Integrand size = 17, antiderivative size = 143 \[ \int x^m (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{1+m}}{1+m}+\frac {11 x^{2+m}}{2+m}+\frac {55 x^{3+m}}{3+m}+\frac {165 x^{4+m}}{4+m}+\frac {330 x^{5+m}}{5+m}+\frac {462 x^{6+m}}{6+m}+\frac {462 x^{7+m}}{7+m}+\frac {330 x^{8+m}}{8+m}+\frac {165 x^{9+m}}{9+m}+\frac {55 x^{10+m}}{10+m}+\frac {11 x^{11+m}}{11+m}+\frac {x^{12+m}}{12+m} \]
x^(1+m)/(1+m)+11*x^(2+m)/(2+m)+55*x^(3+m)/(3+m)+165*x^(4+m)/(4+m)+330*x^(5 +m)/(5+m)+462*x^(6+m)/(6+m)+462*x^(7+m)/(7+m)+330*x^(8+m)/(8+m)+165*x^(9+m )/(9+m)+55*x^(10+m)/(10+m)+11*x^(11+m)/(11+m)+x^(12+m)/(12+m)
Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.83 \[ \int x^m (1+x) \left (1+2 x+x^2\right )^5 \, dx=x^{1+m} \left (\frac {1}{1+m}+\frac {11 x}{2+m}+\frac {55 x^2}{3+m}+\frac {165 x^3}{4+m}+\frac {330 x^4}{5+m}+\frac {462 x^5}{6+m}+\frac {462 x^6}{7+m}+\frac {330 x^7}{8+m}+\frac {165 x^8}{9+m}+\frac {55 x^9}{10+m}+\frac {11 x^{10}}{11+m}+\frac {x^{11}}{12+m}\right ) \]
x^(1 + m)*((1 + m)^(-1) + (11*x)/(2 + m) + (55*x^2)/(3 + m) + (165*x^3)/(4 + m) + (330*x^4)/(5 + m) + (462*x^5)/(6 + m) + (462*x^6)/(7 + m) + (330*x ^7)/(8 + m) + (165*x^8)/(9 + m) + (55*x^9)/(10 + m) + (11*x^10)/(11 + m) + x^11/(12 + m))
Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1184, 53, 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.
\(\displaystyle \int (x+1) \left (x^2+2 x+1\right )^5 x^m \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int (x+1)^{11} x^mdx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (11 x^{m+1}+55 x^{m+2}+165 x^{m+3}+330 x^{m+4}+462 x^{m+5}+462 x^{m+6}+330 x^{m+7}+165 x^{m+8}+55 x^{m+9}+11 x^{m+10}+x^{m+11}+x^m\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^{m+1}}{m+1}+\frac {11 x^{m+2}}{m+2}+\frac {55 x^{m+3}}{m+3}+\frac {165 x^{m+4}}{m+4}+\frac {330 x^{m+5}}{m+5}+\frac {462 x^{m+6}}{m+6}+\frac {462 x^{m+7}}{m+7}+\frac {330 x^{m+8}}{m+8}+\frac {165 x^{m+9}}{m+9}+\frac {55 x^{m+10}}{m+10}+\frac {11 x^{m+11}}{m+11}+\frac {x^{m+12}}{m+12}\) |
x^(1 + m)/(1 + m) + (11*x^(2 + m))/(2 + m) + (55*x^(3 + m))/(3 + m) + (165 *x^(4 + m))/(4 + m) + (330*x^(5 + m))/(5 + m) + (462*x^(6 + m))/(6 + m) + (462*x^(7 + m))/(7 + m) + (330*x^(8 + m))/(8 + m) + (165*x^(9 + m))/(9 + m ) + (55*x^(10 + m))/(10 + m) + (11*x^(11 + m))/(11 + m) + x^(12 + m)/(12 + m)
3.9.41.3.1 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, n}, x] && 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])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(1094\) vs. \(2(143)=286\).
