Integrand size = 9, antiderivative size = 25 \[ \int x^{100} \Gamma (3,a x) \, dx=\frac {1}{101} x^{101} \Gamma (3,a x)-\frac {\Gamma (104,a x)}{101 a^{101}} \] Output:
2/101*x^101*exp(-a*x)*(1+a*x+1/2*a^2*x^2)-1/101*GAMMA(104,a*x)/a^101
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^{100} \Gamma (3,a x) \, dx=\frac {1}{101} x^{101} \Gamma (3,a x)-\frac {\Gamma (104,a x)}{101 a^{101}} \] Input:
Integrate[x^100*Gamma[3, a*x],x]
Output:
(x^101*Gamma[3, a*x])/101 - Gamma[104, a*x]/(101*a^101)
Time = 0.27 (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 |
| \(\displaystyle \frac {1}{101} x^{101} \Gamma (3,a x)-\frac {\Gamma (104,a x)}{101 a^{101}}\) |
Input:
Int[x^100*Gamma[3, a*x],x]
Output:
(x^101*Gamma[3, a*x])/101 - Gamma[104, 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. \(823\) vs. \(2(34)=68\).
Time = 81.73 (sec) , antiderivative size = 824, normalized size of antiderivative = 32.96
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(824\) |
| risch | \(\text {Expression too large to display}\) | \(824\) |
| orering | \(\text {Expression too large to display}\) | \(852\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1335\) |
| default | \(\text {Expression too large to display}\) | \(1335\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1335\) |
| meijerg | \(\text {Expression too large to display}\) | \(2461\) |
| parts | \(\text {Expression too large to display}\) | \(3968\) |
Input:
int(2*x^100*exp(-x*a)*(1+x*a+1/2*a^2*x^2),x,method=_RETURNVERBOSE)
Output:
-exp(-x*a)*(a^102*x^102+104*a^101*x^101+10506*a^100*x^100+1050600*a^99*x^9 9+104009400*a^98*x^98+10192921200*a^97*x^97+988713356400*a^96*x^96+9491648 2214400*a^95*x^95+9017065810368000*a^94*x^94+847604186174592000*a^93*x^93+ 78827189314237056000*a^92*x^92+7252101416909809152000*a^91*x^91+6599412289 38792632832000*a^90*x^90+59394710604491336954880000*a^89*x^89+528612924379 9728988984320000*a^88*x^88+465179373454376151030620160000*a^87*x^87+404706 05490530725139663953920000*a^86*x^86+3480472072185642362011100037120000*a^ 85*x^85+295840126135779600770943503155200000*a^84*x^84+2485057059540548646 4759254265036800000*a^83*x^83+2062597359418655376575018103998054400000*a^8 2*x^82+169132983472329740879151484527840460800000*a^81*x^81+13699771661258 709011211270246755077324800000*a^80*x^80+109598173290069672089690161974040 6185984000000*a^79*x^79+86582556899155040950855227959492088692736000000*a^ 78*x^78+6753439438134093194166707780840382918033408000000*a^77*x^77+520014 836736325175950836499124709484688572416000000*a^76*x^76+395211275919607133 72263573933477920836331503616000000*a^75*x^75+2964084569397053502919768045 010844062724862771200000000*a^74*x^74+219342258135381959216062835330802460 641639845068800000000*a^73*x^73+160119848438828830227725869791485796268397 08690022400000000*a^72*x^72+1152862908759567577639626262498697733132459025 681612800000000*a^71*x^71+818532665219292980124134646374075390524045908233 94508800000000*a^70*x^70+5729728656535050860868942524618527733668321357...
Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (21) = 42\).
