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 Exp

 http://functions.wolfram.com/01.03.21.0686.01

 Input Form

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

 Standard Form

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 MathML Form

 z n z b + d z + e ( z c + g ) ν z 2 - 2 n - 1 d - 2 ( n + 1 ) - ( b + c ν ) 2 4 d + e - c z ν ( z c + g ) ν k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b + c ν ) - h - k + 2 n ( b + c ν + 2 d z ) h + k ( - ( b + c ν + 2 d z ) 2 d ) 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 + c ν ) ( b + c ν + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + c ν + 2 d z ) 2 4 d ) + 2 - ( b + c ν + 2 d z ) 2 d d Γ ( 1 2 ( h + k + 2 ) , - ( b + c ν + 2 d z ) 2 4 d ) ) /; n Condition z z n z 1 2 b d z e z 1 2 c g ν 2 -2 n -1 d -2 n 1 -1 b c ν 2 4 d -1 e -1 c z 1 2 ν z 1 2 c g ν h 0 k k 0 n -1 k -1 h 4 k b c ν -1 h -1 k 2 n b c ν 2 d z 1 2 h k -1 b c ν 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b c ν b c ν 2 d z 1 2 Gamma 1 2 h k 1 -1 b c ν 2 d z 1 2 2 4 d -1 2 -1 b c ν 2 d z 1 2 2 d -1 1 2 d Gamma 1 2 h k 2 -1 b c ν 2 d z 1 2 2 4 d -1 n [/itex]

 Rule Form

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

 2002-12-18