Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Exp






Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Arguments involving algebraic functions > Involving a z+b z1/2+c





http://functions.wolfram.com/01.03.21.0164.01









  


  










Input Form





Integrate[d^(a z + b Sqrt[z] + c), z] == d^(c + b Sqrt[z] + a z)/(a Log[d]) - ((b Sqrt[Pi])/(2 a^(3/2) Sqrt[Log[d]])) d^(-(b^2/(4 a)) + c) Erfi[((b + 2 a Sqrt[z]) Sqrt[Log[d]])/(2 Sqrt[a])]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["d", RowBox[List[RowBox[List["a", " ", "z"]], "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", "c"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["d", RowBox[List["c", "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["a", " ", "z"]]]]], RowBox[List["a", " ", RowBox[List["Log", "[", "d", "]"]]]]], "-", RowBox[List[FractionBox[RowBox[List["b", " ", SqrtBox["\[Pi]"]]], RowBox[List["2", " ", SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List["Log", "[", "d", "]"]]]]]], SuperscriptBox["d", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "a"]]]]], "+", "c"]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", SqrtBox["z"]]]]], ")"]], " ", SqrtBox[RowBox[List["Log", "[", "d", "]"]]]]], RowBox[List["2", " ", SqrtBox["a"]]]], "]"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <msup> <mi> d </mi> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msup> <mi> d </mi> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> </msup> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> d </mi> <mrow> <mi> c </mi> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <ci> d </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> b </ci> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <ci> c </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> d </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> b </ci> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <ci> c </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> b </ci> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> d </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> d </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <ln /> <ci> d </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["d_", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", RowBox[List["b_", " ", SqrtBox["z_"]]], "+", "c_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox["d", RowBox[List["c", "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["a", " ", "z"]]]]], RowBox[List["a", " ", RowBox[List["Log", "[", "d", "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", " ", SqrtBox["\[Pi]"]]], ")"]], " ", SuperscriptBox["d", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "a"]]]]], "+", "c"]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", SqrtBox["z"]]]]], ")"]], " ", SqrtBox[RowBox[List["Log", "[", "d", "]"]]]]], RowBox[List["2", " ", SqrtBox["a"]]]], "]"]]]], RowBox[List["2", " ", SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List["Log", "[", "d", "]"]]]]]]]]]]]]










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





2002-12-18





© 1998-2014 Wolfram Research, Inc.