\(\int x^{100} \Gamma (0,a x) \, dx\) [1]

Optimal result

Integrand size = 9, antiderivative size = 25 \[ \int x^{100} \Gamma (0,a x) \, dx=\frac {1}{101} x^{101} \Gamma (0,a x)-\frac {\Gamma (101,a x)}{101 a^{101}} \] Output:

1/101*x^101*Ei(1,a*x)-1/101*GAMMA(101,a*x)/a^101
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^{100} \Gamma (0,a x) \, dx=\frac {1}{101} x^{101} \Gamma (0,a x)-\frac {\Gamma (101,a x)}{101 a^{101}} \] Input:

Integrate[x^100*Gamma[0, a*x],x]
 

Output:

(x^101*Gamma[0, a*x])/101 - Gamma[101, a*x]/(101*a^101)
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

\(\displaystyle \int x^{100} \Gamma (0,a x) \, dx\)

\(\Big \downarrow \) 7116

\(\displaystyle \frac {1}{101} x^{101} \Gamma (0,a x)-\frac {\Gamma (101,a x)}{101 a^{101}}\)

Input:

Int[x^100*Gamma[0, a*x],x]
 

Output:

(x^101*Gamma[0, a*x])/101 - Gamma[101, 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. \(1321\) vs. \(2(21)=42\).

Time = 0.03 (sec) , antiderivative size = 1322, normalized size of antiderivative = 52.88

\[\text {Expression too large to display}\]

Input:

int(x^100*Ei(1,x*a),x)
 

Output:

1/a^101*(-1055516033003632107099290624742660954185524161191648021896839298 
5597157379491687762424432261634564520258043904000000000000000000/101*x^29* 
a^29*exp(-x*a)-90317312543607677060255118337640856431754876050390050366161 
55614149386741065171711031801018481922539520000000000000000/101*x^35*a^35* 
exp(-x*a)-331284225412682501619179520000/101*x^85*a^85*exp(-x*a)-113496347 
63479915130099899190781300582640044743996215289213325790964685354292137378 
951002431865123140062412800000000000000000/101*x^31*a^31*exp(-x*a)-2365369 
369446553061560941772800000/101*x^83*a^83*exp(-x*a)-9332621544394415268169 
92388562667004907159682643816214685929638952175999932299156089414639761565 
18286253697920827223758251185210916864000000000000000000000000/101*exp(-x* 
a)-16098703928453240136983769705676800000/101*x^81*a^81*exp(-x*a)-69028187 
8632192000/101*x^91*a^91*exp(-x*a)-104319601456376996087654827692785664000 
000/101*x^79*a^79*exp(-x*a)-2508814237322435473895975509378912678659857668 
06639028794893211504149631696254769750883361624497848320000000000000000/10 
1*x^36*a^36*exp(-x*a)-3060996495710533110587942811753716767138020067455779 
26350083396582317564005258945110308535587402371087483273216000000000000000 
000/101*x^28*a^28*exp(-x*a)-1498729784938957529766067890061756345678433710 
45720003807587994398890718324290671405724972006126137054582663222880552700 
66094080000000000000000000000/101*x^13*a^13*exp(-x*a)-19632565766406390410 
9558167142400000/101*x^82*a^82*exp(-x*a)-231411335075716303160448476568...
 

Fricas [F(-2)]

Exception generated. \[ \int x^{100} \Gamma (0,a x) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^100*exp_integral_e(1,a*x),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >> An error occurred when FriCAS evaluated ((x 
)^(((100)::EXPR INT)))*(exp_integral_e(((1)::EXPR INT),(a)*(x))):   There 
are no library operations named exp_integral_e       Use HyperDoc Browse o 
r issue
 

Sympy [F(-1)]

Timed out. \[ \int x^{100} \Gamma (0,a x) \, dx=\text {Timed out} \] Input:

integrate(x**100*expint(1,a*x),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{100} \Gamma (0,a x) \, dx=\int { x^{100} E_{1}\left (a x\right ) \,d x } \] Input:

integrate(x^100*exp_integral_e(1,a*x),x, algorithm="maxima")
 

Output:

integrate(x^100*exp_integral_e(1, a*x), x)
 

Giac [F]

\[ \int x^{100} \Gamma (0,a x) \, dx=\int { x^{100} E_{1}\left (a x\right ) \,d x } \] Input:

integrate(x^100*exp_integral_e(1,a*x),x, algorithm="giac")
 

Output:

integrate(x^100*exp_integral_e(1, a*x), x)
 

Mupad [B] (verification not implemented)

Time = 2.51 (sec) , antiderivative size = 826, normalized size of antiderivative = 33.04 \[ \int x^{100} \Gamma (0,a x) \, dx=\text {Too large to display} \] Input:

int(x^100*expint(a*x),x)
 

Output:

-(x^101*(exp(-a*x)*(1/(a*x) + 100/(a^2*x^2) + 9900/(a^3*x^3) + 970200/(a^4 
*x^4) + 94109400/(a^5*x^5) + 9034502400/(a^6*x^6) + 858277728000/(a^7*x^7) 
 + 80678106432000/(a^8*x^8) + 7503063898176000/(a^9*x^9) + 690281878632192 
000/(a^10*x^10) + 62815650955529472000/(a^11*x^11) + 565340858599765248000 
0/(a^12*x^12) + 503153364153791070720000/(a^13*x^13) + 4427749604553361422 
3360000/(a^14*x^14) + 3852142155961424437432320000/(a^15*x^15) + 331284225 
412682501619179520000/(a^16*x^16) + 28159159160078012637630259200000/(a^17 
*x^17) + 2365369369446553061560941772800000/(a^18*x^18) + 1963256576640639 
04109558167142400000/(a^19*x^19) + 16098703928453240136983769705676800000/ 
(a^20*x^20) + 1303995018204712451095685346159820800000/(a^21*x^21) + 10431 
9601456376996087654827692785664000000/(a^22*x^22) + 8241248515053782690924 
731387730067456000000/(a^23*x^23) + 64281738417419504989212904824294526156 
8000000/(a^24*x^24) + 49496938581413018841693936714706785140736000000/(a^2 
5*x^25) + 3761767332187389431968739190317715670695936000000/(a^26*x^26) + 
282132549914054207397655439273828675302195200000000/(a^27*x^27) + 20877808 
693640011347426502506263321972362444800000000/(a^28*x^28) + 15240800346357 
20828362134682957222503982458470400000000/(a^29*x^29) + 109733762493771899 
642073697172920020286737009868800000000/(a^30*x^30) + 77910971370578048745 
87232499277321440358327700684800000000/(a^31*x^31) + 545376799594046341221 
106274949412500825082939047936000000000/(a^32*x^32) + 37630999171989197...
 

Reduce [F]

\[ \int x^{100} \Gamma (0,a x) \, dx=\int \mathit {ei} \left (1, a x \right ) x^{100}d x \] Input:

int(x^100*Ei(1,a*x),x)
 

Output:

int(ei(1,a*x)*x**100,x)