3.841 \(\int x^m (1+x) (1+2 x+x^2)^5 \, dx\)
Optimal. Leaf size=143 \[ \frac{x^{m+1}}{m+1}+\frac{11 x^{m+2}}{m+2}+\frac{55 x^{m+3}}{m+3}+\frac{165 x^{m+4}}{m+4}+\frac{330 x^{m+5}}{m+5}+\frac{462 x^{m+6}}{m+6}+\frac{462 x^{m+7}}{m+7}+\frac{330 x^{m+8}}{m+8}+\frac{165 x^{m+9}}{m+9}+\frac{55 x^{m+10}}{m+10}+\frac{11 x^{m+11}}{m+11}+\frac{x^{m+12}}{m+12} \]
[Out]
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)
________________________________________________________________________________________
Rubi [A] time = 0.042878, antiderivative size = 143, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used =
{27, 43} \[ \frac{x^{m+1}}{m+1}+\frac{11 x^{m+2}}{m+2}+\frac{55 x^{m+3}}{m+3}+\frac{165 x^{m+4}}{m+4}+\frac{330 x^{m+5}}{m+5}+\frac{462 x^{m+6}}{m+6}+\frac{462 x^{m+7}}{m+7}+\frac{330 x^{m+8}}{m+8}+\frac{165 x^{m+9}}{m+9}+\frac{55 x^{m+10}}{m+10}+\frac{11 x^{m+11}}{m+11}+\frac{x^{m+12}}{m+12} \]
Antiderivative was successfully verified.
[In]
Int[x^m*(1 + x)*(1 + 2*x + x^2)^5,x]
[Out]
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)
Rule 27
Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Rule 43
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])
Rubi steps
\begin{align*} \int x^m (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^m (1+x)^{11} \, dx\\ &=\int \left (x^m+11 x^{1+m}+55 x^{2+m}+165 x^{3+m}+330 x^{4+m}+462 x^{5+m}+462 x^{6+m}+330 x^{7+m}+165 x^{8+m}+55 x^{9+m}+11 x^{10+m}+x^{11+m}\right ) \, 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}\\ \end{align*}
Mathematica [A] time = 0.0531371, size = 119, normalized size = 0.83 \[ x^{m+1} \left (\frac{x^{11}}{m+12}+\frac{11 x^{10}}{m+11}+\frac{55 x^9}{m+10}+\frac{165 x^8}{m+9}+\frac{330 x^7}{m+8}+\frac{462 x^6}{m+7}+\frac{462 x^5}{m+6}+\frac{330 x^4}{m+5}+\frac{165 x^3}{m+4}+\frac{55 x^2}{m+3}+\frac{11 x}{m+2}+\frac{1}{m+1}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^m*(1 + x)*(1 + 2*x + x^2)^5,x]
[Out]
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))
________________________________________________________________________________________
Maple [B] time = 0.007, size = 1096, normalized size = 7.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(1+x)*(x^2+2*x+1)^5,x)
[Out]
x^(1+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+21750465*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+156657490*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+83986050
5*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+216
148086*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+4125*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+1800387072*m^2*x^10+120543840*m*x^11+m^11+836*m^10*x+137060*m^9*x^2+7
604190*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+14122
57770*m^6*x^4+9852674370*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^2+769916070*m^6*x^3+7785487380*m^5*x^4+36088363080*m^4*x^5
+77023113552*m^3*x^6+72321091920*m^2*x^7+26275708800*m*x^8+2634508800*x^9+53130*m^8+7130013*m^7*x+281209005*m^
6*x^2+4343723835*m^5*x^3+29075712600*m^4*x^4+87099379752*m^3*x^5+114113083392*m^2*x^6+58845916800*m*x^7+878169
6000*x^8+696333*m^7+61932948*m^6*x+1630835690*m^5*x^2+16626679410*m^4*x^3+71499692880*m^3*x^4+130678599744*m^2
