This is not too hard. I came up with this tellsimp rule that allows integration of gamma_incomplete of powers of x. I
think it is correct. You have to ratsimp() the result though.
In pw.mac.
matchdeclare(a,true,x,true,n,nonnegintegerp)$
block([simp:false],tellsimp('integrate(gamma_incomplete(s, x^n), x),x*(gamma_incomplete(s, x^n)) -
(x*(gamma_incomplete(1/n + s, x^n)))/(x^n)^(1/n)))$
remove([a,x,n],matchdeclare)$
/* end of pw */
(%i1) display2d:false;
(out1) false
(%i2) integrate(gamma_incomplete(s,x^15),x);
(out2) gamma_incomplete(s,x^15)*x-gamma_incomplete(s+1/15,x^15)
(%i3) diff(%,x);
(out3) 15*x^(15*(s-14/15)+14)*%e^-x^15-15*x^(15*(s-1)+15)*%e^-x^15+gamma_incomplete(s,x^15)
(%i4) ratsimp(%);
(out4) gamma_incomplete(s,x^15)
(%i5) integrate(gamma_incomplete(s,x^5),x);
(out5) gamma_incomplete(s,x^5)*x-gamma_incomplete(s+1/5,x^5)
(%i6) diff(%,x);
(out6) 5*x^(5*(s-4/5)+4)*%e^-x^5-5*x^(5*(s-1)+5)*%e^-x^5+gamma_incomplete(s,x^5)
(%i7) ratsimp(%);
(out7) gamma_incomplete(s,x^5)
HTH,
Rich
--------------------------------------------------
From: "Richard Hennessy" <rich.hennessy at verizon.net>
Sent: Saturday, March 06, 2010 2:10 PM
To: "Dieter Kaiser" <drdieterkaiser at web.de>
Cc: "Maxima List" <maxima at math.utexas.edu>
Subject: Re: [Maxima] Integrating gamma_incomplete
> I also noticed that maxima cannot integrate gamma_incomplete(s, +-x^n) for n=1,2,3,4, etc. which Mathematica can do.
> I hope that can be done too.
>
> Rich
>
>
> --------------------------------------------------
> From: "Dieter Kaiser" <drdieterkaiser at web.de>
> Sent: Saturday, March 06, 2010 7:49 AM
> To: "Richard Hennessy" <rich.hennessy at verizon.net>
> Cc: "Maxima List" <maxima at math.utexas.edu>
> Subject: Re: [Maxima] Integrating gamma_incomplete
>
>> Am Freitag, den 05.03.2010, 22:07 -0500 schrieb Richard Hennessy:
>>> I noticed Maxima can only integrate gamma_incomplete a couple times.
>>> Mathematica can do it as many times as you want. Is this a weakness
>>> in integrate()?
>>
>> Maxima can only integrate the direct function gamma_incomplete, but not
>> the case when a power is involved. Therefore, we get:
>>
>> (%i2) integrate(gamma_incomplete(a,x),x);
>> (%o2) gamma_incomplete(a,x)*x-gamma_incomplete(a+1,x)
>>
>> We get a noun form, when we repeat the integration:
>>
>> (%i3) integrate(%,x);
>> (%o3) 'integrate(gamma_incomplete(a,x)*x,x)
>> -gamma_incomplete(a+1,x)*x+gamma_incomplete(a+2,x)
>>
>> I have already proposed an extension on the mailing list
>> http://www.math.utexas.edu/pipermail/maxima/2010/020534.html to add the
>> integrals of the type x^v*gamma_incomplete(a,x). With this extension we
>> will get:
>>
>> (%i5) integrate(gamma_incomplete(a,x),x);
>> (%o5) gamma_incomplete(a,x)*x-gamma_incomplete(a+1,x)
>>
>> (%i6) integrate(%,x);
>> (%o6) (gamma_incomplete(a,x)*x^2-gamma_incomplete(a+2,x))/2
>> -gamma_incomplete(a+1,x)*x+gamma_incomplete(a+2,x)
>>
>> (%i7) integrate(%,x);
>> (%o7) ((gamma_incomplete(a,x)*x^3-gamma_incomplete(a+3,x))/3
>> -gamma_incomplete(a+2,x)*x+gamma_incomplete(a+3,x))
>> /2
>> -(gamma_incomplete(a+1,x)*x^2-gamma_incomplete(a+3,x))/2
>> +gamma_incomplete(a+2,x)*x-gamma_incomplete(a+3,x)
>>
>> All integrals are solved by Maxima.
>>
>> Dieter Kaiser
>>
>>
>>
>
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