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 Exp

 http://functions.wolfram.com/01.03.21.0791.01

 Input Form

 Integrate[z^3/(a E^(2 d z) + b E^(d z) + c), z] == (-(1/(2 Sqrt[b^2 - 4 a c]))) (a (z^4/(-b + Sqrt[b^2 - 4 a c]) + z^4/(b + Sqrt[b^2 - 4 a c]) + (4 z^3 Log[1 + (2 a E^(d z))/(b - Sqrt[b^2 - 4 a c])])/ ((b - Sqrt[b^2 - 4 a c]) d) - (4 z^3 Log[1 + (2 a E^(d z))/(b + Sqrt[b^2 - 4 a c])])/ ((b + Sqrt[b^2 - 4 a c]) d) + (12 z^2 PolyLog[2, (2 a E^(d z))/(-b + Sqrt[b^2 - 4 a c])])/ ((b - Sqrt[b^2 - 4 a c]) d^2) - (12 z^2 PolyLog[2, -((2 a E^(d z))/(b + Sqrt[b^2 - 4 a c]))])/ ((b + Sqrt[b^2 - 4 a c]) d^2) + (24 z PolyLog[3, (2 a E^(d z))/(-b + Sqrt[b^2 - 4 a c])])/ ((-b + Sqrt[b^2 - 4 a c]) d^3) + (24 z PolyLog[3, -((2 a E^(d z))/(b + Sqrt[b^2 - 4 a c]))])/ ((b + Sqrt[b^2 - 4 a c]) d^3) + (24 PolyLog[4, (2 a E^(d z))/(-b + Sqrt[b^2 - 4 a c])])/ ((b - Sqrt[b^2 - 4 a c]) d^4) - (24 PolyLog[4, -((2 a E^(d z))/(b + Sqrt[b^2 - 4 a c]))])/ ((b + Sqrt[b^2 - 4 a c]) d^4)))

 Standard Form

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RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]], ")"]], " ", SuperscriptBox["d", "4"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z 3 a 2 d z + b d z + c z - 1 2 b 2 - 4 a c ( a ( z 4 b 2 - 4 a c - b + z 4 b + b 2 - 4 a c + 4 log ( 2 d z a b - b 2 - 4 a c + 1 ) z 3 ( b - b 2 - 4 a c ) d - 4 log ( 2 d z a b + b 2 - 4 a c + 1 ) z 3 ( b + b 2 - 4 a c ) d + 12 Li PolyLog 2 ( 2 a d z b 2 - 4 a c - b ) z 2 ( b - b 2 - 4 a c ) d 2 - 12 Li PolyLog 2 ( - 2 a d z b + b 2 - 4 a c ) z 2 ( b + b 2 - 4 a c ) d 2 + 24 Li PolyLog 3 ( 2 a d z b 2 - 4 a c - b ) z ( b 2 - 4 a c - b ) d 3 + 24 Li PolyLog 3 ( - 2 a d z b + b 2 - 4 a c ) z ( b + b 2 - 4 a c ) d 3 + 24 Li PolyLog 4 ( 2 a d z b 2 - 4 a c - b ) ( b - b 2 - 4 a c ) d 4 - 24 Li PolyLog 4 ( - 2 a d z b + b 2 - 4 a c ) ( b + b 2 - 4 a c ) d 4 ) ) z z 3 a 2 d z b d z c -1 -1 1 2 b 2 -1 4 a c 1 2 -1 a z 4 b 2 -1 4 a c 1 2 -1 b -1 z 4 b b 2 -1 4 a c 1 2 -1 4 2 d z a b -1 b 2 -1 4 a c 1 2 -1 1 z 3 b -1 b 2 -1 4 a c 1 2 d -1 -1 4 2 d z a b b 2 -1 4 a c 1 2 -1 1 z 3 b b 2 -1 4 a c 1 2 d -1 12 PolyLog 2 2 a d z b 2 -1 4 a c 1 2 -1 b -1 z 2 b -1 b 2 -1 4 a c 1 2 d 2 -1 -1 12 PolyLog 2 -1 2 a d z b b 2 -1 4 a c 1 2 -1 z 2 b b 2 -1 4 a c 1 2 d 2 -1 24 PolyLog 3 2 a d z b 2 -1 4 a c 1 2 -1 b -1 z b 2 -1 4 a c 1 2 -1 b d 3 -1 24 PolyLog 3 -1 2 a d z b b 2 -1 4 a c 1 2 -1 z b b 2 -1 4 a c 1 2 d 3 -1 24 PolyLog 4 2 a d z b 2 -1 4 a c 1 2 -1 b -1 b -1 b 2 -1 4 a c 1 2 d 4 -1 -1 24 PolyLog 4 -1 2 a d z b b 2 -1 4 a c 1 2 -1 b b 2 -1 4 a c 1 2 d 4 -1 [/itex]

 Rule Form

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

 2002-12-18