\(\int x^{100} \Gamma (-1,a x) \, dx\) [33]

Optimal result

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

1/101*x^100/a*Ei(2,a*x)-1/101*GAMMA(100,a*x)/a^101
 

Mathematica [A] (verified)

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

Integrate[x^100*Gamma[-1, a*x],x]
 

Output:

(x^101*Gamma[-1, a*x])/101 - Gamma[100, a*x]/(101*a^101)
 

Rubi [A] (verified)

Time = 0.20 (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.

\(\displaystyle \int x^{100} \Gamma (-1,a x) \, dx\)

\(\Big \downarrow \) 7116

\(\displaystyle \frac {1}{101} x^{101} \Gamma (-1,a x)-\frac {\Gamma (100,a x)}{101 a^{101}}\)

Input:

Int[x^100*Gamma[-1, a*x],x]
 

Output:

(x^101*Gamma[-1, a*x])/101 - Gamma[100, 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. \(865\) vs. \(2(24)=48\).

Time = 0.07 (sec) , antiderivative size = 866, normalized size of antiderivative = 34.64

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

Input:

int(x^99/a*Ei(2,x*a),x)
 

Output:

1/a^101*(1/101*(Psi(101)+gamma-1-Psi(102)+ln(x)+ln(a))*x^101*a^101+102/102 
01*x^101*a^101+93326215443944152681699238856266700490715968264381621468592 
96389521759999322991560894146397615651828625369792082722375825118521091686 
40000000000000000000000/101-1/20604*(-204*a^100*x^100+204*a^99*x^99+20196* 
a^98*x^98+1979208*a^97*x^97+191983176*a^96*x^96+18430384896*a^95*x^95+1750 
886565120*a^94*x^94+164583337121280*a^93*x^93+15306250352279040*a^92*x^92+ 
1408175032409671680*a^91*x^91+128143927949280122880*a^90*x^90+115329535154 
35211059200*a^89*x^89+1026432862873733784268800*a^88*x^88+9032609193288857 
3015654400*a^87*x^87+7858369998161305852361932800*a^86*x^86+67581981984187 
2303303126220800*a^85*x^85+57444684686559145780765728768000*a^84*x^84+4825 
353513670968245584321216512000*a^83*x^83+400504341634690364383498660970496 
000*a^82*x^82+32841356014044609879446890199580672000*a^81*x^81+26601498371 
37613400235198106166034432000*a^80*x^80+2128119869710090720188158484932827 
54560000*a^79*x^79+16812146970709716689486452030969337610240000*a^78*x^78+ 
1311347463715357901779943258415608333598720000*a^77*x^77+10097375470608255 
8437055630898001841687101440000*a^76*x^76+76740053576622744412162279482481 
39968219709440000*a^75*x^75+5755504018246705830912170961186104976164782080 
00000*a^74*x^74+42590729735025623148750065112777176823619387392000000*a^73 
*x^73+3109123270656870489858754753232733908124215279616000000*a^72*x^72+22 
3856875487294675269830342232756841384943500132352000000*a^71*x^71+15893...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (21) = 42\).

Time = 0.11 (sec) , antiderivative size = 814, normalized size of antiderivative = 32.56 \[ \int x^{100} \Gamma (-1,a x) \, dx=\text {Too large to display} \] Input:

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

Output:

