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 Exp

 http://functions.wolfram.com/01.03.21.0553.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c"]]], ")"]], "\[Nu]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", "\[Nu]"]], RowBox[List["4", " ", "a"]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]]]], ")"]], "\[Nu]"]]]]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c"]]], ")"]], "\[Nu]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]], "2"], " ", "\[Nu]"]], "a"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]], "2"], " ", "\[Nu]"]], RowBox[List["4", " ", "a"]]]]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18