\(\int (1-x)^{2014} x \, dx\) [105]

Optimal result

Integrand size = 9, antiderivative size = 23 \[ \int (1-x)^{2014} x \, dx=-\frac {(1-x)^{2015}}{2015}+\frac {(1-x)^{2016}}{2016} \] Output:

-1/2015*(1-x)^2015+1/2016*(1-x)^2016
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(12138\) vs. \(2(23)=46\).

Time = 0.07 (sec) , antiderivative size = 12138, normalized size of antiderivative = 527.74 \[ \int (1-x)^{2014} x \, dx=\text {Result too large to show} \] Input:

Integrate[(1 - x)^2014*x,x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.19 (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.

\(\displaystyle \int (1-x)^{2014} x \, dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left ((1-x)^{2014}-(1-x)^{2015}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(1-x)^{2016}}{2016}-\frac {(1-x)^{2015}}{2015}\)

Input:

Int[(1 - x)^2014*x,x]
 

Output:

-1/2015*(1 - x)^2015 + (1 - x)^2016/2016
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(10075\) vs. \(2(19)=38\).

Time = 7.77 (sec) , antiderivative size = 10076, normalized size of antiderivative = 438.09

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	\(\text {Expression too large to display}\)	\(10077\)
orering	\(\text {Expression too large to display}\)	\(10088\)

Input:

int((1-x)^2014*x,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-2)]

Exception generated. \[ \int (1-x)^{2014} x \, dx=\text {Exception raised: RecursionError} \] Input:

integrate((1-x)^2014*x,x, algorithm="fricas")
 

Output:

Exception raised: RecursionError >> maximum recursion depth exceeded
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 12024 vs. \(2 (12) = 24\).

Time = 2.79 (sec) , antiderivative size = 12024, normalized size of antiderivative = 522.78 \[ \int (1-x)^{2014} x \, dx=\text {Too large to display} \] Input:

integrate((1-x)**2014*x,x)
 

Output:

x**2016/2016 - 2014*x**2015/2015 + 2013*x**2014/2 - 2026084*x**2013/3 + 13 
58826667*x**2012/4 - 136629987582*x**2011 + 1373131035323509*x**2010/30 - 
91953985549170536*x**2009/7 + 26377651026988133103*x**2008/8 - 66174915585 
42444915874*x**2007/9 + 294992994835264731661117*x**2006/2 - 1344229107990 
97740606580164*x**2005/5 + 53876689232818214844524454823*x**2004/12 - 6917 
62702790623489451562620638*x**2003 + 2571973087266166342850029070691063*x* 
*2002/26 - 39588591248274824267756569403940400*x**2001/3 + 131961937837090 
040663501674882660163383*x**2000/80 - 193964593913505209402992927651733959 
950*x**1999 + 387541161559807075301489477559812273935075*x**1998/18 - 2262 
922530915969121458419278661196219696900*x**1997 + 903358447595910292527771 
972191922410542659825*x**1996/4 - 2145475773979272745021812337899023630563 
0977010*x**1995 + 3889161470317168646281025630946632817062982660525*x**199 
4/2 - 168502105628944447818705199670044187130493533353400*x**1993 + 111885 
369941148961252482398045233357984870743212666075*x**1992/8 - 1113818576786 
312843600553552092286915421060813044865230*x**1991 + 170499877545918079073 
432876733170209565984828166917578371*x**1990/2 - 7347690365544198594596833 
03160816263037237362099897570964140*x**1989/117 + 178354148359957950795405 
5928245847409394570340210724349640535*x**1988/4 - 305508278474757676047756 
49853836796394905428348789090875481890*x**1987 + 1213478574439825631499678 
6936077927810352871698504752093214631735*x**1986/6 - 129502922523593280...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10076 vs. \(2 (15) = 30\).

Time = 2.52 (sec) , antiderivative size = 10076, normalized size of antiderivative = 438.09 \[ \int (1-x)^{2014} x \, dx=\text {Too large to display} \] Input:

integrate((1-x)^2014*x,x, algorithm="maxima")
 

Output:

1/2016*x^2016 - 2014/2015*x^2015 + 2013/2*x^2014 - 2026084/3*x^2013 + 1358 
826667/4*x^2012 - 136629987582*x^2011 + 1373131035323509/30*x^2010 - 91953 
985549170536/7*x^2009 + 26377651026988133103/8*x^2008 - 661749155854244491 
5874/9*x^2007 + 294992994835264731661117/2*x^2006 - 1344229107990977406065 
80164/5*x^2005 + 53876689232818214844524454823/12*x^2004 - 691762702790623 
489451562620638*x^2003 + 2571973087266166342850029070691063/26*x^2002 - 39 
588591248274824267756569403940400/3*x^2001 + 13196193783709004066350167488 
2660163383/80*x^2000 - 193964593913505209402992927651733959950*x^1999 + 38 
7541161559807075301489477559812273935075/18*x^1998 - 226292253091596912145 
8419278661196219696900*x^1997 + 903358447595910292527771972191922410542659 
825/4*x^1996 - 21454757739792727450218123378990236305630977010*x^1995 + 38 
89161470317168646281025630946632817062982660525/2*x^1994 - 168502105628944 
447818705199670044187130493533353400*x^1993 + 1118853699411489612524823980 
45233357984870743212666075/8*x^1992 - 111381857678631284360055355209228691 
5421060813044865230*x^1991 + 170499877545918079073432876733170209565984828 
166917578371/2*x^1990 - 73476903655441985945968330316081626303723736209989 
7570964140/117*x^1989 + 17835414835995795079540559282458474093945703402107 
24349640535/4*x^1988 - 305508278474757676047756498538367963949054283487890 
90875481890*x^1987 + 12134785744398256314996786936077927810352871698504752 
093214631735/6*x^1986 - 12950292252359328049213699468709059079824445040...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10076 vs. \(2 (15) = 30\).