1/101*(a^101*x^101*gamma(-1, a*x) - (a^99*x^99 + 99*a^98*x^98 + 9702*a^97* 
x^97 + 941094*a^96*x^96 + 90345024*a^95*x^95 + 8582777280*a^94*x^94 + 8067 
81064320*a^93*x^93 + 75030638981760*a^92*x^92 + 6902818786321920*a^91*x^91 
 + 628156509555294720*a^90*x^90 + 56534085859976524800*a^89*x^89 + 5031533 
641537910707200*a^88*x^88 + 442774960455336142233600*a^87*x^87 + 385214215 
59614244374323200*a^86*x^86 + 3312842254126825016191795200*a^85*x^85 + 281 
591591600780126376302592000*a^84*x^84 + 23653693694465530615609417728000*a 
^83*x^83 + 1963256576640639041095581671424000*a^82*x^82 + 1609870392845324 
01369837697056768000*a^81*x^81 + 13039950182047124510956853461598208000*a^ 
80*x^80 + 1043196014563769960876548276927856640000*a^79*x^79 + 82412485150 
537826909247313877300674560000*a^78*x^78 + 6428173841741950498921290482429 
452615680000*a^77*x^77 + 494969385814130188416939367147067851407360000*a^7 
6*x^76 + 37617673321873894319687391903177156706959360000*a^75*x^75 + 28213 
25499140542073976554392738286753021952000000*a^74*x^74 + 20877808693640011 
3474265025062633219723624448000000*a^73*x^73 + 152408003463572082836213468 
29572225039824584704000000*a^72*x^72 + 10973376249377189964207369717292002 
02867370098688000000*a^71*x^71 + 77910971370578048745872324992773214403583 
277006848000000*a^70*x^70 + 5453767995940463412211062749494125008250829390 
479360000000*a^69*x^69 + 3763099917198919754425633297150946255693072279430 
75840000000*a^68*x^68 + 25589079436952654330094306420626434538712891500...
 

Sympy [F(-1)]

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

integrate(x**100*uppergamma(-1,a*x),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate(x^100*gamma(-1, a*x), x)
 

Giac [F]

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

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

Output:

integrate(x^100*gamma(-1, a*x), x)
 

Mupad [B] (verification not implemented)

Time = 2.34 (sec) , antiderivative size = 822, normalized size of antiderivative = 32.88 \[ \int x^{100} \Gamma (-1,a x) \, dx=\text {Too large to display} \] Input:

int((x^99*expint(2, a*x))/a,x)
 

Output:

-(x^100*(exp(-a*x)*(1/(a*x) + 99/(a^2*x^2) + 9702/(a^3*x^3) + 941094/(a^4* 
x^4) + 90345024/(a^5*x^5) + 8582777280/(a^6*x^6) + 806781064320/(a^7*x^7) 
+ 75030638981760/(a^8*x^8) + 6902818786321920/(a^9*x^9) + 6281565095552947 
20/(a^10*x^10) + 56534085859976524800/(a^11*x^11) + 5031533641537910707200 
/(a^12*x^12) + 442774960455336142233600/(a^13*x^13) + 38521421559614244374 
323200/(a^14*x^14) + 3312842254126825016191795200/(a^15*x^15) + 2815915916 
00780126376302592000/(a^16*x^16) + 23653693694465530615609417728000/(a^17* 
x^17) + 1963256576640639041095581671424000/(a^18*x^18) + 16098703928453240 
1369837697056768000/(a^19*x^19) + 13039950182047124510956853461598208000/( 
a^20*x^20) + 1043196014563769960876548276927856640000/(a^21*x^21) + 824124 
85150537826909247313877300674560000/(a^22*x^22) + 642817384174195049892129 
0482429452615680000/(a^23*x^23) + 4949693858141301884169393671470678514073 
60000/(a^24*x^24) + 37617673321873894319687391903177156706959360000/(a^25* 
x^25) + 2821325499140542073976554392738286753021952000000/(a^26*x^26) + 20 
8778086936400113474265025062633219723624448000000/(a^27*x^27) + 1524080034 
6357208283621346829572225039824584704000000/(a^28*x^28) + 1097337624937718 
996420736971729200202867370098688000000/(a^29*x^29) + 77910971370578048745 
872324992773214403583277006848000000/(a^30*x^30) + 54537679959404634122110 
62749494125008250829390479360000000/(a^31*x^31) + 376309991719891975442563 
329715094625569307227943075840000000/(a^32*x^32) + 25589079436952654330...
 

Reduce [F]

\[ \int x^{100} \Gamma (-1,a x) \, dx=\frac {\int \mathit {ei} \left (2, a x \right ) x^{99}d x}{a} \] Input:

int(x^99/a*Ei(2,a*x),x)
 

Output:

int(ei(2,a*x)*x**99,x)/a