\(\int x^{100} \Gamma (-3,a x) \, dx\) [51]

Optimal result

Integrand size = 9, antiderivative size = 25 \[ \int x^{100} \Gamma (-3,a x) \, dx=\frac {1}{101} x^{101} \Gamma (-3,a x)-\frac {\Gamma (98,a x)}{101 a^{101}} \] Output:

1/101*x^98/a^3*Ei(4,a*x)-1/101*GAMMA(98,a*x)/a^101
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^{100} \Gamma (-3,a x) \, dx=\frac {1}{101} x^{101} \Gamma (-3,a x)-\frac {\Gamma (98,a x)}{101 a^{101}} \] Input:

Integrate[x^100*Gamma[-3, a*x],x]
 

Output:

(x^101*Gamma[-3, a*x])/101 - Gamma[98, 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.

\(\displaystyle \int x^{100} \Gamma (-3,a x) \, dx\)

\(\Big \downarrow \) 7116

\(\displaystyle \frac {1}{101} x^{101} \Gamma (-3,a x)-\frac {\Gamma (98,a x)}{101 a^{101}}\)

Input:

Int[x^100*Gamma[-3, a*x],x]
 

Output:

(x^101*Gamma[-3, a*x])/101 - Gamma[98, 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^97/a^3*Ei(4,x*a),x)
 

Output:

1/a^101*(1/606*(Psi(101)+gamma-11/6-Psi(102)+ln(x)+ln(a))*x^101*a^101+1117 
/367236*x^101*a^101+961927596824821198533284259495636987123438139191729761 
58104477319333745612481875498805879175589072651261284189679678167647067832 
320000000000000000000000/101-1/247248*(-408*a^100*x^100+408*a^99*x^99-816* 
a^98*x^98+2448*a^97*x^97+237456*a^96*x^96+22795776*a^95*x^95+2165598720*a^ 
94*x^94+203566279680*a^93*x^93+18931664010240*a^92*x^92+1741713088942080*a 
^91*x^91+158495891093729280*a^90*x^90+14264630198435635200*a^89*x^89+12695 
52087660771532800*a^88*x^88+111720583714147894886400*a^87*x^87+97196907831 
30866855116800*a^86*x^86+835893407349254549540044800*a^85*x^85+71050939624 
686636710903808000*a^84*x^84+5968278928473677483715919872000*a^83*x^83+495 
367151063315231148421349376000*a^82*x^82+406201063871918489541705506488320 
00*a^81*x^81+3290228617362539765287814602555392000*a^80*x^80+2632182893890 
03181223025168204431360000*a^79*x^79+2079424486173125131661898828815007744 
0000*a^78*x^78+1621951099215037602696281086475706040320000*a^77*x^77+12489 
0234639557895407613643658629365104640000*a^76*x^76+94916578326064000509786 
36918055831747952640000*a^75*x^75+7118743374454800038233977688541873810964 
48000000*a^74*x^74+52678700970965520282931434895209866201137152000000*a^73 
*x^73+3845545170880482980653994747350320232683012096000000*a^72*x^72+27687 
9252303394774607087621809223056753176870912000000*a^71*x^71+19658426913541 
028997103221148454837029475557834752000000*a^70*x^70+137608988394787202...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 798 vs. \(2 (21) = 42\).

Time = 0.10 (sec) , antiderivative size = 798, normalized size of antiderivative = 31.92 \[ \int x^{100} \Gamma (-3,a x) \, dx=\text {Too large to display} \] Input:

integrate(x^100*gamma(-3,a*x),x, algorithm="fricas")
 

Output:

