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 Exp

 http://functions.wolfram.com/01.03.21.0636.01

 Input Form

 Integrate[z^n E^(b Sqrt[z] + d z) (E^(c z))^\[Nu], z] == (2^(-1 - 2 n) E^((-c) z \[Nu] - b^2/(4 (d + c \[Nu]))) (E^(c z))^\[Nu] Sum[(-1)^(-h + k) 4^k b^(-h - k + 2 n) (b + 2 Sqrt[z] (d + c \[Nu]))^ (h + k) (-((b + 2 Sqrt[z] (d + c \[Nu]))^2/(d + c \[Nu])))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 Sqrt[z] (d + c \[Nu])) Gamma[(1/2) (1 + h + k), -((b + 2 Sqrt[z] (d + c \[Nu]))^2/(4 (d + c \[Nu])))] + 2 (d + c \[Nu]) Sqrt[-((b + 2 Sqrt[z] (d + c \[Nu]))^2/(d + c \[Nu]))] Gamma[(1/2) (2 + h + k), -((b + 2 Sqrt[z] (d + c \[Nu]))^2/ (4 (d + c \[Nu])))]), {k, 0, n}, {h, 0, k}])/ (d + c \[Nu])^(2 (1 + n)) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n z b + d z ( c z ) ν z 2 - 2 n - 1 - b 2 4 ( d + c ν ) - c z ν ( c z ) ν ( d + c ν ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k b - h - k + 2 n ( b + 2 z ( d + c ν ) ) h + k ( - ( b + 2 z ( d + c ν ) ) 2 d + c ν ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( b + 2 z ( d + c ν ) ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 z ( d + c ν ) ) 2 4 ( d + c ν ) ) + 2 - ( b + 2 z ( d + c ν ) ) 2 d + c ν ( d + c ν ) Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 z ( d + c ν ) ) 2 4 ( d + c ν ) ) ) /; n Condition z z n z 1 2 b d z c z ν 2 -2 n -1 -1 b 2 4 d c ν -1 -1 c z ν c z ν d c ν -2 n 1 h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 z 1 2 d c ν h k -1 b 2 z 1 2 d c ν 2 d c ν -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 z 1 2 d c ν Gamma 1 2 h k 1 -1 b 2 z 1 2 d c ν 2 4 d c ν -1 2 -1 b 2 z 1 2 d c ν 2 d c ν -1 1 2 d c ν Gamma 1 2 h k 2 -1 b 2 z 1 2 d c ν 2 4 d c ν -1 n [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2002-12-18