3.1.64 \(\int (1-x)^{99} x \, dx\) [64]

3.1.64.1 Optimal result

Integrand size = 9, antiderivative size = 23 \[ \int (1-x)^{99} x \, dx=-\frac {1}{100} (1-x)^{100}+\frac {1}{101} (1-x)^{101} \]

output

-1/100*(1-x)^100+1/101*(1-x)^101
 

3.1.64.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(567\) vs. \(2(23)=46\).

Time = 0.01 (sec) , antiderivative size = 567, normalized size of antiderivative = 24.65 \[ \int (1-x)^{99} x \, dx =\text {Too large to display} \]

input

Integrate[(1 - x)^99*x,x]
 

output

x^2/2 - 33*x^3 + (4851*x^4)/4 - (156849*x^5)/5 + 627396*x^6 - 10217592*x^7 
 + 140066157*x^8 - 1654114616*x^9 + (85600431378*x^10)/5 - 157366449604*x^ 
11 + 1298273209233*x^12 - 9696194317908*x^13 + 66026466069564*x^14 - (2062 
057324941768*x^15)/5 + (4750096337812287*x^16)/2 - 12666923567499432*x^17 
+ 62806829355518017*x^18 - 290505891817783026*x^19 + (12572449429225165403 
*x^20)/10 - 5104603527655330314*x^21 + 19490304378320352108*x^22 - 7013281 
3684308016488*x^23 + 238292173768273828749*x^24 - (19146258135816088501224 
*x^25)/25 + 2331916055003241548226*x^26 - 6736646381120475583764*x^27 + 18 
488763007525700846649*x^28 - 48264408157576669898532*x^29 + (5998576442441 
67183024612*x^30)/5 - 284248449886557530554488*x^31 + (2570079734390957672 
096829*x^32)/4 - 1386787891870780679371896*x^33 + (57205000539669703024090 
71*x^34)/2 - (28206275157871771317939099*x^35)/5 + (4258594484619855669571 
1973*x^36)/4 - 19237666204653402059452899*x^37 + 3330028769928308192747402 
4*x^38 - 55246631151825154632274992*x^39 + (439428796464188236515924114*x^ 
40)/5 - 134109601422466453670901168*x^41 + 196374773511468735732390996*x^4 
2 - 276016272695076305811040776*x^43 + 372502480574415750374856978*x^44 - 
(2414047083412492769871166152*x^45)/5 + 601126348833940887359223192*x^46 - 
 719077854633508484642475024*x^47 + 826548729646668720118931889*x^48 - 913 
043842041304917057126672*x^49 + (24233705307512968006891237086*x^50)/25 - 
989130828878080326811887228*x^51 + 970109082168886474373197089*x^52 - 9...
 

3.1.64.3 Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {49, 2009}

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.

\(\displaystyle \int (1-x)^{99} x \, dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left ((1-x)^{99}-(1-x)^{100}\right )dx\)

\(\Big \downarrow \) 2009

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

input

Int[(1 - x)^99*x,x]
 

output

-1/100*(1 - x)^100 + (1 - x)^101/101
 

3.1.64.3.1 Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.1.64.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(500\) vs. \(2(19)=38\).

Time = 0.19 (sec) , antiderivative size = 501, normalized size of antiderivative = 21.78

method	result	size
gosper	\(\text {Expression too large to display}\)	\(501\)
default	\(\text {Expression too large to display}\)	\(502\)
risch	\(\text {Expression too large to display}\)	\(502\)
parallelrisch	\(\text {Expression too large to display}\)	\(502\)

input

int(x*(1-x)^99,x,method=_RETURNVERBOSE)
 

output

-1/10100*x^2*(100*x^99-9999*x^98+494900*x^97-16165050*x^96+391960800*x^95- 
7524830775*x^94+119129952480*x^93-1599564027600*x^92+18592781869200*x^91-1 
90037092945700*x^90+1729128713835600*x^89-14145670154903560*x^88+104900475 
306026400*x^87-710003828928813300*x^86+4411583725232560800*x^85-2528966019 
2321540400*x^84+134332724433331476360*x^83-663667626664673365350*x^82+3059 
800945425883628200*x^81-13203492783118238531700*x^80+534659954674417560296 
00*x^79-203648157735809402877030*x^78+731164847106703494794400*x^77-247919 
4963740288382850800*x^76+7952742286283782215118800*x^75-241721508964678117 
32795300*x^74+69714962380376909325764496*x^73-191035745261543332611892200* 
x^72+497964017002692209469746400*x^71-1236085967492793928008474900*x^70+29 
24823134349146195851039200*x^69-6603091490864731434780756240*x^68+14234925 
496610562332226630300*x^67-29326230200904915179092563225*x^66+577770505450 
66400054331617100*x^65-108925997889075399236629520550*x^64+196625391061305 
336057915852480*x^63-340025750167248881400829989825*x^62+56358486911597475 
4134709022400*x^61-895722354016828362340647962400*x^60+1365609490550246519 
634102631200*x^59-1997897787191193993562250150280*x^58+2805764446902887448 
586210864800*x^57-3783394480727510220367051779600*x^56+4899707046811384846 
791142017600*x^55-6095468885616544243924694533800*x^54+7285651347867363554 
793785087040*x^53-8367877730115588952806888703200*x^52+9236242400221923655 
456660172400*x^51-9798101729905753391169290598900*x^50+9990221371668611...
 