1/101*(a^101*x^101*gamma(-3, a*x) - (a^97*x^97 + 97*a^96*x^96 + 9312*a^95* 
x^95 + 884640*a^94*x^94 + 83156160*a^93*x^93 + 7733522880*a^92*x^92 + 7114 
84104960*a^91*x^91 + 64745053551360*a^90*x^90 + 5827054819622400*a^89*x^89 
 + 518607878946393600*a^88*x^88 + 45637493347282636800*a^87*x^87 + 3970461 
921213589401600*a^86*x^86 + 341459725224368688537600*a^85*x^85 + 290240766 
44071338525696000*a^84*x^84 + 2438022438101992436158464000*a^83*x^83 + 202 
355862362465372201152512000*a^82*x^82 + 16593180713722160520494505984000*a 
^81*x^81 + 1344047637811495002160054984704000*a^80*x^80 + 1075238110249196 
00172804398776320000*a^79*x^79 + 8494381070968648413651547503329280000*a^7 
8*x^78 + 662561723535554576264820705259683840000*a^77*x^77 + 5101725271223 
7702372391194304995655680000*a^76*x^76 + 387731120613006538030173076717966 
9831680000*a^75*x^75 + 290798340459754903522629807538475237376000000*a^74* 
x^74 + 21519077194021862860674605757847167565824000000*a^73*x^73 + 1570892 
635163595988829246220322843232305152000000*a^72*x^72 + 1131042697317789111 
95705727863244712725970944000000*a^71*x^71 + 80304031509563026948951066782 
90374603543937024000000*a^70*x^70 + 56212822056694118864265746748032622224 
8075591680000000*a^69*x^69 + 387868472191189420163433652561425093351172158 
25920000000*a^68*x^68 + 26375056109000880571113488374176906347879706761625 
60000000*a^67*x^67 + 17671287593030589982646037210698527253079403530289152 
0000000*a^66*x^66 + 116630498114001893885463845590610279870324063299908...
 

Sympy [F(-1)]

Timed out. \[ \int x^{100} \Gamma (-3,a x) \, dx=\text {Timed out} \] Input:

integrate(x**100*uppergamma(-3,a*x),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{100} \Gamma (-3,a x) \, dx=\int { x^{100} \Gamma \left (-3, a x\right ) \,d x } \] Input:

integrate(x^100*gamma(-3,a*x),x, algorithm="maxima")
 

Output:

integrate(x^100*gamma(-3, a*x), x)
 

Giac [F]

\[ \int x^{100} \Gamma (-3,a x) \, dx=\int { x^{100} \Gamma \left (-3, a x\right ) \,d x } \] Input:

integrate(x^100*gamma(-3,a*x),x, algorithm="giac")
 

Output:

integrate(x^100*gamma(-3, a*x), x)
 

Mupad [B] (verification not implemented)

Time = 2.39 (sec) , antiderivative size = 805, normalized size of antiderivative = 32.20 \[ \int x^{100} \Gamma (-3,a x) \, dx=\text {Too large to display} \] Input:

int((x^97*expint(4, a*x))/a^3,x)
 

Output:

(x^98*(expint(4, a*x) - exp(-a*x)*(1/(a*x) + 97/(a^2*x^2) + 9312/(a^3*x^3) 
 + 884640/(a^4*x^4) + 83156160/(a^5*x^5) + 7733522880/(a^6*x^6) + 71148410 
4960/(a^7*x^7) + 64745053551360/(a^8*x^8) + 5827054819622400/(a^9*x^9) + 5 
18607878946393600/(a^10*x^10) + 45637493347282636800/(a^11*x^11) + 3970461 
921213589401600/(a^12*x^12) + 341459725224368688537600/(a^13*x^13) + 29024 
076644071338525696000/(a^14*x^14) + 2438022438101992436158464000/(a^15*x^1 
5) + 202355862362465372201152512000/(a^16*x^16) + 165931807137221605204945 
05984000/(a^17*x^17) + 1344047637811495002160054984704000/(a^18*x^18) + 10 
7523811024919600172804398776320000/(a^19*x^19) + 8494381070968648413651547 
503329280000/(a^20*x^20) + 662561723535554576264820705259683840000/(a^21*x 
^21) + 51017252712237702372391194304995655680000/(a^22*x^22) + 38773112061 
30065380301730767179669831680000/(a^23*x^23) + 290798340459754903522629807 
538475237376000000/(a^24*x^24) + 21519077194021862860674605757847167565824 
000000/(a^25*x^25) + 1570892635163595988829246220322843232305152000000/(a^ 
26*x^26) + 113104269731778911195705727863244712725970944000000/(a^27*x^27) 
 + 8030403150956302694895106678290374603543937024000000/(a^28*x^28) + 5621 
28220566941188642657467480326222248075591680000000/(a^29*x^29) + 387868472 
19118942016343365256142509335117215825920000000/(a^30*x^30) + 263750561090 
0088057111348837417690634787970676162560000000/(a^31*x^31) + 1767128759303 
05899826460372106985272530794035302891520000000/(a^32*x^32) + 116630498...
 

Reduce [F]

\[ \int x^{100} \Gamma (-3,a x) \, dx=\frac {\int \mathit {ei} \left (4, a x \right ) x^{97}d x}{a^{3}} \] Input:

int(x^97/a^3*Ei(4,a*x),x)
 

Output:

int(ei(4,a*x)*x**97,x)/a**3