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http://functions.wolfram.com/01.03.21.0256.01
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Integrate[(a^(c (z^r)^(1/r) + d))^v, z] ==
((a^(c (z^r)^(1/r) + d))^v z)/(z^r)^r^(-1)/(c v Log[a])
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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> <msup> <mrow> <mo> ( </mo> <msup> <mi> a </mi> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> d </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mi> v </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <msup> <mi> a </mi> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> d </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mi> v </mi> </msup> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> / </mo> <mi> r </mi> </mrow> </msup> </mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> v </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> d </ci> </apply> </apply> <ci> v </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> d </ci> </apply> </apply> <ci> v </ci> </apply> <ci> z </ci> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <ci> v </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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