∫ (1 − x)2014xdx [105]

Optimal. Leaf size=23 \[ -\frac {(1-x)^{2015}}{2015}+\frac {(1-x)^{2016}}{2016} \]

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \begin {gather*} \frac {(1-x)^{2016}}{2016}-\frac {(1-x)^{2015}}{2015} \end {gather*}

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]

&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

\begin {gather*} \begin {aligned} \text {Integral} &=\int \left ((1-x)^{2014}-(1-x)^{2015}\right ) \, dx\\ &=-\frac {(1-x)^{2015}}{2015}+\frac {(1-x)^{2016}}{2016}\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(12138\) vs. \(2(23)=46\).
time = 0.02, size = 12138, normalized size = 527.74 \begin {gather*} \text {Result too large to show} \end {gather*}

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(10076\) vs. \(2(19)=38\).
time = 8.49, size = 10077, normalized size = 438.13


method	result	size

gosper	\(\text {Expression too large to display}\)	\(10076\)
default	\(\text {Expression too large to display}\)	\(10077\)
risch	\(\text {Expression too large to display}\)	\(10077\)
parallelrisch	\(\text {Expression too large to display}\)	\(10077\)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 10076 vs. \(2 (15) = 30\).
time = 2.63, size = 10076, normalized size = 438.09 \begin {gather*} \text {Too large to display} \end {gather*}

1/2016*x^2016 - 2014/2015*x^2015 + 2013/2*x^2014 - 2026084/3*x^2013 + 1358826667/4*x^2012 - 136629987582*x^201

1 + 1373131035323509/30*x^2010 - 91953985549170536/7*x^2009 + 26377651026988133103/8*x^2008 - 6617491558542444

915874/9*x^2007 + 294992994835264731661117/2*x^2006 - 134422910799097740606580164/5*x^2005 + 53876689232818214

844524454823/12*x^2004 - 691762702790623489451562620638*x^2003 + 2571973087266166342850029070691063/26*x^2002

- 39588591248274824267756569403940400/3*x^2001 + 131961937837090040663501674882660163383/80*x^2000 - 193964593

913505209402992927651733959950*x^1999 + 387541161559807075301489477559812273935075/18*x^1998 - 226292253091596

9121458419278661196219696900*x^1997 + 903358447595910292527771972191922410542659825/4*x^1996 - 214547577397927

27450218123378990236305630977010*x^1995 + 3889161470317168646281025630946632817062982660525/2*x^1994 - 1685021

05628944447818705199670044187130493533353400*x^1993 + 111885369941148961252482398045233357984870743212666075/8

*x^1992 - 1113818576786312843600553552092286915421060813044865230*x^1991 + 17049987754591807907343287673317020

9565984828166917578371/2*x^1990 - 734769036554419859459683303160816263037237362099897570964140/117*x^1989 + 17

83541483599579507954055928245847409394570340210724349640535/4*x^1988 - 305508278474757676047756498538367963949

05428348789090875481890*x^1987 + 12134785744398256314996786936077927810352871698504752093214631735/6*x^1986 -

129502922523593280492136994687090590798244450408627036948442699584*x^1985 + 7964945723407916692243386031594277

710664745455362659645679567475797155/992*x^1984 - 144744165624118561261874947183971562119263975542371901322485

6987762340/3*x^1983 + 28125770905652554308973525362902431630707416661449962135634978911861255*x^1982 - 1591918

227609528492506862969541498328323111814372651235402040612137475080*x^1981 + 1575998643333880780933185120941293

056782727079281496851166395104490693223777/18*x^1980 - 4683033812006962024438255993532176273777170532620461943

334529407905781402420*x^1979 + 243764171383939582809061001601068990795547409624002219219883316766423225728865*

x^1978 - 37070895655847111385453234444800707383220081936223576743436134792982165581556720/3*x^1977 + 317425641

23997639240757908526821868866411524435554862316447839037129021749036585525/52*x^1976 - 29405041309901956341188

419072725681104134131163443548404833409099760693787449612268*x^1975 + 4146109760683970391004500755905823687091