Time = 0.10 (sec) , antiderivative size = 846, normalized size of antiderivative = 33.84 \[ \int x^{100} \Gamma (3,a x) \, dx=\text {Too large to display} \] Input:
integrate(x^100*gamma(3,a*x),x, algorithm="fricas")
Output:
1/101*(a^101*x^101*gamma(3, a*x) - (a^103*x^103 + 103*a^102*x^102 + 10506* a^101*x^101 + 1061106*a^100*x^100 + 106110600*a^99*x^99 + 10504949400*a^98 *x^98 + 1029485041200*a^97*x^97 + 99860048996400*a^96*x^96 + 9586564703654 400*a^95*x^95 + 910723646847168000*a^94*x^94 + 85608022803633792000*a^93*x ^93 + 7961546120737942656000*a^92*x^92 + 732462243107890724352000*a^91*x^9 1 + 66654064122818055916032000*a^90*x^90 + 5998865771053625032442880000*a^ 89*x^89 + 533899053623772627887416320000*a^88*x^88 + 469831167188919912540 92636160000*a^87*x^87 + 4087531154543603239106059345920000*a^86*x^86 + 351 527679290749878563121103749120000*a^85*x^85 + 2987985273971373967786529381 8675200000*a^84*x^84 + 2509907630135954132940684680768716800000*a^83*x^83 + 208322333301284193034076828503803494400000*a^82*x^82 + 17082431330705303 828794299937311886540800000*a^81*x^81 + 1383676937787129610132338294922262 809804800000*a^80*x^80 + 110694155022970368810587063593781024784384000000* a^79*x^79 + 8744838246814659136036378023908700957966336000000*a^78*x^78 + 682097383251543412610837485864878674721374208000000*a^77*x^77 + 5252149851 0368842771034486411595657953545814016000000*a^76*x^76 + 399163388678803205 0598620967281270004469481865216000000*a^75*x^75 + 299372541509102403794896 572546095250335211139891200000000*a^74*x^74 + 2215356807167357788082234636 8411048524805624351948800000000*a^73*x^73 + 161721046923217118530003128489 4006542310810577692262400000000*a^72*x^72 + 116439153784716325341602252...
Time = 0.61 (sec) , antiderivative size = 896, normalized size of antiderivative = 35.84 \[ \int x^{100} \Gamma (3,a x) \, dx=\text {Too large to display} \] Input:
integrate(x**100*uppergamma(3,a*x),x)
Output:
Piecewise(((-a**102*x**102 - 104*a**101*x**101 - 10506*a**100*x**100 - 105 0600*a**99*x**99 - 104009400*a**98*x**98 - 10192921200*a**97*x**97 - 98871 3356400*a**96*x**96 - 94916482214400*a**95*x**95 - 9017065810368000*a**94* x**94 - 847604186174592000*a**93*x**93 - 78827189314237056000*a**92*x**92 - 7252101416909809152000*a**91*x**91 - 659941228938792632832000*a**90*x**9 0 - 59394710604491336954880000*a**89*x**89 - 5286129243799728988984320000* a**88*x**88 - 465179373454376151030620160000*a**87*x**87 - 404706054905307 25139663953920000*a**86*x**86 - 3480472072185642362011100037120000*a**85*x **85 - 295840126135779600770943503155200000*a**84*x**84 - 2485057059540548 6464759254265036800000*a**83*x**83 - 2062597359418655376575018103998054400 000*a**82*x**82 - 169132983472329740879151484527840460800000*a**81*x**81 - 13699771661258709011211270246755077324800000*a**80*x**80 - 10959817329006 96720896901619740406185984000000*a**79*x**79 - 865825568991550409508552279 59492088692736000000*a**78*x**78 - 675343943813409319416670778084038291803 3408000000*a**77*x**77 - 5200148367363251759508364991247094846885724160000 00*a**76*x**76 - 39521127591960713372263573933477920836331503616000000*a** 75*x**75 - 2964084569397053502919768045010844062724862771200000000*a**74*x **74 - 219342258135381959216062835330802460641639845068800000000*a**73*x** 73 - 16011984843882883022772586979148579626839708690022400000000*a**72*x** 72 - 1152862908759567577639626262498697733132459025681612800000000*a**7...