Time = 0.25 (sec) , antiderivative size = 1095, normalized size of antiderivative = 7.66
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1095\) |
gosper | \(\text {Expression too large to display}\) | \(1096\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1561\) |
x^m*(m^11*x^11+11*m^11*x^10+66*m^10*x^11+55*m^11*x^9+737*m^10*x^10+1925*m^ 9*x^11+165*m^11*x^8+3740*m^10*x^9+21780*m^9*x^10+32670*m^8*x^11+330*m^11*x ^7+11385*m^10*x^8+112035*m^9*x^9+373890*m^8*x^10+357423*m^7*x^11+462*m^11* x^6+23100*m^10*x^7+345840*m^9*x^8+1947000*m^8*x^9+4131303*m^7*x^10+2637558 *m^6*x^11+462*m^11*x^5+32802*m^10*x^6+711810*m^9*x^7+6089490*m^8*x^8+21750 465*m^7*x^9+30748641*m^6*x^10+13339535*m^5*x^11+330*m^11*x^4+33264*m^10*x^ 5+1025640*m^9*x^6+12709620*m^8*x^7+68855985*m^7*x^8+163460220*m^6*x^9+1566 57490*m^5*x^10+45995730*m^4*x^11+165*m^11*x^3+24090*m^10*x^4+1055670*m^9*x ^5+18586260*m^8*x^6+145645830*m^7*x^7+523190745*m^6*x^8+839860505*m^5*x^9+ 543539260*m^4*x^10+105258076*m^3*x^11+55*m^11*x^2+12210*m^10*x^3+776160*m^ 9*x^4+19431720*m^8*x^5+216148086*m^7*x^6+1120622580*m^6*x^7+2714671410*m^5 *x^8+2935253200*m^4*x^9+1250343336*m^3*x^10+150917976*m^2*x^11+11*m^11*x+4 125*m^10*x^2+399465*m^9*x^3+14523300*m^8*x^4+229661586*m^7*x^5+1687068306* m^6*x^6+5881795590*m^5*x^7+9569532060*m^4*x^8+6793843980*m^3*x^9+180038707 2*m^2*x^10+120543840*m*x^11+m^11+836*m^10*x+137060*m^9*x^2+7604190*m^8*x^3 +174706290*m^7*x^4+1822135392*m^6*x^5+8976008580*m^5*x^6+20948784780*m^4*x ^7+22313339400*m^3*x^8+9832379040*m^2*x^9+1442897280*m*x^10+39916800*x^11+ 77*m^10+28215*m^9*x+2656170*m^8*x^2+93244635*m^7*x^3+1412257770*m^6*x^4+98 52674370*m^5*x^5+32372349240*m^4*x^6+49287977640*m^3*x^7+32492401920*m^2*x ^8+7911984960*m*x^9+479001600*x^10+2640*m^9+557040*m^8*x+33251955*m^7*x...
Leaf count of result is larger than twice the leaf count of optimal. 757 vs. \(2 (143) = 286\).
