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Mathematica can't compute this kind integrals because is not updated.  FoxH function was introduced in 2021 year in version 12.1.

 Integrate[x^a1*Exp[-a2*x^a3 - a4*x^a5], {x, 0, Infinity}] == (a4^(-((1 + a1)/a5))*FoxH[{{{1 - (1 + a1)/a5, a3/a5}}, {}}, {{{0, 1}}, {}}, a2 a4^(-(a3/a5))])/a5

Mathematica can't compute this kind integrals because is not Updated. 

FoxH function was introduced in 2021 year in version 12.1.

 Integrate[x^a1*Exp[-a2*x^a3 - a4*x^a5], {x, 0, Infinity}] == (a4^(-((1 + a1)/a5))*FoxH[{{{1 - (1 + a1)/a5, a3/a5}}, {}}, {{{0, 1}}, {}}, a2 a4^(-(a3/a5))])/a5

EDIT:

This kind integrals is easy to compute using Mellin Transform and Inverse Mellin Transform:

inv = InverseMellinTransform[Integrate[
MellinTransform[Exp[-k*t - b*t^a], b, s], {t, 0, Infinity}, 
Assumptions -> {k > 0, a > 0}][[1]], s, b](*Can't Compute.Weakness!*)

(* InverseMellinTransform[k^(-1 + a s) Gamma[s] Gamma[1 - a s], s, b] *)

From defintion of FoxH function we can easy establish formula by a Mellin–Barnes integral (Inverse Mellin Transform)

 1/(2 Pi  I) ContourIntegrate[k^(-1 + a s)  Gamma[s]  Gamma[1 - a  s]*b^-s /. k -> 1 /. b -> 2/. a -> 4/3, s \[Element] InfiniteLine[{1/3, 0}, {0, 1}]](*Can't Compute.Weakness!*)

 1/(2 Pi  I) NContourIntegrate[k^(-1 + a s)  Gamma[s]  Gamma[1 - a  s]*b^-s /. k -> 1 /. b -> 2 /. a -> 4/3, s \[Element] InfiniteLine[{1/3, 0}, {0, 1}]]
 (* 0.377357 + 0. I *)

General formula exist. Can be expressed by FoxH function:

Integrate[Exp[-k*t - b*t^a], {t, 0, Infinity}] == (b^(-1/a) FoxH[{{{1 - 1/a, 1/a}}, {}}, {{{0, 1}}, {}}, b^(-1/a) k])/a

Mathematica can't compute this kind integrals because is not updated. FoxH function was introduced in 2021 year in version 12.1.

$Version
(* "13.3.0 for Microsoft Windows (64-bit) (June 3, 2023)"*)

FHI[k_, b_, a_] := NIntegrate[Exp[-k*t - b*t^a], {t, 0, Infinity}]
FHI[1, 2, 4/3]
(* 0.377357 *)

FHA[k_, b_, a_] := (b^(-1/a) FoxH[{{{1 - 1/a, 1/a}}, {}}, {{{0, 1}}, {}}, b^(-1/a) k])/a
FHA[1, 2, 4/3]
(*(3 FoxH[{{{1/4, 3/4}}, {}}, {{{0, 1}}, {}}, 1/2^(3/4)])/(4 2^(3/4))*) 

% // FunctionExpand
(*(3 (3/2)^(1/4)*Gamma[7/12] Gamma[11/
 12] HypergeometricPFQ[{7/12, 11/12}, {1/2, 3/4}, 27/2048])/(
 8 Gamma[3/4]) - 
 3/16 Sqrt[\[Pi]/2]
 HypergeometricPFQ[{5/6, 7/6}, {3/4, 5/4}, 27/2048] + (
 9 (3/2)^(3/4)
 Gamma[13/12] Gamma[17/
 12] HypergeometricPFQ[{13/12, 17/12}, {5/4, 3/2}, 27/2048])/(
 128 Gamma[5/4]) - 
 1/32 HypergeometricPFQ[{1, 4/3, 5/3}, {5/4, 3/2, 7/4}, 27/2048]*)
 
 % // N
 (* 0.377357 *)

General formula exist. Can be expressed by FoxH function:

Integrate[Exp[-k*t - b*t^a], {t, 0, Infinity}] == (b^(-1/a) FoxH[{{{1 - 1/a, 1/a}}, {}}, {{{0, 1}}, {}}, b^(-1/a) k])/a

Mathematica can't compute this kind integrals because is not updated. FoxH function was introduced in 2021 year in version 12.1.