180545350679873368223491236430795491757775525/3*x^1974 - 63412964778336184585839720798751223807459136568252263

550556300343764105256910301716600*x^1973 + 5684106108463777496364455720240876368514714260586800253709270268295

316442312451914078575/2*x^1972 - 11203370255912948436822167508990647463957738482038571840497169715415716452436

88771886809300/9*x^1971 + 533107094617355816419170566510634085586038442791694394384531565896504801056252790790

2040665*x^1970 - 223337786925829485322981480991438350221372508041201117237722302034525923462936129127215408800

*x^1969 + 73254775197564099825039778493724892741313585299056307138160140869975075852658475240572087079675/8*x^

1968 - 2573073313780892656182942719993545807672157537925368302641529369583213434895514150007213443740700/7*x^1

967 + 14453316646020616414051419806563949347846356733759521758169481834749898741394522590424547417732285*x^196

6 - 556877644197119279476418494149379621380918035553099592288242272393357711099559188691208229092310776*x^1965

+ 42065669602328983454644988159758954775468201354159354767262170532890818036890454346362487193259926655/2*x^1

964 - 10127108905672074866716134495513810074049503443438355901587873018991904643082375127060735008804049547340

/13*x^1963 + 2547356732218368808068451666327803104686436046670043326856172840236758304253962947034557682799843

62513795/9*x^1962 - 100916469754973309008528860784251421869215186502357434768819760899018319803591237124743943

8789779967687120*x^1961 + 988981146158764563555753732295712954085089773602729130230773532270173159028459930987

702066483515289307875639/28*x^1960 - 3641754812942163592621144153796255272209301318341529098880465874988579919

304712880932120180334504864257742540/3*x^1959 + 40980195768992485650787266723287875856054581998136074879102742

342919244118363195990927194823828420896033237685*x^1958 - 1359291901349214321338405479553195610468798911828009

406507555490127093365391727648159674832991084850660654090760*x^1957 + 2658774264104553320301560515549261476451

40294720076440116895926079723975200408973392729321685058600107994356274635/6*x^1956 - 142019189244195789127950

6313904609247029851564277116872753537985799241047323477570643261488664366848376947188162908*x^1955 + 447589392

09983045977338893334713714920785994860335491205908266637912877071032782750326842694817487942770394173511915*x^

1954 - 3871199637329501447418000582050152588675313103951258803662243942776879796538544404275706258986721565787

26484398554499520/279*x^1953 + 1354225754277894499676937345006632528266873746430900140410227088281348549726611

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RecursionError} \end {gather*}

Exception raised: RecursionError >> maximum recursion depth exceeded

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 12024 vs. \(2 (12) = 24\).
time = 3.22, size = 12024, normalized size = 522.78 \begin {gather*} \text {Too large to display} \end {gather*}

x**2016/2016 - 2014*x**2015/2015 + 2013*x**2014/2 - 2026084*x**2013/3 + 1358826667*x**2012/4 - 136629987582*x*

*2011 + 1373131035323509*x**2010/30 - 91953985549170536*x**2009/7 + 26377651026988133103*x**2008/8 - 661749155

8542444915874*x**2007/9 + 294992994835264731661117*x**2006/2 - 134422910799097740606580164*x**2005/5 + 5387668

9232818214844524454823*x**2004/12 - 691762702790623489451562620638*x**2003 + 257197308726616634285002907069106

3*x**2002/26 - 39588591248274824267756569403940400*x**2001/3 + 131961937837090040663501674882660163383*x**2000

/80 - 193964593913505209402992927651733959950*x**1999 + 387541161559807075301489477559812273935075*x**1998/18

- 2262922530915969121458419278661196219696900*x**1997 + 903358447595910292527771972191922410542659825*x**1996/

4 - 21454757739792727450218123378990236305630977010*x**1995 + 388916147031716864628102563094663281706298266052

5*x**1994/2 - 168502105628944447818705199670044187130493533353400*x**1993 + 1118853699411489612524823980452333

57984870743212666075*x**1992/8 - 1113818576786312843600553552092286915421060813044865230*x**1991 + 17049987754

