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 Exp

 http://functions.wolfram.com/01.03.21.0612.01

 Input Form

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

 Standard Form

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 MathML Form

 z n a z b + d z h z c + f z + g z 2 - 2 n - 1 - ( b log ( a ) + c log ( h ) ) 2 4 ( d log ( a ) + f log ( h ) ) h g ( d log ( a ) + f log ( h ) ) - 2 ( n + 1 ) k = 0 n j = 0 k ( - 1 ) k - j 4 k ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["j", 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 log ( a ) + c log ( h ) ) - j - k + 2 n ( b log ( a ) + c log ( h ) + 2 z ( d log ( a ) + f log ( h ) ) ) j + k ( - ( b log ( a ) + c log ( h ) + 2 z ( d log ( a ) + f log ( h ) ) ) 2 d log ( a ) + f log ( h ) ) 1 2 ( - j - k - 1 ) ( Γ ( 1 2 ( j + k + 1 ) , - ( b log ( a ) + c log ( h ) + 2 z ( d log ( a ) + f log ( h ) ) ) 2 4 ( d log ( a ) + f log ( h ) ) ) ( b log ( a ) + c log ( h ) ) ( b log ( a ) + c log ( h ) + 2 z ( d log ( a ) + f log ( h ) ) ) + 2 Γ ( 1 2 ( j + k + 2 ) , - ( b log ( a ) + c log ( h ) + 2 z ( d log ( a ) + f log ( h ) ) ) 2 4 ( d log ( a ) + f log ( h ) ) ) ( d log ( a ) + f log ( h ) ) - ( b log ( a ) + c log ( h ) + 2 z ( d log ( a ) + f log ( h ) ) ) 2 d log ( a ) + f log ( h ) ) /; n Condition z z n a z 1 2 b d z h z 1 2 c f z g 2 -2 n -1 -1 b a c h 2 4 d a f h -1 h g d a f h -2 n 1 j 0 k k 0 n -1 k -1 j 4 k Binomial k j Binomial n k b a c h -1 j -1 k 2 n b a c h 2 z 1 2 d a f h j k -1 b a c h 2 z 1 2 d a f h 2 d a f h -1 1 2 -1 j -1 k -1 Gamma 1 2 j k 1 -1 b a c h 2 z 1 2 d a f h 2 4 d a f h -1 b a c h b a c h 2 z 1 2 d a f h 2 Gamma 1 2 j k 2 -1 b a c h 2 z 1 2 d a f h 2 4 d a f h -1 d a f h -1 b a c h 2 z 1 2 d a f h 2 d a f h -1 1 2 n [/itex]

 Rule Form

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

 2002-12-18

© 1998-2013 Wolfram Research, Inc.