3.1.64.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (15) = 30\).

Time = 0.24 (sec) , antiderivative size = 501, normalized size of antiderivative = 21.78 \[ \int (1-x)^{99} x \, dx=\text {Too large to display} \]

input

integrate(x*(1-x)^99,x, algorithm="fricas")
 

output

-1/101*x^101 + 99/100*x^100 - 49*x^99 + 3201/2*x^98 - 38808*x^97 + 2980131 
/4*x^96 - 58975224/5*x^95 + 158372676*x^94 - 1840869492*x^93 + 18815553757 
*x^92 - 171200862756*x^91 + 7002807007378/5*x^90 - 10386185673864*x^89 + 7 
0297408804833*x^88 - 436790467844808*x^87 + 2503926751715004*x^86 - 665013 
48729372018/5*x^85 + 131419332012806607/2*x^84 - 302950588656028082*x^83 + 
 1307276513180023617*x^82 - 5293662917568490696*x^81 + 2016318393423855474 
03/10*x^80 - 72392559119475593544*x^79 + 245464847895078057708*x^78 - 7874 
00226364730912388*x^77 + 2393282266977011062653*x^76 - 1725617880702398745 
68724/25*x^75 + 18914430223915181446722*x^74 - 49303368020068535591064*x^7 
3 + 122384749256712270099849*x^72 - 289586448945460019391192*x^71 + 326885 
7173695411601376612/5*x^70 - 1409398564020847755666003*x^69 + 116143485944 
17788189739629/4*x^68 - 5720500053966970302409071*x^67 + 21569504532490178 
066659311/2*x^66 - 97339302505596701018770224/5*x^65 + 1346636634325738144 
16170293/4*x^64 - 55800482090690569716307824*x^63 + 8868538158582459033075 
7224*x^62 - 135208860450519457389515112*x^61 + 989058310490690095822896114 
/5*x^60 - 277798460089394796889723848*x^59 + 374593512943317843600698196*x 
^58 - 485119509585285628395162576*x^57 + 603511770853123192467791538*x^56 
- 3606758093003645324155339152/5*x^55 + 828502745555998906218503832*x^54 - 
 914479445566527094599669324*x^53 + 970109082168886474373197089*x^52 - 989 
130828878080326811887228*x^51 + 24233705307512968006891237086/25*x^50 -...
 

3.1.64.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (12) = 24\).

Time = 0.08 (sec) , antiderivative size = 561, normalized size of antiderivative = 24.39 \[ \int (1-x)^{99} x \, dx=\text {Too large to display} \]

input

integrate(x*(1-x)**99,x)
 

output

-x**101/101 + 99*x**100/100 - 49*x**99 + 3201*x**98/2 - 38808*x**97 + 2980 
131*x**96/4 - 58975224*x**95/5 + 158372676*x**94 - 1840869492*x**93 + 1881 
5553757*x**92 - 171200862756*x**91 + 7002807007378*x**90/5 - 1038618567386 
4*x**89 + 70297408804833*x**88 - 436790467844808*x**87 + 2503926751715004* 
x**86 - 66501348729372018*x**85/5 + 131419332012806607*x**84/2 - 302950588 
656028082*x**83 + 1307276513180023617*x**82 - 5293662917568490696*x**81 + 
201631839342385547403*x**80/10 - 72392559119475593544*x**79 + 245464847895 
078057708*x**78 - 787400226364730912388*x**77 + 2393282266977011062653*x** 
76 - 172561788070239874568724*x**75/25 + 18914430223915181446722*x**74 - 4 
9303368020068535591064*x**73 + 122384749256712270099849*x**72 - 2895864489 
45460019391192*x**71 + 3268857173695411601376612*x**70/5 - 140939856402084 
7755666003*x**69 + 11614348594417788189739629*x**68/4 - 572050005396697030 
2409071*x**67 + 21569504532490178066659311*x**66/2 - 973393025055967010187 
70224*x**65/5 + 134663663432573814416170293*x**64/4 - 55800482090690569716 
307824*x**63 + 88685381585824590330757224*x**62 - 135208860450519457389515 
112*x**61 + 989058310490690095822896114*x**60/5 - 277798460089394796889723 
848*x**59 + 374593512943317843600698196*x**58 - 48511950958528562839516257 
6*x**57 + 603511770853123192467791538*x**56 - 3606758093003645324155339152 
*x**55/5 + 828502745555998906218503832*x**54 - 914479445566527094599669324 
*x**53 + 970109082168886474373197089*x**52 - 98913082887808032681188722...
 