During this time, nothing happened for Mathematica to calculate such integrals.

$Version
(* "13.3.0 for Microsoft Windows (64-bit) (June 3, 2023)"*)

FHI[k_, b_, a_] := NIntegrate[Exp[-k*t - b*t^a], {t, 0, Infinity}]
FHI[1, 2, 4/3]
(* 0.377357 *)

FHA[k_, b_, a_] := (b^(-1/a) FoxH[{{{1 - 1/a, 1/a}}, {}}, {{{0, 1}}, {}}, b^(-1/a) k])/a
FHA[1, 2, 4/3]
(*(3 FoxH[{{{1/4, 3/4}}, {}}, {{{0, 1}}, {}}, 1/2^(3/4)])/(4 2^(3/4))*) 

% // FunctionExpand
(*(3 (3/2)^(1/4)*Gamma[7/12] Gamma[11/
 12] HypergeometricPFQ[{7/12, 11/12}, {1/2, 3/4}, 27/2048])/(
 8 Gamma[3/4]) - 
 3/16 Sqrt[\[Pi]/2]
 HypergeometricPFQ[{5/6, 7/6}, {3/4, 5/4}, 27/2048] + (
 9 (3/2)^(3/4)
 Gamma[13/12] Gamma[17/
 12] HypergeometricPFQ[{13/12, 17/12}, {5/4, 3/2}, 27/2048])/(
 128 Gamma[5/4]) - 
 1/32 HypergeometricPFQ[{1, 4/3, 5/3}, {5/4, 3/2, 7/4}, 27/2048]*)
 
 % // N
 (* 0.377357 *)

Generalization for more exotic integral:

 Integrate[x^a1*Exp[-a2*x^a3 - a4*x^a5], {x, 0, Infinity}] == (a4^(-((1 + a1)/a5))*FoxH[{{{1 - (1 + a1)/a5, a3/a5}}, {}}, {{{0, 1}}, {}}, a2 a4^(-(a3/a5))])/a5

General formula exist. Can be expressed by FoxH function:

Integrate[Exp[-k*t - b*t^a], {t, 0, Infinity}] == (b^(-1/a) FoxH[{{{1 - 1/a, 1/a}}, {}}, {{{0, 1}}, {}}, b^(-1/a) k])/a

Mathematica can't compute this kind integrals because is not updated. FoxH function was introduced in 2021 year in version 12.1.

General formula exist. Can be expressed by FoxH function:

Integrate[Exp[-k*t - b*t^a], {t, 0, Infinity}] == (b^(-1/a) FoxH[{{{1 - 1/a, 1/a}}, {}}, {{{0, 1}}, {}}, b^(-1/a) k])/a

Mathematica can't compute this kind integrals because is not updated. FoxH function was introduced in 2021 year in version 12.1.

$Version
(* "13.3.0 for Microsoft Windows (64-bit) (June 3, 2023)"*)

FHI[k_, b_, a_] := NIntegrate[Exp[-k*t - b*t^a], {t, 0, Infinity}]
FHI[1, 2, 4/3]
(* 0.377357 *)

FHA[k_, b_, a_] := (b^(-1/a) FoxH[{{{1 - 1/a, 1/a}}, {}}, {{{0, 1}}, {}}, b^(-1/a) k])/a
FHA[1, 2, 4/3]
(*(3 FoxH[{{{1/4, 3/4}}, {}}, {{{0, 1}}, {}}, 1/2^(3/4)])/(4 2^(3/4))*) 

% // FunctionExpand
(*(3 (3/2)^(1/4)*Gamma[7/12] Gamma[11/
 12] HypergeometricPFQ[{7/12, 11/12}, {1/2, 3/4}, 27/2048])/(
 8 Gamma[3/4]) - 
 3/16 Sqrt[\[Pi]/2]
 HypergeometricPFQ[{5/6, 7/6}, {3/4, 5/4}, 27/2048] + (
 9 (3/2)^(3/4)
 Gamma[13/12] Gamma[17/
 12] HypergeometricPFQ[{13/12, 17/12}, {5/4, 3/2}, 27/2048])/(
 128 Gamma[5/4]) - 
 1/32 HypergeometricPFQ[{1, 4/3, 5/3}, {5/4, 3/2, 7/4}, 27/2048]*)
 
 % // N
 (* 0.377357 *)