Integrand size = 9, antiderivative size = 25 \[ \int x^{100} \Gamma (2,a x) \, dx=\frac {1}{101} x^{101} \Gamma (2,a x)-\frac {\Gamma (103,a x)}{101 a^{101}} \] Output:
1/101*x^101*exp(-a*x)*(a*x+1)-1/101*GAMMA(103,a*x)/a^101
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^{100} \Gamma (2,a x) \, dx=\frac {1}{101} x^{101} \Gamma (2,a x)-\frac {\Gamma (103,a x)}{101 a^{101}} \] Input:
Integrate[x^100*Gamma[2, a*x],x]
Output:
(x^101*Gamma[2, a*x])/101 - Gamma[103, a*x]/(101*a^101)
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.
| \(\displaystyle \int x^{100} \Gamma (2,a x) \, dx\) |
| \(\Big \downarrow \) 7116 |
| \(\displaystyle \frac {1}{101} x^{101} \Gamma (2,a x)-\frac {\Gamma (103,a x)}{101 a^{101}}\) |
Input:
Int[x^100*Gamma[2, a*x],x]
Output:
(x^101*Gamma[2, a*x])/101 - Gamma[103, a*x]/(101*a^101)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(815\) vs. \(2(26)=52\).
Time = 75.90 (sec) , antiderivative size = 816, normalized size of antiderivative = 32.64
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(816\) |
| risch | \(\text {Expression too large to display}\) | \(816\) |
| orering | \(\text {Expression too large to display}\) | \(816\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1322\) |
| default | \(\text {Expression too large to display}\) | \(1322\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1322\) |
| meijerg | \(\text {Expression too large to display}\) | \(1632\) |
| parts | \(\text {Expression too large to display}\) | \(2635\) |
Input:
int(x^100*exp(-x*a)*(a*x+1),x,method=_RETURNVERBOSE)
Output:
-(a^101*x^101+102*a^100*x^100+10200*a^99*x^99+1009800*a^98*x^98+98960400*a ^97*x^97+9599158800*a^96*x^96+921519244800*a^95*x^95+87544328256000*a^94*x ^94+8229166856064000*a^93*x^93+765312517613952000*a^92*x^92+70408751620483 584000*a^91*x^91+6407196397464006144000*a^90*x^90+576647675771760552960000 *a^89*x^89+51321643143686689213440000*a^88*x^88+45163045966444286507827200 00*a^87*x^87+392918499908065292618096640000*a^86*x^86+33790990992093615165 156311040000*a^85*x^85+2872234234327957289038286438400000*a^84*x^84+241267 675683548412279216060825600000*a^83*x^83+200252170817345182191749330485248 00000*a^82*x^82+1642067800702230493972344509979033600000*a^81*x^81+1330074 91856880670011759905308301721600000*a^80*x^80+1064059934855045360094079242 4664137728000000*a^79*x^79+840607348535485834474322601548466880512000000*a ^78*x^78+65567373185767895088997162920780416679936000000*a^77*x^77+5048687 735304127921852781544900092084355072000000*a^76*x^76+383700267883113722060 811397412406998410985472000000*a^75*x^75+287775200912335291545608548059305 24880823910400000000*a^74*x^74+2129536486751281157437503255638858841180969 369600000000*a^73*x^73+155456163532843524492937737661636695406210763980800 000000*a^72*x^72+111928437743647337634915171116378420692471750066176000000 00*a^71*x^71+794691907979896097207897714926286786916549425469849600000000* a^70*x^70+55628433558592726804552840044840075084158459782889472000000000*a ^69*x^69+38383619155428981495141459630939651808069337250193735680000000...
Leaf count of result is larger than twice the leaf count of optimal. 838 vs. \(2 (21) = 42\).
