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Exp






Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Integration > Indefinite integration > Involving functions of the direct function and a power function > Involving algebraic functions of the direct function and a power function > Involving zn ef z(a+b cd z)beta





http://functions.wolfram.com/01.03.21.0798.01









  


  










Input Form





Integrate[z^4 e^(f z) (a + b c^(d z))^\[Beta], z] == (1/(f^5 Log[e]^5)) (((a + b c^(d z))^\[Beta] e^(f z) (24 HypergeometricPFQ[{-\[Beta], (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c])}, {1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c])}, -((b c^(d z))/a)] + f z Log[e] (-24 HypergeometricPFQ[{-\[Beta], (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c])}, {1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c])}, -((b c^(d z))/a)] + f z Log[e] (12 HypergeometricPFQ[{-\[Beta], (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c])}, {1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c])}, -((b c^(d z))/a)] + f z Log[e] (-4 HypergeometricPFQ[{-\[Beta], (f Log[e])/(d Log[c]), (f Log[e])/(d Log[c])}, {1 + (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c])}, -((b c^(d z))/a)] + f z Hypergeometric2F1[-\[Beta], (f Log[e])/(d Log[c]), 1 + (f Log[e])/(d Log[c]), -((b c^(d z))/a)] Log[e])))))/ (1 + (b c^(d z))/a)^\[Beta])










Standard Form





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MathML Form







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</mo> <mo> ( </mo> <mi> e </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> f </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> e </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mi> a </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;5&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;4&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, &quot;\[Beta]&quot;]], HypergeometricPFQ], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;f&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;e&quot;, &quot;)&quot;]]]], RowBox[List[&quot;d&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;c&quot;, &quot;)&quot;]]]]], HypergeometricPFQ], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;f&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;e&quot;, &quot;)&quot;]]]], RowBox[List[&quot;d&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;c&quot;, &quot;)&quot;]]]]], HypergeometricPFQ], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;f&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;e&quot;, &quot;)&quot;]]]], RowBox[List[&quot;d&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;c&quot;, &quot;)&quot;]]]]], HypergeometricPFQ], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;f&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;e&quot;, &quot;)&quot;]]]], RowBox[List[&quot;d&quot;, &quot; 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Rule Form





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





2002-12-18