 1/2}, {0}}, (b r^2)/4]/(2 b^(7/2) Sqrt[π]) *)

Using FunctionExpand to convert the MeijerG

ift2 = ift // FunctionExpand // FullSimplify[#, {b > 0}] &

(* (1/(7200 b^4 π r))(b r^2 (7200 (-1 + EulerGamma) + 
     200 b (-11 + 6 EulerGamma) r^2 + b^2 (-137 + 60 EulerGamma) r^4 + 
     60 (120 + b r^2 (20 + b r^2)) Log[r]) - 
  3600 Sqrt[b r^2] ((2 CoshIntegral[Sqrt[b] r] - 
        2 (Log[r] + Floor[Arg[r]/(2 π)] (Log[1/r] + Log[r])) + 
        Log[r^2]) Sinh[Sqrt[b] r] - 
     2 Cosh[Sqrt[b] r] SinhIntegral[Sqrt[b] r])) *)

The function is defined only in the limit at r == 0

{ift, ift2} /. r -> 0 // Quiet

(* {Indeterminate, Indeterminate} *)

Limit[ift2, r -> 0]

(* 0 *)

The function is real when

FunctionDomain[{ift2, b > 0}, {b, r}] // FullSimplify

(* Sqrt[b] r > 0 *)

Then for r > 0

ift3 = ift2 // FullSimplify[#, {r > 0, b > 0}] &

(* (1/(7200 b^(
 7/2) π))(Sqrt[b]
    r (7200 (-1 + EulerGamma) + 200 b (-11 + 6 EulerGamma) r^2 + 
     b^2 (-137 + 60 EulerGamma) r^4 + 60 (120 + b r^2 (20 + b r^2)) Log[r]) - 
  7200 CoshIntegral[Sqrt[b] r] Sinh[Sqrt[b] r] + 
  7200 Cosh[Sqrt[b] r] SinhIntegral[Sqrt[b] r]) *)

Plot3D[ift3, {b, 0, 5}, {r, 0, 10},
 PlotPoints -> 50,
 MaxRecursion -> 5,
 ClippingStyle -> None,
 AxesLabel -> Automatic,
 WorkingPrecision -> 20]