\(\int x^{100} \Gamma (\frac {1}{2},a x) \, dx\) [60]

Optimal result

Integrand size = 11, antiderivative size = 29 \[ \int x^{100} \Gamma \left (\frac {1}{2},a x\right ) \, dx=\frac {1}{101} x^{101} \Gamma \left (\frac {1}{2},a x\right )-\frac {\Gamma \left (\frac {203}{2},a x\right )}{101 a^{101}} \] Output:

1/101*x^101*Pi^(1/2)*erfc((a*x)^(1/2))-1/101*GAMMA(203/2,a*x)/a^101
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int x^{100} \Gamma \left (\frac {1}{2},a x\right ) \, dx=\frac {1}{101} x^{101} \Gamma \left (\frac {1}{2},a x\right )-\frac {\Gamma \left (\frac {203}{2},a x\right )}{101 a^{101}} \] Input:

Integrate[x^100*Gamma[1/2, a*x],x]
 

Output:

(x^101*Gamma[1/2, a*x])/101 - Gamma[203/2, a*x]/(101*a^101)
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {7116}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^{100} \Gamma \left (\frac {1}{2},a x\right ) \, dx\)

\(\Big \downarrow \) 7116

\(\displaystyle \frac {1}{101} x^{101} \Gamma \left (\frac {1}{2},a x\right )-\frac {\Gamma \left (\frac {203}{2},a x\right )}{101 a^{101}}\)

Input:

Int[x^100*Gamma[1/2, a*x],x]
 

Output:

(x^101*Gamma[1/2, a*x])/101 - Gamma[203/2, a*x]/(101*a^101)
 

Defintions of rubi rules used

rule 7116

Int[Gamma[n_, (b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 
1)*(Gamma[n, b*x]/(d*(m + 1))), x] - Simp[(d*x)^m*(Gamma[m + n + 1, b*x]/(b 
*(m + 1)*(b*x)^m)), x] /; FreeQ[{b, d, m, n}, x] && NeQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1345\) vs. \(2(25)=50\).

Time = 0.19 (sec) , antiderivative size = 1346, normalized size of antiderivative = 46.41

method	result	size
parts	\(\text {Expression too large to display}\)	\(1346\)
derivativedivides	\(\text {Expression too large to display}\)	\(1354\)
default	\(\text {Expression too large to display}\)	\(1354\)

Input:

int(x^100*Pi^(1/2)*erfc((x*a)^(1/2)),x,method=_RETURNVERBOSE)
 

Output:

1/101*x^101*Pi^(1/2)*erfc((x*a)^(1/2))+2/101/a^101*(-481452080155966748007 
633880476899108122537146653367093423140625/536870912*(x*a)^(145/2)/exp(x*a 
)+133992804268466370262665421635374186682822236732455213600900017590393037 
39739562002013398089719088133024853378593435555958802327570687024940685390 
27184400769291974725364464819431304931640625/50706024009129176059868128215 
04*Pi^(1/2)*erf((x*a)^(1/2))-133992804268466370262665421635374186682822236 
73245521360090001759039303739739562002013398089719088133024853378593435555 
95880232757068702494068539027184400769291974725364464819431304931640625/25 
35301200456458802993406410752*(x*a)^(1/2)/exp(x*a)-21645665733525127277259 
87903663070822182164893629324197134542032080668044388757651684870093650125 
52081051815931846822673833447497233617582916665391768573616754055023193359 
375/154742504910672534362390528*(x*a)^(29/2)/exp(x*a)-44852123721630074770 
64823209929735140263860411364401779216831618206677014383991832602406973417 
51009722721429443359375/36028797018963968*(x*a)^(93/2)/exp(x*a)-5280756135 
18318762367820462505858445619523667969761004515923009468510000811065748442 
74393397042974579078560179163823409966274340229687013435795848903656005859 
375/604462909807314587353088*(x*a)^(45/2)/exp(x*a)-20465381695240335899649 
09098246082727925626897391951221975288291426389417598349668832623339618618 
30147452263038623045050485994188828211500164176919618912718866498287916183 
4716796875/4951760157141521099596496896*(x*a)^(19/2)/exp(x*a)-974541985...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (25) = 50\).

