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 Exp

 http://functions.wolfram.com/01.03.21.0520.01

 Input Form

 Integrate[Sqrt[(a + b E^(e z))/(c + d E^(e z))], z] == (Sqrt[(a + b E^(e z))/(c + d E^(e z))] Sqrt[c + d E^(e z)] ((-Sqrt[a]) Sqrt[d] Log[(1/(a^(3/2) Sqrt[c])) ((e (2 a c + b c E^(e z) + a d E^(e z) + 2 Sqrt[a] Sqrt[c] Sqrt[a + b E^(e z)] Sqrt[c + d E^(e z)]))/E^(e z))] + Sqrt[b] Sqrt[c] Log[2 Sqrt[a + b E^(e z)] Sqrt[c + d E^(e z)] + (a d + b (c + 2 d E^(e z)))/(Sqrt[b] Sqrt[d])]))/ (Sqrt[c] Sqrt[d] e Sqrt[a + b E^(e z)])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]], RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]], RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox["a"]]], " ", SqrtBox["d"], " ", RowBox[List["Log", "[", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SqrtBox["c"]]]], RowBox[List["(", RowBox[List["e", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", "c"]], "+", RowBox[List["b", " ", "c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], "+", RowBox[List["a", " ", "d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox["c"], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]]]]], ")"]]]], ")"]]]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", SqrtBox["c"], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["a", " ", "d"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]], ")"]]]]]], RowBox[List[SqrtBox["b"], " ", SqrtBox["d"]]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox["c"], " ", SqrtBox["d"], " ", "e", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]]], ")"]]]]]]]]

 MathML Form

 a + b e z c + d e z z 1 c d e a + b e z ( a + b e z c + d e z c + d e z ( b c log ( a d + b ( c + 2 d e z ) b d + 2 a + b e z c + d e z ) - a d log ( e - e z ( b e z c + 2 a c + 2 a a + b e z c + d e z c + a d e z ) a 3 / 2 c ) ) ) z a b e z c d e z -1 1 2 1 c 1 2 d 1 2 e a b e z 1 2 -1 a b e z c d e z -1 1 2 c d e z 1 2 b 1 2 c 1 2 a d b c 2 d e z b 1 2 d 1 2 -1 2 a b e z 1 2 c d e z 1 2 -1 a 1 2 d 1 2 e -1 e z b e z c 2 a c 2 a 1 2 a b e z 1 2 c d e z 1 2 c 1 2 a d e z a 3 2 c 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SqrtBox[FractionBox[RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e_", " ", "z_"]]]]]]], RowBox[List["c_", "+", RowBox[List["d_", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e_", " ", "z_"]]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]], RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox["a"]]], " ", SqrtBox["d"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["e", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", "c"]], "+", RowBox[List["b", " ", "c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], "+", RowBox[List["a", " ", "d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox["c"], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SqrtBox["c"]]]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", SqrtBox["c"], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["a", " ", "d"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]], ")"]]]]]], RowBox[List[SqrtBox["b"], " ", SqrtBox["d"]]]]]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox["c"], " ", SqrtBox["d"], " ", "e", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]]]]]]]]

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

 2002-12-18