Integrand size = 11, antiderivative size = 29 \[ \int x^{100} \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {1}{101} x^{101} \Gamma \left (\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {205}{2},a x\right )}{101 a^{101}} \] Output:
1/101*x^101*((a*x)^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2)))-1/101*G AMMA(205/2,a*x)/a^101
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int x^{100} \Gamma \left (\frac {3}{2},a x\right ) \, dx=\frac {1}{101} x^{101} \Gamma \left (\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {205}{2},a x\right )}{101 a^{101}} \] Input:
Integrate[x^100*Gamma[3/2, a*x],x]
Output:
(x^101*Gamma[3/2, a*x])/101 - Gamma[205/2, a*x]/(101*a^101)
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int x^{100} \Gamma \left (\frac {3}{2},a x\right ) \, dx\) |
| \(\Big \downarrow \) 7116 |
| \(\displaystyle \frac {1}{101} x^{101} \Gamma \left (\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {205}{2},a x\right )}{101 a^{101}}\) |
Input:
Int[x^100*Gamma[3/2, a*x],x]
Output:
(x^101*Gamma[3/2, a*x])/101 - Gamma[205/2, 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. \(2678\) vs. \(2(39)=78\).
Time = 0.80 (sec) , antiderivative size = 2679, normalized size of antiderivative = 92.38
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2679\) |
| default | \(\text {Expression too large to display}\) | \(2679\) |
| parts | \(\text {Expression too large to display}\) | \(2684\) |
Input:
int(x^100*((x*a)^(1/2)*exp(-x*a)+1/2*Pi^(1/2)*erfc((x*a)^(1/2))),x,method= _RETURNVERBOSE)
Output:
1/a^101*(-481452080155966748007633880476899108122537146653367093423140625/ 268435456*(x*a)^(145/2)/exp(x*a)+13399280426846637026266542163537418668282 22367324552136009000175903930373973956200201339808971908813302485337859343 55559588023275706870249406853902718440076929197472536446481943130493164062 5/2535301200456458802993406410752*Pi^(1/2)*erf((x*a)^(1/2))-13399280426846 63702626654216353741866828222367324552136009000175903930373973956200201339 80897190881330248533785934355559588023275706870249406853902718440076929197 4725364464819431304931640625/1267650600228229401496703205376*(x*a)^(1/2)/e xp(x*a)-216456657335251272772598790366307082218216489362932419713454203208 06680443887576516848700936501255208105181593184682267383344749723361758291 6665391768573616754055023193359375/77371252455336267181195264*(x*a)^(29/2) /exp(x*a)-4485212372163007477064823209929735140263860411364401779216831618 20667701438399183260240697341751009722721429443359375/18014398509481984*(x *a)^(93/2)/exp(x*a)-528075613518318762367820462505858445619523667969761004 51592300946851000081106574844274393397042974579078560179163823409966274340 229687013435795848903656005859375/302231454903657293676544*(x*a)^(45/2)/ex p(x*a)-2046538169524033589964909098246082727925626897391951221975288291426 38941759834966883262333961861830147452263038623045050485994188828211500164 1769196189127188664982879161834716796875/2475880078570760549798248448*(x*a )^(19/2)/exp(x*a)-97454198548763504284043290392670606091696518923426248...
Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (39) = 78\).
Time = 0.48 (sec) , antiderivative size = 848, normalized size of antiderivative = 29.24 \[ \int x^{100} \Gamma \left (\frac {3}{2},a x\right ) \, dx=\text {Too large to display} \] Input:
integrate(x^100*((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2))),x, algorithm="fricas")
Output:
1/512130842492204678204668094971904*(2535301200456458802993406410752*sqrt( pi)*a^101*x^101 - sqrt(pi)*(2535301200456458802993406410752*a^101*x^101 - 27200539266498673163321080591980959896612914056688408360982703570849786591 67131086408719812212974891004045235854467417859636872496849466062959134225 18433356166270869248986358344554901123046875)*erf(sqrt(a*x)) - 406*(126765 0600228229401496703205376*a^100*x^100 + 127398885322937054850418672140288* a^99*x^99 + 12676189089632236957616657877958656*a^98*x^98 + 12486046253287 75340325240800978927616*a^97*x^97 + 12173895096955559568171097809544544256 0*a^96*x^96 + 11747808768562114983285109386210485207040*a^95*x^95 + 112191 5737397681980903727946383101337272320*a^94*x^94 + 106021037184080947195402 290933203076372234240*a^93*x^93 + 9912966976711568562770114202254487640803 901440*a^92*x^92 + 916949445345820092056235563708540106774360883200*a^91*x ^91 + 83900874249142538423145554079331419769854020812800*a^90*x^90 + 75930 29119547399727294672644179493489171788883558400*a^89*x^89 + 67957610619949 2275592873201654064667280875105078476800*a^88*x^88 + 601424853986550663899 69278346384723054357446799445196800*a^87*x^87 + 52624674723823183091223118 55308663267256276594951454720000*a^86*x^86 + 45520343636107053373907997548 4199372617667925463300833280000*a^85*x^85 + 389198938088715306346913379038 99046358810607627112221245440000*a^84*x^84 + 32887310268496443386314180528 79469417319496344490982695239680000*a^83*x^83 + 27460904074194530227572...