Time = 0.15 (sec) , antiderivative size = 838, normalized size of antiderivative = 33.52 \[ \int x^{100} \Gamma (2,a x) \, dx=\text {Too large to display} \] Input:
integrate(x^100*gamma(2,a*x),x, algorithm="fricas")
Output:
1/101*(a^101*x^101*gamma(2, a*x) - (a^102*x^102 + 102*a^101*x^101 + 10302* a^100*x^100 + 1030200*a^99*x^99 + 101989800*a^98*x^98 + 9995000400*a^97*x^ 97 + 969515038800*a^96*x^96 + 93073443724800*a^95*x^95 + 8841977153856000* a^94*x^94 + 831145852462464000*a^93*x^93 + 77296564279009152000*a^92*x^92 + 7111283913668841984000*a^91*x^91 + 647126836143864620544000*a^90*x^90 + 58241415252947815848960000*a^89*x^89 + 5183485957512355610557440000*a^88*x ^88 + 456146764261087293729054720000*a^87*x^87 + 3968476849071459455442776 0640000*a^86*x^86 + 3412890090201455131680787415040000*a^85*x^85 + 2900956 57667123686192866930278400000*a^84*x^84 + 24368035244038389640200822143385 600000*a^83*x^83 + 2022546925255186340136668237901004800000*a^82*x^82 + 16 5848847870925279891206795507882393600000*a^81*x^81 + 134337566775449476711 87750436138473881600000*a^80*x^80 + 10747005342035958136950200348910779105 28000000*a^79*x^79 + 84901342202084069281906582756395154931712000000*a^78* x^78 + 6622304691762557403988713454998822084673536000000*a^77*x^77 + 50991 7461265716920107130936034909300519862272000000*a^76*x^76 + 387537270561944 85928141951138653106839509532672000000*a^75*x^75 + 29065295292145864446106 46335398983012963214950400000000*a^74*x^74 + 21508318516187939690118782881 9524742959277906329600000000*a^73*x^73 + 157010725168171959737867115038253 06236027287162060800000000*a^72*x^72 + 11304772212108381101126432282754220 48993964675668377600000000*a^71*x^71 + 80263882705969505817997669207554...
Time = 0.50 (sec) , antiderivative size = 877, normalized size of antiderivative = 35.08 \[ \int x^{100} \Gamma (2,a x) \, dx=\text {Too large to display} \] Input:
integrate(x**100*uppergamma(2,a*x),x)
Output:
Piecewise(((-a**101*x**101 - 102*a**100*x**100 - 10200*a**99*x**99 - 10098 00*a**98*x**98 - 98960400*a**97*x**97 - 9599158800*a**96*x**96 - 921519244 800*a**95*x**95 - 87544328256000*a**94*x**94 - 8229166856064000*a**93*x**9 3 - 765312517613952000*a**92*x**92 - 70408751620483584000*a**91*x**91 - 64 07196397464006144000*a**90*x**90 - 576647675771760552960000*a**89*x**89 - 51321643143686689213440000*a**88*x**88 - 4516304596644428650782720000*a**8 7*x**87 - 392918499908065292618096640000*a**86*x**86 - 3379099099209361516 5156311040000*a**85*x**85 - 2872234234327957289038286438400000*a**84*x**84 - 241267675683548412279216060825600000*a**83*x**83 - 20025217081734518219 174933048524800000*a**82*x**82 - 1642067800702230493972344509979033600000* a**81*x**81 - 133007491856880670011759905308301721600000*a**80*x**80 - 106 40599348550453600940792424664137728000000*a**79*x**79 - 840607348535485834 474322601548466880512000000*a**78*x**78 - 65567373185767895088997162920780 416679936000000*a**77*x**77 - 50486877353041279218527815449000920843550720 00000*a**76*x**76 - 383700267883113722060811397412406998410985472000000*a* *75*x**75 - 28777520091233529154560854805930524880823910400000000*a**74*x* *74 - 2129536486751281157437503255638858841180969369600000000*a**73*x**73 - 155456163532843524492937737661636695406210763980800000000*a**72*x**72 - 11192843774364733763491517111637842069247175006617600000000*a**71*x**71 - 794691907979896097207897714926286786916549425469849600000000*a**70*x**7...