\[ \int x^{100} \Gamma (3,a x) \, dx=\int { x^{100} \Gamma \left (3, a x\right ) \,d x } \] Input:
integrate(x^100*gamma(3,a*x),x, algorithm="maxima")
Output:
integrate(x^100*gamma(3, a*x), x)
\[ \int x^{100} \Gamma (3,a x) \, dx=\int { x^{100} \Gamma \left (3, a x\right ) \,d x } \] Input:
integrate(x^100*gamma(3,a*x),x, algorithm="giac")
Output:
integrate(x^100*gamma(3, a*x), x)
Time = 2.61 (sec) , antiderivative size = 821, normalized size of antiderivative = 32.84 \[ \int x^{100} \Gamma (3,a x) \, dx=\text {Too large to display} \] Input:
int(2*x^100*exp(-a*x)*(a*x + (a^2*x^2)/2 + 1),x)
Output:
-exp(-a*x)*((9804852194540772680739322034239379553554619625855933151490376 78683156105528873493387539020533500381115381350356210812804186951825892573 184000000000000000000000000*x)/a^100 + a*x^102 + 9804852194540772680739322 03423937955355461962585593315149037678683156105528873493387539020533500381 115381350356210812804186951825892573184000000000000000000000000/a^101 + 10 4*x^101 + (10506*x^100)/a + (1050600*x^99)/a^2 + (104009400*x^98)/a^3 + (1 0192921200*x^97)/a^4 + (988713356400*x^96)/a^5 + (94916482214400*x^95)/a^6 + (9017065810368000*x^94)/a^7 + (847604186174592000*x^93)/a^8 + (78827189 314237056000*x^92)/a^9 + (7252101416909809152000*x^91)/a^10 + (65994122893 8792632832000*x^90)/a^11 + (59394710604491336954880000*x^89)/a^12 + (52861 29243799728988984320000*x^88)/a^13 + (465179373454376151030620160000*x^87) /a^14 + (40470605490530725139663953920000*x^86)/a^15 + (348047207218564236 2011100037120000*x^85)/a^16 + (295840126135779600770943503155200000*x^84)/ a^17 + (24850570595405486464759254265036800000*x^83)/a^18 + (2062597359418 655376575018103998054400000*x^82)/a^19 + (16913298347232974087915148452784 0460800000*x^81)/a^20 + (13699771661258709011211270246755077324800000*x^80 )/a^21 + (1095981732900696720896901619740406185984000000*x^79)/a^22 + (865 82556899155040950855227959492088692736000000*x^78)/a^23 + (675343943813409 3194166707780840382918033408000000*x^77)/a^24 + (5200148367363251759508364 99124709484688572416000000*x^76)/a^25 + (395211275919607133722635739334...
Time = 0.20 (sec) , antiderivative size = 825, normalized size of antiderivative = 33.00 \[ \int x^{100} \Gamma (3,a x) \, dx =\text {Too large to display} \] Input:
int(2*x^100*exp(-a*x)*(1+a*x+1/2*a^2*x^2),x)
Output:
( - a**102*x**102 - 104*a**101*x**101 - 10506*a**100*x**100 - 1050600*a**9 9*x**99 - 104009400*a**98*x**98 - 10192921200*a**97*x**97 - 988713356400*a **96*x**96 - 94916482214400*a**95*x**95 - 9017065810368000*a**94*x**94 - 8 47604186174592000*a**93*x**93 - 78827189314237056000*a**92*x**92 - 7252101 416909809152000*a**91*x**91 - 659941228938792632832000*a**90*x**90 - 59394 710604491336954880000*a**89*x**89 - 5286129243799728988984320000*a**88*x** 88 - 465179373454376151030620160000*a**87*x**87 - 404706054905307251396639 53920000*a**86*x**86 - 3480472072185642362011100037120000*a**85*x**85 - 29 5840126135779600770943503155200000*a**84*x**84 - 2485057059540548646475925 4265036800000*a**83*x**83 - 2062597359418655376575018103998054400000*a**82 *x**82 - 169132983472329740879151484527840460800000*a**81*x**81 - 13699771 661258709011211270246755077324800000*a**80*x**80 - 10959817329006967208969 01619740406185984000000*a**79*x**79 - 865825568991550409508552279594920886 92736000000*a**78*x**78 - 675343943813409319416670778084038291803340800000 0*a**77*x**77 - 520014836736325175950836499124709484688572416000000*a**76* x**76 - 39521127591960713372263573933477920836331503616000000*a**75*x**75 - 2964084569397053502919768045010844062724862771200000000*a**74*x**74 - 21 9342258135381959216062835330802460641639845068800000000*a**73*x**73 - 1601 1984843882883022772586979148579626839708690022400000000*a**72*x**72 - 1152 862908759567577639626262498697733132459025681612800000000*a**71*x**71 -...