I'm an amateur to Laplace transforms, but it looks to me that you can express your eventual solution in terms of integrals of type
LaplaceTransform[Gamma[m, C x^(1/n)], x, z]
with various m, C, and n.
Assuming that your n's are positive integers and looking at the first few such results, it looks to me as if there is pattern lurking, which I leave to you to determine:
In[1]:= Table[
n -> LaplaceTransform[Gamma[m, C x^(1/n)], x, z], {n, 1, 6}]
Out[1]= {1 -> ((1 - ((C + z)/C)^-m) Gamma[m])/z,
2 -> 1/2 z^(
1/2 (-3 - m)) (2 z^((1 + m)/2) Gamma[m] +
C^m (-Sqrt[z] Gamma[m/2] Hypergeometric1F1[m/2, 1/2, C^2/(4 z)] +
C Gamma[(1 + m)/2] Hypergeometric1F1[(1 + m)/2, 3/2, C^2/(
4 z)])),
3 -> 1/6 z^(
1/3 (-5 - m)) (6 z^((2 + m)/3) Gamma[m] +
2 C^(1 + m) z^(1/3)
Gamma[(1 + m)/
3] HypergeometricPFQ[{1/3 + m/3}, {2/3, 4/3}, -(C^3/(27 z))] -
C^(2 + m)
Gamma[(2 + m)/
3] HypergeometricPFQ[{2/3 + m/3}, {4/3, 5/3}, -(C^3/(27 z))] -
2 C^m z^(2/3)
Gamma[m/
3] HypergeometricPFQ[{m/3}, {1/3, 2/3}, -(C^3/(27 z))]),
4 -> 1/24 z^(
1/4 (-7 - m)) (24 z^((3 + m)/4) Gamma[m] +
6 C^(1 + m) Sqrt[z]
Gamma[(1 + m)/
4] HypergeometricPFQ[{1/4 + m/4}, {1/2, 3/4, 5/4}, C^4/(
256 z)] -
3 C^(2 + m) z^(1/4)
Gamma[(2 + m)/
4] HypergeometricPFQ[{1/2 + m/4}, {3/4, 5/4, 3/2}, C^4/(
256 z)] +
C^(3 + m)
Gamma[(3 + m)/
4] HypergeometricPFQ[{3/4 + m/4}, {5/4, 3/2, 7/4}, C^4/(
256 z)] -
6 C^m z^(3/4)
Gamma[m/4] HypergeometricPFQ[{m/4}, {1/4, 1/2, 3/4}, C^4/(
256 z)]),
5 -> 1/120 z^(
1/5 (-9 - m)) (120 z^((4 + m)/5) Gamma[m] +
24 C^(1 + m) z^(3/5)
Gamma[(1 + m)/
5] HypergeometricPFQ[{1/5 + m/5}, {2/5, 3/5, 4/5, 6/5}, -(C^5/(
3125 z))] -
12 C^(2 + m) z^(2/5)
Gamma[(2 + m)/
5] HypergeometricPFQ[{2/5 + m/5}, {3/5, 4/5, 6/5, 7/5}, -(C^5/(
3125 z))] +
4 C^(3 + m) z^(1/5)
Gamma[(3 + m)/
5] HypergeometricPFQ[{3/5 + m/5}, {4/5, 6/5, 7/5, 8/5}, -(C^5/(
3125 z))] -
C^(4 + m)
Gamma[(4 + m)/
5] HypergeometricPFQ[{4/5 + m/5}, {6/5, 7/5, 8/5, 9/5}, -(C^5/(
3125 z))] -
24 C^m z^(4/5)
Gamma[m/
5] HypergeometricPFQ[{m/5}, {1/5, 2/5, 3/5, 4/5}, -(C^5/(
3125 z))]),
6 -> 1/720 z^(
1/6 (-11 - m)) (720 z^((5 + m)/6) Gamma[m] +
120 C^(1 + m) z^(2/3)
Gamma[(1 + m)/
6] HypergeometricPFQ[{1/6 + m/6}, {1/3, 1/2, 2/3, 5/6, 7/6},
C^6/(46656 z)] -
60 C^(2 + m) Sqrt[z]
Gamma[(2 + m)/
6] HypergeometricPFQ[{1/3 + m/6}, {1/2, 2/3, 5/6, 7/6, 4/3},
C^6/(46656 z)] +
20 C^(3 + m) z^(1/3)
Gamma[(3 + m)/
6] HypergeometricPFQ[{1/2 + m/6}, {2/3, 5/6, 7/6, 4/3, 3/2},
C^6/(46656 z)] -
5 C^(4 + m) z^(1/6)
Gamma[(4 + m)/
6] HypergeometricPFQ[{2/3 + m/6}, {5/6, 7/6, 4/3, 3/2, 5/3},
C^6/(46656 z)] +
C^(5 + m)
Gamma[(5 + m)/
6] HypergeometricPFQ[{5/6 + m/6}, {7/6, 4/3, 3/2, 5/3, 11/6},
C^6/(46656 z)] -
120 C^m z^(5/6)
Gamma[m/6] HypergeometricPFQ[{m/6}, {1/6, 1/3, 1/2, 2/3, 5/6},
C^6/(46656 z)])}