5918079073432876733170209565984828166917578371*x**1990/2 - 734769036554419859459683303160816263037237362099897

570964140*x**1989/117 + 1783541483599579507954055928245847409394570340210724349640535*x**1988/4 - 305508278474

75767604775649853836796394905428348789090875481890*x**1987 + 1213478574439825631499678693607792781035287169850

4752093214631735*x**1986/6 - 129502922523593280492136994687090590798244450408627036948442699584*x**1985 + 7964

945723407916692243386031594277710664745455362659645679567475797155*x**1984/992 - 14474416562411856126187494718

39715621192639755423719013224856987762340*x**1983/3 + 28125770905652554308973525362902431630707416661449962135

634978911861255*x**1982 - 1591918227609528492506862969541498328323111814372651235402040612137475080*x**1981 +

1575998643333880780933185120941293056782727079281496851166395104490693223777*x**1980/18 - 46830338120069620244

38255993532176273777170532620461943334529407905781402420*x**1979 + 2437641713839395828090610016010689907955474

09624002219219883316766423225728865*x**1978 - 3707089565584711138545323444480070738322008193622357674343613479

2982165581556720*x**1977/3 + 317425641239976392407579085268218688664115244355548623164478390371290217490365855

25*x**1976/52 - 29405041309901956341188419072725681104134131163443548404833409099760693787449612268*x**1975 +

4146109760683970391004500755905823687091180545350679873368223491236430795491757775525*x**1974/3 - 634129647783

36184585839720798751223807459136568252263550556300343764105256910301716600*x**1973 + 5684106108463777496364455

720240876368514714260586800253709270268295316442312451914078575*x**1972/2 - 1120337025591294843682216750899064

746395773848203857184049716971541571645243688771886809300*x**1971/9 + 5331070946173558164191705665106340855860

384427916943943845315658965048010562527907902040665*x**1970 - 223337786925829485322981480991438350221372508041

201117237722302034525923462936129127215408800*x**1969 + 732547751975640998250397784937248927413135852990563071

38160140869975075852658475240572087079675*x**1968/8 - 25730733137808926561829427199935458076721575379253683026

41529369583213434895514150007213443740700*x**1967/7 + 14453316646020616414051419806563949347846356733759521758

169481834749898741394522590424547417732285*x**1966 - 556877644197119279476418494149379621380918035553099592288

242272393357711099559188691208229092310776*x**1965 + 420656696023289834546449881597589547754682013541593547672

62170532890818036890454346362487193259926655*x**1964/2 - 10127108905672074866716134495513810074049503443438355

901587873018991904643082375127060735008804049547340*x**1963/13 + 254735673221836880806845166632780310468643604

667004332685617284023675830425396294703455768279984362513795*x**1962/9 - 1009164697549733090085288607842514218

692151865023574347688197608990183198035912371247439438789779967687120*x**1961 + 988981146158764563555753732295

712954085089773602729130230773532270173159028459930987702066483515289307875639*x**1960/28 - 364175481294216359

2621144153796255272209301318341529098880465874988579919304712880932120180334504864257742540*x**1959/3 + 409801

95768992485650787266723287875856054581998136074879102742342919244118363195990927194823828420896033237685*x**19

58 - 135929190134921432133840547955319561046879891182800940650755549012709336539172764815967483299108485066065

4090760*x**1957 + 26587742641045533203015605155492614764514029472007644011689592607972397520040897339272932168

5058600107994356274635*x**1956/6 - 142019189244195789127950631390460924702985156427711687275353798579924104732

3477570643261488664366848376947188162908*x**1955 + 44758939209983045977338893334713714920785994860335491205908

266637912877071032782750326842694817487942770394173511915*x**1954 - 387119963732950144741800058205015258867531

310395125880366224394277687979653854440427570625898672156578726484398554499520*x**1953/279 + 13542257542778944

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 10076 vs. \(2 (15) = 30\).
time = 1.68, size = 10076, normalized size = 438.09 \begin {gather*} \text {Too large to display} \end {gather*}