Integrand size = 9, antiderivative size = 23 \[ \int (1-x)^{2014} x \, dx=-\frac {(1-x)^{2015}}{2015}+\frac {(1-x)^{2016}}{2016} \]
Leaf count is larger than twice the leaf count of optimal. \(12138\) vs. \(2(23)=46\).
Time = 0.13 (sec) , antiderivative size = 12138, normalized size of antiderivative = 527.74 \[ \int (1-x)^{2014} x \, dx=\text {Result too large to show} \]
Time = 0.87 (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 |
| \(\displaystyle \frac {(1-x)^{2016}}{2016}-\frac {(1-x)^{2015}}{2015}\) |
3.2.5.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(10075\) vs. \(2(19)=38\).
Time = 7.62 (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 | \(\text {Expression too large to display}\) | \(10077\) |
Exception generated. \[ \int (1-x)^{2014} x \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> System error: Heap exhausted (no more space for allocation).2293760 bytes available, 2557536 requested.PROCEED WITH CAUTION.
Leaf count of result is larger than twice the leaf count of optimal. 12024 vs. \(2 (12) = 24\).
Time = 2.41 (sec) , antiderivative size = 12024, normalized size of antiderivative = 522.78 \[ \int (1-x)^{2014} x \, dx=\text {Too large to display} \]
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...
Leaf count of result is larger than twice the leaf count of optimal. 10076 vs. \(2 (15) = 30\).
Time = 2.29 (sec) , antiderivative size = 10076, normalized size of antiderivative = 438.09 \[ \int (1-x)^{2014} x \, dx=\text {Too large to display} \]
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...
Leaf count of result is larger than twice the leaf count of optimal. 10076 vs. \(2 (15) = 30\).
Time = 1.64 (sec) , antiderivative size = 10076, normalized size of antiderivative = 438.09 \[ \int (1-x)^{2014} x \, dx=\text {Too large to display} \]
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...
Time = 34.75 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int (1-x)^{2014} x \, dx=\frac {{\left (x-1\right )}^{2015}}{2015}+\frac {{\left (x-1\right )}^{2016}}{2016} \]