3.1.64.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (15) = 30\).

Time = 0.20 (sec) , antiderivative size = 501, normalized size of antiderivative = 21.78 \[ \int (1-x)^{99} x \, dx=\text {Too large to display} \]

input

integrate(x*(1-x)^99,x, algorithm="maxima")
 

output

-1/101*x^101 + 99/100*x^100 - 49*x^99 + 3201/2*x^98 - 38808*x^97 + 2980131 
/4*x^96 - 58975224/5*x^95 + 158372676*x^94 - 1840869492*x^93 + 18815553757 
*x^92 - 171200862756*x^91 + 7002807007378/5*x^90 - 10386185673864*x^89 + 7 
0297408804833*x^88 - 436790467844808*x^87 + 2503926751715004*x^86 - 665013 
48729372018/5*x^85 + 131419332012806607/2*x^84 - 302950588656028082*x^83 + 
 1307276513180023617*x^82 - 5293662917568490696*x^81 + 2016318393423855474 
03/10*x^80 - 72392559119475593544*x^79 + 245464847895078057708*x^78 - 7874 
00226364730912388*x^77 + 2393282266977011062653*x^76 - 1725617880702398745 
68724/25*x^75 + 18914430223915181446722*x^74 - 49303368020068535591064*x^7 
3 + 122384749256712270099849*x^72 - 289586448945460019391192*x^71 + 326885 
7173695411601376612/5*x^70 - 1409398564020847755666003*x^69 + 116143485944 
17788189739629/4*x^68 - 5720500053966970302409071*x^67 + 21569504532490178 
066659311/2*x^66 - 97339302505596701018770224/5*x^65 + 1346636634325738144 
16170293/4*x^64 - 55800482090690569716307824*x^63 + 8868538158582459033075 
7224*x^62 - 135208860450519457389515112*x^61 + 989058310490690095822896114 
/5*x^60 - 277798460089394796889723848*x^59 + 374593512943317843600698196*x 
^58 - 485119509585285628395162576*x^57 + 603511770853123192467791538*x^56 
- 3606758093003645324155339152/5*x^55 + 828502745555998906218503832*x^54 - 
 914479445566527094599669324*x^53 + 970109082168886474373197089*x^52 - 989 
130828878080326811887228*x^51 + 24233705307512968006891237086/25*x^50 -...
 

3.1.64.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (15) = 30\).

Time = 0.27 (sec) , antiderivative size = 501, normalized size of antiderivative = 21.78 \[ \int (1-x)^{99} x \, dx=\text {Too large to display} \]

input

integrate(x*(1-x)^99,x, algorithm="giac")
 

output

-1/101*x^101 + 99/100*x^100 - 49*x^99 + 3201/2*x^98 - 38808*x^97 + 2980131 
/4*x^96 - 58975224/5*x^95 + 158372676*x^94 - 1840869492*x^93 + 18815553757 
*x^92 - 171200862756*x^91 + 7002807007378/5*x^90 - 10386185673864*x^89 + 7 
0297408804833*x^88 - 436790467844808*x^87 + 2503926751715004*x^86 - 665013 
48729372018/5*x^85 + 131419332012806607/2*x^84 - 302950588656028082*x^83 + 
 1307276513180023617*x^82 - 5293662917568490696*x^81 + 2016318393423855474 
03/10*x^80 - 72392559119475593544*x^79 + 245464847895078057708*x^78 - 7874 
00226364730912388*x^77 + 2393282266977011062653*x^76 - 1725617880702398745 
68724/25*x^75 + 18914430223915181446722*x^74 - 49303368020068535591064*x^7 
3 + 122384749256712270099849*x^72 - 289586448945460019391192*x^71 + 326885 
7173695411601376612/5*x^70 - 1409398564020847755666003*x^69 + 116143485944 
17788189739629/4*x^68 - 5720500053966970302409071*x^67 + 21569504532490178 
066659311/2*x^66 - 97339302505596701018770224/5*x^65 + 1346636634325738144 
16170293/4*x^64 - 55800482090690569716307824*x^63 + 8868538158582459033075 
7224*x^62 - 135208860450519457389515112*x^61 + 989058310490690095822896114 
/5*x^60 - 277798460089394796889723848*x^59 + 374593512943317843600698196*x 
^58 - 485119509585285628395162576*x^57 + 603511770853123192467791538*x^56 
- 3606758093003645324155339152/5*x^55 + 828502745555998906218503832*x^54 - 
 914479445566527094599669324*x^53 + 970109082168886474373197089*x^52 - 989 
130828878080326811887228*x^51 + 24233705307512968006891237086/25*x^50 -...
 

3.1.64.9 Mupad [B] (verification not implemented)

Time = 13.92 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int (1-x)^{99} x \, dx=-\frac {\left (100\,x+1\right )\,{\left (x-1\right )}^{100}}{10100} \]

input

int(-x*(x - 1)^99,x)
 

output

-((100*x + 1)*(x - 1)^100)/10100