\[ \int x^{100} \Gamma (2,a x) \, dx=\int { x^{100} \Gamma \left (2, a x\right ) \,d x } \] Input:
integrate(x^100*gamma(2,a*x),x, algorithm="maxima")
Output:
integrate(x^100*gamma(2, a*x), x)
\[ \int x^{100} \Gamma (2,a x) \, dx=\int { x^{100} \Gamma \left (2, a x\right ) \,d x } \] Input:
integrate(x^100*gamma(2,a*x),x, algorithm="giac")
Output:
integrate(x^100*gamma(2, a*x), x)
Time = 2.61 (sec) , antiderivative size = 814, normalized size of antiderivative = 32.56 \[ \int x^{100} \Gamma (2,a x) \, dx=\text {Too large to display} \] Input:
int(x^100*exp(-a*x)*(a*x + 1),x)
Output:
-exp(-a*x)*((9519273975282303573533322363339203450053028762966925389796482 31731219519930945139211202932556796486519787718792437682334162089151352012 8000000000000000000000000*x)/a^100 + 9519273975282303573533322363339203450 05302876296692538979648231731219519930945139211202932556796486519787718792 4376823341620891513520128000000000000000000000000/a^101 + x^101 + (102*x^1 00)/a + (10200*x^99)/a^2 + (1009800*x^98)/a^3 + (98960400*x^97)/a^4 + (959 9158800*x^96)/a^5 + (921519244800*x^95)/a^6 + (87544328256000*x^94)/a^7 + (8229166856064000*x^93)/a^8 + (765312517613952000*x^92)/a^9 + (70408751620 483584000*x^91)/a^10 + (6407196397464006144000*x^90)/a^11 + (5766476757717 60552960000*x^89)/a^12 + (51321643143686689213440000*x^88)/a^13 + (4516304 596644428650782720000*x^87)/a^14 + (392918499908065292618096640000*x^86)/a ^15 + (33790990992093615165156311040000*x^85)/a^16 + (28722342343279572890 38286438400000*x^84)/a^17 + (241267675683548412279216060825600000*x^83)/a^ 18 + (20025217081734518219174933048524800000*x^82)/a^19 + (164206780070223 0493972344509979033600000*x^81)/a^20 + (1330074918568806700117599053083017 21600000*x^80)/a^21 + (10640599348550453600940792424664137728000000*x^79)/ a^22 + (840607348535485834474322601548466880512000000*x^78)/a^23 + (655673 73185767895088997162920780416679936000000*x^77)/a^24 + (504868773530412792 1852781544900092084355072000000*x^76)/a^25 + (3837002678831137220608113974 12406998410985472000000*x^75)/a^26 + (287775200912335291545608548059305...
Time = 0.19 (sec) , antiderivative size = 817, normalized size of antiderivative = 32.68 \[ \int x^{100} \Gamma (2,a x) \, dx =\text {Too large to display} \] Input:
int(x^100*exp(-a*x)*(a*x+1),x)
Output:
( - a**101*x**101 - 102*a**100*x**100 - 10200*a**99*x**99 - 1009800*a**98* x**98 - 98960400*a**97*x**97 - 9599158800*a**96*x**96 - 921519244800*a**95 *x**95 - 87544328256000*a**94*x**94 - 8229166856064000*a**93*x**93 - 76531 2517613952000*a**92*x**92 - 70408751620483584000*a**91*x**91 - 64071963974 64006144000*a**90*x**90 - 576647675771760552960000*a**89*x**89 - 513216431 43686689213440000*a**88*x**88 - 4516304596644428650782720000*a**87*x**87 - 392918499908065292618096640000*a**86*x**86 - 3379099099209361516515631104 0000*a**85*x**85 - 2872234234327957289038286438400000*a**84*x**84 - 241267 675683548412279216060825600000*a**83*x**83 - 20025217081734518219174933048 524800000*a**82*x**82 - 1642067800702230493972344509979033600000*a**81*x** 81 - 133007491856880670011759905308301721600000*a**80*x**80 - 106405993485 50453600940792424664137728000000*a**79*x**79 - 840607348535485834474322601 548466880512000000*a**78*x**78 - 65567373185767895088997162920780416679936 000000*a**77*x**77 - 5048687735304127921852781544900092084355072000000*a** 76*x**76 - 383700267883113722060811397412406998410985472000000*a**75*x**75 - 28777520091233529154560854805930524880823910400000000*a**74*x**74 - 212 9536486751281157437503255638858841180969369600000000*a**73*x**73 - 1554561 63532843524492937737661636695406210763980800000000*a**72*x**72 - 111928437 74364733763491517111637842069247175006617600000000*a**71*x**71 - 794691907 979896097207897714926286786916549425469849600000000*a**70*x**70 - 55628...