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http://functions.wolfram.com/01.03.21.0296.01
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Integrate[a^(d z + e) h^(c Sqrt[z] + f z + g), z] ==
(1/2) a^e h^g
(-((c Sqrt[Pi] Erfi[(c Log[h] + 2 Sqrt[z] (d Log[a] + f Log[h]))/
(2 Sqrt[d Log[a] + f Log[h]])] Log[h])/
E^((c^2 Log[h]^2)/(4 (d Log[a] + f Log[h])))/(d Log[a] + f Log[h])^
(3/2)) + (2 a^(d z) h^(c Sqrt[z] + f z))/(d Log[a] + f Log[h]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["d", " ", "z"]], "+", "e"]]], SuperscriptBox["h", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]], "+", "g"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "e"], " ", SuperscriptBox["h", "g"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "h", "]"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]]], "]"]], " ", RowBox[List["Log", "[", "h", "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["a", RowBox[List["d", " ", "z"]]], " ", SuperscriptBox["h", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]]]]], RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> a </mi> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> e </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> h </mi> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> g </mi> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> a </mi> <mi> e </mi> </msup> <mo> ⁢ </mo> <msup> <mi> h </mi> <mi> g </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> h </mi> <mrow> <mrow> <msqrt> <mi> 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</mo> <mi> h </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> h </mi> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> h </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <ci> e </ci> </apply> </apply> <apply> <power /> <ci> h </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <ci> f </ci> <ci> z </ci> </apply> <ci> g </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <ci> a </ci> <ci> e </ci> </apply> <apply> <power /> <ci> h </ci> <ci> g </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> h </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <ci> f </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> d </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> <apply> <times /> <ci> f </ci> <apply> <ln /> <ci> h </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> h </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <ci> d </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> <apply> <times /> <ci> f </ci> <apply> <ln /> <ci> h </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> c </ci> <apply> <ln /> <ci> h </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times 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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["a_", RowBox[List[RowBox[List["d_", " ", "z_"]], "+", "e_"]]], " ", SuperscriptBox["h_", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", RowBox[List["f_", " ", "z_"]], "+", "g_"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "e"], " ", SuperscriptBox["h", "g"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "h", "]"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]]], "]"]], " ", RowBox[List["Log", "[", "h", "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["a", RowBox[List["d", " ", "z"]]], " ", SuperscriptBox["h", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]]]]], RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