Time = 0.51 (sec) , antiderivative size = 848, normalized size of antiderivative = 29.24 \[ \int x^{100} \Gamma \left (\frac {1}{2},a x\right ) \, dx=\text {Too large to display} \] Input:

integrate(x^100*pi^(1/2)*erfc((a*x)^(1/2)),x, algorithm="fricas")
 

Output:

1/256065421246102339102334047485952*(2535301200456458802993406410752*sqrt( 
pi)*a^101*x^101 - sqrt(pi)*(2535301200456458802993406410752*a^101*x^101 - 
13399280426846637026266542163537418668282223673245521360090001759039303739 
73956200201339808971908813302485337859343555595880232757068702494068539027 
184400769291974725364464819431304931640625)*erf(sqrt(a*x)) - 2*(1267650600 
228229401496703205376*a^100*x^100 + 127398885322937054850418672140288*a^99 
*x^99 + 12676189089632236957616657877958656*a^98*x^98 + 124860462532877534 
0325240800978927616*a^97*x^97 + 121738950969555595681710978095445442560*a^ 
96*x^96 + 11747808768562114983285109386210485207040*a^95*x^95 + 1121915737 
397681980903727946383101337272320*a^94*x^94 + 1060210371840809471954022909 
33203076372234240*a^93*x^93 + 99129669767115685627701142022544876408039014 
40*a^92*x^92 + 916949445345820092056235563708540106774360883200*a^91*x^91 
+ 83900874249142538423145554079331419769854020812800*a^90*x^90 + 759302911 
9547399727294672644179493489171788883558400*a^89*x^89 + 679576106199492275 
592873201654064667280875105078476800*a^88*x^88 + 6014248539865506638996927 
8346384723054357446799445196800*a^87*x^87 + 526246747238231830912231185530 
8663267256276594951454720000*a^86*x^86 + 455203436361070533739079975484199 
372617667925463300833280000*a^85*x^85 + 3891989380887153063469133790389904 
6358810607627112221245440000*a^84*x^84 + 328873102684964433863141805287946 
9417319496344490982695239680000*a^83*x^83 + 274609040741945302275723407...
 

Sympy [F(-1)]

Timed out. \[ \int x^{100} \Gamma \left (\frac {1}{2},a x\right ) \, dx=\text {Timed out} \] Input:

integrate(x**100*pi**(1/2)*erfc((a*x)**(1/2)),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (25) = 50\).

Time = 0.04 (sec) , antiderivative size = 755, normalized size of antiderivative = 26.03 \[ \int x^{100} \Gamma \left (\frac {1}{2},a x\right ) \, dx=\text {Too large to display} \] Input:

integrate(x^100*pi^(1/2)*erfc((a*x)^(1/2)),x, algorithm="maxima")
 

Output:

1/256065421246102339102334047485952*sqrt(pi)*(2535301200456458802993406410 
752*a^101*x^101*erfc(sqrt(a*x)) - (2*(1267650600228229401496703205376*(a*x 
)^(201/2) + 127398885322937054850418672140288*(a*x)^(199/2) + 126761890896 
32236957616657877958656*(a*x)^(197/2) + 1248604625328775340325240800978927 
616*(a*x)^(195/2) + 121738950969555595681710978095445442560*(a*x)^(193/2) 
+ 11747808768562114983285109386210485207040*(a*x)^(191/2) + 11219157373976 
81980903727946383101337272320*(a*x)^(189/2) + 1060210371840809471954022909 
33203076372234240*(a*x)^(187/2) + 9912966976711568562770114202254487640803 
901440*(a*x)^(185/2) + 916949445345820092056235563708540106774360883200*(a 
*x)^(183/2) + 83900874249142538423145554079331419769854020812800*(a*x)^(18 
1/2) + 7593029119547399727294672644179493489171788883558400*(a*x)^(179/2) 
+ 679576106199492275592873201654064667280875105078476800*(a*x)^(177/2) + 6 
0142485398655066389969278346384723054357446799445196800*(a*x)^(175/2) + 52 
62467472382318309122311855308663267256276594951454720000*(a*x)^(173/2) + 4 
55203436361070533739079975484199372617667925463300833280000*(a*x)^(171/2) 
+ 38919893808871530634691337903899046358810607627112221245440000*(a*x)^(16 
9/2) + 3288731026849644338631418052879469417319496344490982695239680000*(a 
*x)^(167/2) + 274609040741945302275723407415435696346177944764997055052513 
280000*(a*x)^(165/2) + 226552458612104874377471811117734449485596804431122 
57041832345600000*(a*x)^(163/2) + 1846402537688654726176395260609535763...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1362 vs. \(2 (25) = 50\).