Timed out. \[ \int x^{100} \Gamma \left (\frac {3}{2},a x\right ) \, dx=\text {Timed out} \] Input:
integrate(x**100*((a*x)**(1/2)*exp(-a*x)+1/2*pi**(1/2)*erfc((a*x)**(1/2))) ,x)
Output:
Timed out
\[ \int x^{100} \Gamma \left (\frac {3}{2},a x\right ) \, dx=\int { \frac {1}{2} \, {\left (\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right ) + 2 \, \sqrt {a x} e^{\left (-a x\right )}\right )} x^{100} \,d x } \] Input:
integrate(x^100*((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2))),x, algorithm="maxima")
Output:
1/2*integrate((sqrt(pi)*erfc(sqrt(a*x)) + 2*sqrt(a*x)*e^(-a*x))*x^100, x)
Leaf count of result is larger than twice the leaf count of optimal. 2693 vs. \(2 (39) = 78\).
Time = 0.14 (sec) , antiderivative size = 2693, normalized size of antiderivative = 92.86 \[ \int x^{100} \Gamma \left (\frac {3}{2},a x\right ) \, dx=\text {Too large to display} \] Input:
integrate(x^100*((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2))),x, algorithm="giac")
Output:
1/512130842492204678204668094971904*(2535301200456458802993406410752*sqrt( pi)*a*x^101 - sqrt(pi)*(2535301200456458802993406410752*a^101*x^101*erf(sq rt(a*x)) + (2*(1267650600228229401496703205376*sqrt(a*x)*a^100*x^100 + 127 398885322937054850418672140288*sqrt(a*x)*a^99*x^99 + 126761890896322369576 16657877958656*sqrt(a*x)*a^98*x^98 + 1248604625328775340325240800978927616 *sqrt(a*x)*a^97*x^97 + 121738950969555595681710978095445442560*sqrt(a*x)*a ^96*x^96 + 11747808768562114983285109386210485207040*sqrt(a*x)*a^95*x^95 + 1121915737397681980903727946383101337272320*sqrt(a*x)*a^94*x^94 + 1060210 37184080947195402290933203076372234240*sqrt(a*x)*a^93*x^93 + 9912966976711 568562770114202254487640803901440*sqrt(a*x)*a^92*x^92 + 916949445345820092 056235563708540106774360883200*sqrt(a*x)*a^91*x^91 + 839008742491425384231 45554079331419769854020812800*sqrt(a*x)*a^90*x^90 + 7593029119547399727294 672644179493489171788883558400*sqrt(a*x)*a^89*x^89 + 679576106199492275592 873201654064667280875105078476800*sqrt(a*x)*a^88*x^88 + 601424853986550663 89969278346384723054357446799445196800*sqrt(a*x)*a^87*x^87 + 5262467472382 318309122311855308663267256276594951454720000*sqrt(a*x)*a^86*x^86 + 455203 436361070533739079975484199372617667925463300833280000*sqrt(a*x)*a^85*x^85 + 38919893808871530634691337903899046358810607627112221245440000*sqrt(a*x )*a^84*x^84 + 328873102684964433863141805287946941731949634449098269523968 0000*sqrt(a*x)*a^83*x^83 + 27460904074194530227572340741543569634617794...
Timed out. \[ \int x^{100} \Gamma \left (\frac {3}{2},a x\right ) \, dx=\int x^{100}\,\left (\frac {\sqrt {\Pi }\,\mathrm {erfc}\left (\sqrt {a\,x}\right )}{2}+{\mathrm {e}}^{-a\,x}\,\sqrt {a\,x}\right ) \,d x \] Input:
int(x^100*((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2)),x)
Output:
int(x^100*((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2)), x)
\[ \int x^{100} \Gamma \left (\frac {3}{2},a x\right ) \, dx=\text {too large to display} \] Input:
int(x^100*((a*x)^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2))),x)
Output:
( - 2535301200456458802993406410752*sqrt(pi)*e**(a*x)*erf(sqrt(x)*sqrt(a)) *a**101*x**101 + 133992804268466370262665421635374186682822236732455213600 90001759039303739739562002013398089719088133024853378593435555958802327570 68702494068539027184400769291974725364464819431304931640625*sqrt(pi)*e**(a *x)*erf(sqrt(x)*sqrt(a)) + 27066546462230206793058415170345585709930091819 95595314738180355325939355427391524406706414123255802871020382475873982303 67807016927877903801844883491248955396978894523621893525123596191406250*e* *(a*x)*sqrt(a)*int(sqrt(x)/(e**(a*x)*x),x) - 51466614369266113700766150138 2656*sqrt(x)*sqrt(a)*a**100*x**100 - 51723947441112444269269980888956928*s qrt(x)*sqrt(a)*a**99*x**99 - 5146532770390688204792363098451214336*sqrt(x) *sqrt(a)*a**98*x**98 - 506933477883482788172047765197444612096*sqrt(x)*sqr t(a)*a**97*x**97 - 49426014093639571846774657106750849679360*sqrt(x)*sqrt( a)*a**96*x**96 - 4769610360036218683213754410801456994058240*sqrt(x)*sqrt( a)*a**95*x**95 - 455497789383458884246913546231539142932561920*sqrt(x)*sqr t(a)*a**94*x**94 - 43044541096736864561333330118880449007127101440*sqrt(x) *sqrt(a)*a**93*x**93 - 4024664592544896836484666366115321982166383984640*s qrt(x)*sqrt(a)*a**92*x**92 - 372281474810402957374831638865667283350390518 579200*sqrt(x)*sqrt(a)*a**91*x**91 - 3406375494515187059979709495620855642 6560732449996800*sqrt(x)*sqrt(a)*a**90*x**90 - 308276982253624428928163709 3536874356603746286724710400*sqrt(x)*sqrt(a)*a**89*x**89 - 275907899116...