Time = 1.68 (sec) , antiderivative size = 10076, normalized size of antiderivative = 438.09 \[ \int (1-x)^{2014} x \, dx=\text {Too large to display} \] Input:

integrate((1-x)^2014*x,x, algorithm="giac")
 

Output:

1/2016*x^2016 - 2014/2015*x^2015 + 2013/2*x^2014 - 2026084/3*x^2013 + 1358 
826667/4*x^2012 - 136629987582*x^2011 + 1373131035323509/30*x^2010 - 91953 
985549170536/7*x^2009 + 26377651026988133103/8*x^2008 - 661749155854244491 
5874/9*x^2007 + 294992994835264731661117/2*x^2006 - 1344229107990977406065 
80164/5*x^2005 + 53876689232818214844524454823/12*x^2004 - 691762702790623 
489451562620638*x^2003 + 2571973087266166342850029070691063/26*x^2002 - 39 
588591248274824267756569403940400/3*x^2001 + 13196193783709004066350167488 
2660163383/80*x^2000 - 193964593913505209402992927651733959950*x^1999 + 38 
7541161559807075301489477559812273935075/18*x^1998 - 226292253091596912145 
8419278661196219696900*x^1997 + 903358447595910292527771972191922410542659 
825/4*x^1996 - 21454757739792727450218123378990236305630977010*x^1995 + 38 
89161470317168646281025630946632817062982660525/2*x^1994 - 168502105628944 
447818705199670044187130493533353400*x^1993 + 1118853699411489612524823980 
45233357984870743212666075/8*x^1992 - 111381857678631284360055355209228691 
5421060813044865230*x^1991 + 170499877545918079073432876733170209565984828 
166917578371/2*x^1990 - 73476903655441985945968330316081626303723736209989 
7570964140/117*x^1989 + 17835414835995795079540559282458474093945703402107 
24349640535/4*x^1988 - 305508278474757676047756498538367963949054283487890 
90875481890*x^1987 + 12134785744398256314996786936077927810352871698504752 
093214631735/6*x^1986 - 12950292252359328049213699468709059079824445040...
 

Mupad [B] (verification not implemented)

Time = 3.62 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int (1-x)^{2014} x \, dx=\frac {{\left (x-1\right )}^{2015}}{2015}+\frac {{\left (x-1\right )}^{2016}}{2016} \] Input:

int(x*(x - 1)^2014,x)
 

Output:

(x - 1)^2015/2015 + (x - 1)^2016/2016
 

Reduce [B] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 10075, normalized size of antiderivative = 438.04 \[ \int (1-x)^{2014} x \, dx =\text {Too large to display} \] Input:

int((1-x)^2014*x,x)
                                                                                    
                                                                                    
 

Output:

(x**2*(2015*x**2014 - 4060224*x**2013 + 4088644560*x**2012 - 2743479822720 
*x**2011 + 1379970009938520*x**2010 - 555023800755103680*x**2009 + 1859329 
27231085706672*x**2008 - 53362736893894645451520*x**2007 + 133940436384840 
34227041340*x**2006 - 2986870989863717937228888640*x**2005 + 5991661716698 
02901771527961040*x**2004 - 109211625032905361160334841081472*x**2003 + 18 
238336839093622089168418446681960*x**2002 - 281010612178418236378971574006 
0509120*x**2001 + 401845075154465829406888542004771683120*x**2000 - 536061 
19637463974044483815498487616832000*x**1999 + 6700763279491758084811288047 
191717776261974*x**1998 - 787930731979197401845213990423979761467288000*x* 
*1997 + 87460289340817260754040145295698433981667726000*x**1996 - 91925344 
21988086403953249130548657731501535056000*x**1995 + 9174147050405026566795 
04104079228723250703611877000*x**1994 - 8715437508089560915737406951506929 
7530186380049102400*x**1993 + 78993536455906075808343167795283248473929653 
41445538000*x**1992 - 6844959935701232937070570103076402987289760509295156 
16000*x**1991 + 5681315314871661954478551207940859451755766598852757956350 
0*x**1990 - 4524598375364431485787912661451371599300050077183373331915200* 
x**1989 + 346305711281065128767630984590276676053663104186389631780905520* 
x**1988 - 2551118094916945752044020428574354065265288121210844366387494080 
0*x**1987 + 18112933890843889650978210384893527950847498547044032205209417 
24600*x**1986 - 1241047949151299621948238358622499877872406272555849965...