Time = 0.29 (sec) , antiderivative size = 757, normalized size of antiderivative = 5.29 \[ \int x^m (1+x) \left (1+2 x+x^2\right )^5 \, dx=\text {Too large to display} \]
((m^11 + 66*m^10 + 1925*m^9 + 32670*m^8 + 357423*m^7 + 2637558*m^6 + 13339 535*m^5 + 45995730*m^4 + 105258076*m^3 + 150917976*m^2 + 120543840*m + 399 16800)*x^12 + 11*(m^11 + 67*m^10 + 1980*m^9 + 33990*m^8 + 375573*m^7 + 279 5331*m^6 + 14241590*m^5 + 49412660*m^4 + 113667576*m^3 + 163671552*m^2 + 1 31172480*m + 43545600)*x^11 + 55*(m^11 + 68*m^10 + 2037*m^9 + 35400*m^8 + 395463*m^7 + 2972004*m^6 + 15270191*m^5 + 53368240*m^4 + 123524436*m^3 + 1 78770528*m^2 + 143854272*m + 47900160)*x^10 + 165*(m^11 + 69*m^10 + 2096*m ^9 + 36906*m^8 + 417309*m^7 + 3170853*m^6 + 16452554*m^5 + 57997164*m^4 + 135232360*m^3 + 196923648*m^2 + 159246720*m + 53222400)*x^9 + 330*(m^11 + 70*m^10 + 2157*m^9 + 38514*m^8 + 441351*m^7 + 3395826*m^6 + 17823623*m^5 + 63481166*m^4 + 149357508*m^3 + 219154824*m^2 + 178320960*m + 59875200)*x^ 8 + 462*(m^11 + 71*m^10 + 2220*m^9 + 40230*m^8 + 467853*m^7 + 3651663*m^6 + 19428590*m^5 + 70070020*m^4 + 166716696*m^3 + 246998016*m^2 + 202573440* m + 68428800)*x^7 + 462*(m^11 + 72*m^10 + 2285*m^9 + 42060*m^8 + 497103*m^ 7 + 3944016*m^6 + 21326135*m^5 + 78113340*m^4 + 188526796*m^3 + 282854112* m^2 + 234434880*m + 79833600)*x^6 + 330*(m^11 + 73*m^10 + 2352*m^9 + 44010 *m^8 + 529413*m^7 + 4279569*m^6 + 23592386*m^5 + 88108220*m^4 + 216665736* m^3 + 330686208*m^2 + 278128512*m + 95800320)*x^5 + 165*(m^11 + 74*m^10 + 2421*m^9 + 46086*m^8 + 565119*m^7 + 4666158*m^6 + 26325599*m^5 + 100767754 *m^4 + 254135820*m^3 + 397471608*m^2 + 341673120*m + 119750400)*x^4 + 5...
Leaf count of result is larger than twice the leaf count of optimal. 11008 vs. \(2 (117) = 234\).
Time = 1.04 (sec) , antiderivative size = 11008, normalized size of antiderivative = 76.98 \[ \int x^m (1+x) \left (1+2 x+x^2\right )^5 \, dx=\text {Too large to display} \]
Piecewise((log(x) - 11/x - 55/(2*x**2) - 55/x**3 - 165/(2*x**4) - 462/(5*x **5) - 77/x**6 - 330/(7*x**7) - 165/(8*x**8) - 55/(9*x**9) - 11/(10*x**10) - 1/(11*x**11), Eq(m, -12)), (x + 11*log(x) - 55/x - 165/(2*x**2) - 110/x **3 - 231/(2*x**4) - 462/(5*x**5) - 55/x**6 - 165/(7*x**7) - 55/(8*x**8) - 11/(9*x**9) - 1/(10*x**10), Eq(m, -11)), (x**2/2 + 11*x + 55*log(x) - 165 /x - 165/x**2 - 154/x**3 - 231/(2*x**4) - 66/x**5 - 55/(2*x**6) - 55/(7*x* *7) - 11/(8*x**8) - 1/(9*x**9), Eq(m, -10)), (x**3/3 + 11*x**2/2 + 55*x + 165*log(x) - 330/x - 231/x**2 - 154/x**3 - 165/(2*x**4) - 33/x**5 - 55/(6* x**6) - 11/(7*x**7) - 1/(8*x**8), Eq(m, -9)), (x**4/4 + 11*x**3/3 + 55*x** 2/2 + 165*x + 330*log(x) - 462/x - 231/x**2 - 110/x**3 - 165/(4*x**4) - 11 /x**5 - 11/(6*x**6) - 1/(7*x**7), Eq(m, -8)), (x**5/5 + 11*x**4/4 + 55*x** 3/3 + 165*x**2/2 + 330*x + 462*log(x) - 462/x - 165/x**2 - 55/x**3 - 55/(4 *x**4) - 11/(5*x**5) - 1/(6*x**6), Eq(m, -7)), (x**6/6 + 11*x**5/5 + 55*x* *4/4 + 55*x**3 + 165*x**2 + 462*x + 462*log(x) - 330/x - 165/(2*x**2) - 55 /(3*x**3) - 11/(4*x**4) - 1/(5*x**5), Eq(m, -6)), (x**7/7 + 11*x**6/6 + 11 *x**5 + 165*x**4/4 + 110*x**3 + 231*x**2 + 462*x + 330*log(x) - 165/x - 55 /(2*x**2) - 11/(3*x**3) - 1/(4*x**4), Eq(m, -5)), (x**8/8 + 11*x**7/7 + 55 *x**6/6 + 33*x**5 + 165*x**4/2 + 154*x**3 + 231*x**2 + 330*x + 165*log(x) - 55/x - 11/(2*x**2) - 1/(3*x**3), Eq(m, -4)), (x**9/9 + 11*x**8/8 + 55*x* *7/7 + 55*x**6/2 + 66*x**5 + 231*x**4/2 + 154*x**3 + 165*x**2 + 165*x +...
Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00 \[ \int x^m (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{m + 12}}{m + 12} + \frac {11 \, x^{m + 11}}{m + 11} + \frac {55 \, x^{m + 10}}{m + 10} + \frac {165 \, x^{m + 9}}{m + 9} + \frac {330 \, x^{m + 8}}{m + 8} + \frac {462 \, x^{m + 7}}{m + 7} + \frac {462 \, x^{m + 6}}{m + 6} + \frac {330 \, x^{m + 5}}{m + 5} + \frac {165 \, x^{m + 4}}{m + 4} + \frac {55 \, x^{m + 3}}{m + 3} + \frac {11 \, x^{m + 2}}{m + 2} + \frac {x^{m + 1}}{m + 1} \]
x^(m + 12)/(m + 12) + 11*x^(m + 11)/(m + 11) + 55*x^(m + 10)/(m + 10) + 16 5*x^(m + 9)/(m + 9) + 330*x^(m + 8)/(m + 8) + 462*x^(m + 7)/(m + 7) + 462* x^(m + 6)/(m + 6) + 330*x^(m + 5)/(m + 5) + 165*x^(m + 4)/(m + 4) + 55*x^( m + 3)/(m + 3) + 11*x^(m + 2)/(m + 2) + x^(m + 1)/(m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1560 vs. \(2 (143) = 286\).
Time = 0.29 (sec) , antiderivative size = 1560, normalized size of antiderivative = 10.91 \[ \int x^m (1+x) \left (1+2 x+x^2\right )^5 \, dx=\text {Too large to display} \]
(m^11*x^12*x^m + 11*m^11*x^11*x^m + 66*m^10*x^12*x^m + 55*m^11*x^10*x^m + 737*m^10*x^11*x^m + 1925*m^9*x^12*x^m + 165*m^11*x^9*x^m + 3740*m^10*x^10* x^m + 21780*m^9*x^11*x^m + 32670*m^8*x^12*x^m + 330*m^11*x^8*x^m + 11385*m ^10*x^9*x^m + 112035*m^9*x^10*x^m + 373890*m^8*x^11*x^m + 357423*m^7*x^12* x^m + 462*m^11*x^7*x^m + 23100*m^10*x^8*x^m + 345840*m^9*x^9*x^m + 1947000 *m^8*x^10*x^m + 4131303*m^7*x^11*x^m + 2637558*m^6*x^12*x^m + 462*m^11*x^6 *x^m + 32802*m^10*x^7*x^m + 711810*m^9*x^8*x^m + 6089490*m^8*x^9*x^m + 217 50465*m^7*x^10*x^m + 30748641*m^6*x^11*x^m + 13339535*m^5*x^12*x^m + 330*m ^11*x^5*x^m + 33264*m^10*x^6*x^m + 1025640*m^9*x^7*x^m + 12709620*m^8*x^8* x^m + 68855985*m^7*x^9*x^m + 163460220*m^6*x^10*x^m + 156657490*m^5*x^11*x ^m + 45995730*m^4*x^12*x^m + 165*m^11*x^4*x^m + 24090*m^10*x^5*x^m + 10556 70*m^9*x^6*x^m + 18586260*m^8*x^7*x^m + 145645830*m^7*x^8*x^m + 523190745* m^6*x^9*x^m + 839860505*m^5*x^10*x^m + 543539260*m^4*x^11*x^m + 105258076* m^3*x^12*x^m + 55*m^11*x^3*x^m + 12210*m^10*x^4*x^m + 776160*m^9*x^5*x^m + 19431720*m^8*x^6*x^m + 216148086*m^7*x^7*x^m + 1120622580*m^6*x^8*x^m + 2 714671410*m^5*x^9*x^m + 2935253200*m^4*x^10*x^m + 1250343336*m^3*x^11*x^m + 150917976*m^2*x^12*x^m + 11*m^11*x^2*x^m + 4125*m^10*x^3*x^m + 399465*m^ 9*x^4*x^m + 14523300*m^8*x^5*x^m + 229661586*m^7*x^6*x^m + 1687068306*m^6* x^7*x^m + 5881795590*m^5*x^8*x^m + 9569532060*m^4*x^9*x^m + 6793843980*m^3 *x^10*x^m + 1800387072*m^2*x^11*x^m + 120543840*m*x^12*x^m + m^11*x*x^m...
Time = 10.76 (sec) , antiderivative size = 1459, normalized size of antiderivative = 10.20 \[ \int x^m (1+x) \left (1+2 x+x^2\right )^5 \, dx=\text {Too large to display} \]
(x^m*x^8*(58845916800*m + 72321091920*m^2 + 49287977640*m^3 + 20948784780* m^4 + 5881795590*m^5 + 1120622580*m^6 + 145645830*m^7 + 12709620*m^8 + 711 810*m^9 + 23100*m^10 + 330*m^11 + 19758816000))/(1486442880*m + 1931559552 *m^2 + 1414014888*m^3 + 657206836*m^4 + 206070150*m^5 + 44990231*m^6 + 692 6634*m^7 + 749463*m^8 + 55770*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600 ) + (x^m*x^10*(7911984960*m + 9832379040*m^2 + 6793843980*m^3 + 2935253200 *m^4 + 839860505*m^5 + 163460220*m^6 + 21750465*m^7 + 1947000*m^8 + 112035 *m^9 + 3740*m^10 + 55*m^11 + 2634508800))/(1486442880*m + 1931559552*m^2 + 1414014888*m^3 + 657206836*m^4 + 206070150*m^5 + 44990231*m^6 + 6926634*m ^7 + 749463*m^8 + 55770*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600) + (x ^m*x^2*(6858181440*m + 7194486816*m^2 + 4179838476*m^3 + 1524718360*m^4 + 371026645*m^5 + 61932948*m^6 + 7130013*m^7 + 557040*m^8 + 28215*m^9 + 836* m^10 + 11*m^11 + 2634508800))/(1486442880*m + 1931559552*m^2 + 1414014888* m^3 + 657206836*m^4 + 206070150*m^5 + 44990231*m^6 + 6926634*m^7 + 749463* m^8 + 55770*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600) + (x^m*x^11*(144 2897280*m + 1800387072*m^2 + 1250343336*m^3 + 543539260*m^4 + 156657490*m^ 5 + 30748641*m^6 + 4131303*m^7 + 373890*m^8 + 21780*m^9 + 737*m^10 + 11*m^ 11 + 479001600))/(1486442880*m + 1931559552*m^2 + 1414014888*m^3 + 6572068 36*m^4 + 206070150*m^5 + 44990231*m^6 + 6926634*m^7 + 749463*m^8 + 55770*m ^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600) + (x^m*x^6*(108308914560*m...