Time = 0.12 (sec) , antiderivative size = 1362, normalized size of antiderivative = 46.97 \[ \int x^{100} \Gamma \left (\frac {1}{2},a x\right ) \, dx=\text {Too large to display} \] Input:

integrate(x^100*pi^(1/2)*erfc((a*x)^(1/2)),x, algorithm="giac")
 

Output:

1/256065421246102339102334047485952*sqrt(pi)*(2535301200456458802993406410 
752*a*x^101 - (2535301200456458802993406410752*a^101*x^101*erf(sqrt(a*x)) 
+ (2*(1267650600228229401496703205376*sqrt(a*x)*a^100*x^100 + 127398885322 
937054850418672140288*sqrt(a*x)*a^99*x^99 + 126761890896322369576166578779 
58656*sqrt(a*x)*a^98*x^98 + 1248604625328775340325240800978927616*sqrt(a*x 
)*a^97*x^97 + 121738950969555595681710978095445442560*sqrt(a*x)*a^96*x^96 
+ 11747808768562114983285109386210485207040*sqrt(a*x)*a^95*x^95 + 11219157 
37397681980903727946383101337272320*sqrt(a*x)*a^94*x^94 + 1060210371840809 
47195402290933203076372234240*sqrt(a*x)*a^93*x^93 + 9912966976711568562770 
114202254487640803901440*sqrt(a*x)*a^92*x^92 + 916949445345820092056235563 
708540106774360883200*sqrt(a*x)*a^91*x^91 + 839008742491425384231455540793 
31419769854020812800*sqrt(a*x)*a^90*x^90 + 7593029119547399727294672644179 
493489171788883558400*sqrt(a*x)*a^89*x^89 + 679576106199492275592873201654 
064667280875105078476800*sqrt(a*x)*a^88*x^88 + 601424853986550663899692783 
46384723054357446799445196800*sqrt(a*x)*a^87*x^87 + 5262467472382318309122 
311855308663267256276594951454720000*sqrt(a*x)*a^86*x^86 + 455203436361070 
533739079975484199372617667925463300833280000*sqrt(a*x)*a^85*x^85 + 389198 
93808871530634691337903899046358810607627112221245440000*sqrt(a*x)*a^84*x^ 
84 + 3288731026849644338631418052879469417319496344490982695239680000*sqrt 
(a*x)*a^83*x^83 + 27460904074194530227572340741543569634617794476499705...
 

Mupad [F(-1)]

Timed out. \[ \int x^{100} \Gamma \left (\frac {1}{2},a x\right ) \, dx=\int \sqrt {\Pi }\,x^{100}\,\mathrm {erfc}\left (\sqrt {a\,x}\right ) \,d x \] Input:

int(Pi^(1/2)*x^100*erfc((a*x)^(1/2)),x)
 

Output:

int(Pi^(1/2)*x^100*erfc((a*x)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 2.59 (sec) , antiderivative size = 1266, normalized size of antiderivative = 43.66 \[ \int x^{100} \Gamma \left (\frac {1}{2},a x\right ) \, dx =\text {Too large to display} \] Input:

int(x^100*Pi^(1/2)*erfc((a*x)^(1/2)),x)
 

Output:

( - 2535301200456458802993406410752*sqrt(pi)*e**(a*x)*erf(sqrt(x)*sqrt(a)) 
*a**101*x**101 + 133992804268466370262665421635374186682822236732455213600 
90001759039303739739562002013398089719088133024853378593435555958802327570 
68702494068539027184400769291974725364464819431304931640625*sqrt(pi)*e**(a 
*x)*erf(sqrt(x)*sqrt(a)) - 2535301200456458802993406410752*sqrt(x)*sqrt(a) 
*a**100*x**100 - 254797770645874109700837344280576*sqrt(x)*sqrt(a)*a**99*x 
**99 - 25352378179264473915233315755917312*sqrt(x)*sqrt(a)*a**98*x**98 - 2 
497209250657550680650481601957855232*sqrt(x)*sqrt(a)*a**97*x**97 - 2434779 
01939111191363421956190890885120*sqrt(x)*sqrt(a)*a**96*x**96 - 23495617537 
124229966570218772420970414080*sqrt(x)*sqrt(a)*a**95*x**95 - 2243831474795 
363961807455892766202674544640*sqrt(x)*sqrt(a)*a**94*x**94 - 2120420743681 
61894390804581866406152744468480*sqrt(x)*sqrt(a)*a**93*x**93 - 19825933953 
423137125540228404508975281607802880*sqrt(x)*sqrt(a)*a**92*x**92 - 1833898 
890691640184112471127417080213548721766400*sqrt(x)*sqrt(a)*a**91*x**91 - 1 
67801748498285076846291108158662839539708041625600*sqrt(x)*sqrt(a)*a**90*x 
**90 - 15186058239094799454589345288358986978343577767116800*sqrt(x)*sqrt( 
a)*a**89*x**89 - 1359152212398984551185746403308129334561750210156953600*s 
qrt(x)*sqrt(a)*a**88*x**88 - 120284970797310132779938556692769446108714893 
598890393600*sqrt(x)*sqrt(a)*a**87*x**87 - 1052493494476463661824462371061 
7326534512553189902909440000*sqrt(x)*sqrt(a)*a**86*x**86 - 910406872722...