Infinity appears as a factor in Integrate result!

• To: mathgroup at smc.vnet.net
• Subject: [mg74862] Infinity appears as a factor in Integrate result!
• From: "dimitris" <dimmechan at yahoo.com>
• Date: Mon, 9 Apr 2007 06:09:42 -0400 (EDT)

```Consider the integral

In[13]:=
f = HoldForm[Integrate[1/(x^3 + a*x^2 + b*x + 1), {x, 0, Infinity}]]

Then

In[17]:=
Integrate[1/(x^3 + a*x^2 + b*x + 1), {x, 0, Infinity}]

Out[17]=
If[(Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 1]] != 0) &&
(Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 2]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 2]] != 0) &&
(Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 3]] != 0),
-(Log[-Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]]/(b + 2*a*Root[1 + b*#1
+ a*#1^2 + #1^3 & , 1] +
3*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]^2)) - Log[-Root[1 + b*#1
+ a*#1^2 + #1^3 & , 2]]/
(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 2] + 3*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 2]^2) -
Log[-Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]]/(b + 2*a*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 3] +
3*Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]^2) +
Infinity*(1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1] +
3*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]^2) +
1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 2] + 3*Root[1 + b*#1
+ a*#1^2 + #1^3 & , 2]^2) +
1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 3] + 3*Root[1 + b*#1
+ a*#1^2 + #1^3 & , 3]^2)),
Integrate[1/(1 + b*x + a*x^2 + x^3), {x, 0, Infinity},
Assumptions ->  !((Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] <= 0 ||
Im[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] != 0) &&
(Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 2]] <= 0 || Im[Root[1 +
b*#1 + a*#1^2 + #1^3 & , 2]] != 0) &&
(Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]] <= 0 || Im[Root[1 +
b*#1 + a*#1^2 + #1^3 & , 3]] != 0))]]

In the results it appears Infinity!. For anyone who dosn't believe me
try:

In[18]:=
%[[2]]

Out[18]=
-(Log[-Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]]/(b + 2*a*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 1] +
3*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]^2)) - Log[-Root[1 + b*#1 +
a*#1^2 + #1^3 & , 2]]/
(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 2] + 3*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 2]^2) -
Log[-Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]]/(b + 2*a*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 3] +
3*Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]^2) +
Infinity*(1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1] + 3*Root[1
+ b*#1 + a*#1^2 + #1^3 & , 1]^2) +
1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 2] + 3*Root[1 + b*#1
+ a*#1^2 + #1^3 & , 2]^2) +
1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 3] + 3*Root[1 + b*#1
+ a*#1^2 + #1^3 & , 3]^2))

The following term is multiplied with Infinity and the result of this
product
appeared in the results!

In[19]:=
Cases[%, (a_)*Infinity -> a]

Out[19]=
{1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 1] + 3*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 1]^2) +
1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 2] + 3*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 2]^2) +
1/(b + 2*a*Root[1 + b*#1 + a*#1^2 + #1^3 & , 3] + 3*Root[1 + b*#1 +
a*#1^2 + #1^3 & , 3]^2)}

Obviolusly Integarate algorithm failes for this integral.

What is funnier is that

In[20]:=
Integrate[1/(x^3 + a*x^2 + b*x + 1), {x, 0, Infinity}]

Out[20]=
If[(Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 1]] != 0) &&
(Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 2]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 2]] != 0) &&
(Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]] <= 0 || Im[Root[1 + b*#1
+ a*#1^2 + #1^3 & , 3]] != 0), ComplexInfinity,
Integrate[1/(1 + x*(b + x*(a + x))), {x, 0, Infinity},
Assumptions -> (Im[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] == 0 &&
Re[Root[1 + b*#1 + a*#1^2 + #1^3 & , 1]] > 0) ||
(Im[Root[1 + b*#1 + a*#1^2 + #1^3 & , 2]] == 0 && Re[Root[1 +
b*#1 + a*#1^2 + #1^3 & , 2]] > 0) ||
(Im[Root[1 + b*#1 + a*#1^2 + #1^3 & , 3]] == 0 && Re[Root[1 +
b*#1 + a*#1^2 + #1^3 & , 3]] > 0)]]

I.e. if you try again to obtain the integral Mathematica returns
ComplexInfinity!

Dimitris

```

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