*x^5+93588929280*m*x^6+19758816000*x^7+6230301*m^6+371026645*m^5*x+6441351180*m^4*x^2+41932410300*m^3*x^3+1091
26448640*m^2*x^4+108308914560*m*x^5+31614105600*x^6+38759930*m^5+1524718360*m^4*x+16822322440*m^3*x^2+65582815
320*m^2*x^3+91782408960*m*x^4+36883123200*x^5+167310220*m^4+4179838476*m^3*x+27303851520*m^2*x^2+56376064800*m
*x^3+31614105600*x^4+489896616*m^3+7194486816*m^2*x+24324220800*m*x^2+19758816000*x^3+924118272*m^2+6858181440
*m*x+8781696000*x^2+1007441280*m+2634508800*x+479001600)/(12+m)/(11+m)/(10+m)/(9+m)/(8+m)/(7+m)/(6+m)/(5+m)/(4
+m)/(3+m)/(2+m)/(1+m)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.35167, size = 2898, normalized size = 20.27 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")
[Out]
((m^11 + 66*m^10 + 1925*m^9 + 32670*m^8 + 357423*m^7 + 2637558*m^6 + 13339535*m^5 + 45995730*m^4 + 105258076*m
^3 + 150917976*m^2 + 120543840*m + 39916800)*x^12 + 11*(m^11 + 67*m^10 + 1980*m^9 + 33990*m^8 + 375573*m^7 + 2
795331*m^6 + 14241590*m^5 + 49412660*m^4 + 113667576*m^3 + 163671552*m^2 + 131172480*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
+ 178770528*m^2 + 143854272*m + 47900160)*x^10 + 165*(m^11 + 69*m^10 + 2096*m^9 + 36906*m^8 + 417309*m^7 + 31
70853*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 + 3651
663*m^6 + 19428590*m^5 + 70070020*m^4 + 166716696*m^3 + 246998016*m^2 + 202573440*m + 68428800)*x^7 + 462*(m^1
1 + 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 + 427956
9*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 + 3
97471608*m^2 + 341673120*m + 119750400)*x^4 + 55*(m^11 + 75*m^10 + 2492*m^9 + 48294*m^8 + 604581*m^7 + 5112891
*m^6 + 29651558*m^5 + 117115476*m^4 + 305860408*m^3 + 496433664*m^2 + 442258560*m + 159667200)*x^3 + 11*(m^11
+ 76*m^10 + 2565*m^9 + 50640*m^8 + 648183*m^7 + 5630268*m^6 + 33729695*m^5 + 138610760*m^4 + 379985316*m^3 + 6
54044256*m^2 + 623471040*m + 239500800)*x^2 + (m^11 + 77*m^10 + 2640*m^9 + 53130*m^8 + 696333*m^7 + 6230301*m^
6 + 38759930*m^5 + 167310220*m^4 + 489896616*m^3 + 924118272*m^2 + 1007441280*m + 479001600)*x)*x^m/(m^12 + 78
*m^11 + 2717*m^10 + 55770*m^9 + 749463*m^8 + 6926634*m^7 + 44990231*m^6 + 206070150*m^5 + 657206836*m^4 + 1414
014888*m^3 + 1931559552*m^2 + 1486442880*m + 479001600)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m*(1+x)*(x**2+2*x+1)**5,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.37412, size = 2106, normalized size = 14.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")
[Out]
(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^1
0*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 + 21750465*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 + 1
65*m^11*x^4*x^m + 24090*m^10*x^5*x^m + 1055670*m^9*x^6*x^m + 18586260*m^8*x^7*x^m + 145645830*m^7*x^8*x^m + 52
3190745*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 + 2714671410*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 + 168
7068306*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 + 836*m^10*x^2*x^m + 137060*m^9*x^3*x^m + 7604190*m^8*x^4*x^m
+ 174706290*m^7*x^5*x^m + 1822135392*m^6*x^6*x^m + 8976008580*m^5*x^7*x^m + 20948784780*m^4*x^8*x^m + 22313339
400*m^3*x^9*x^m + 9832379040*m^2*x^10*x^m + 1442897280*m*x^11*x^m + 39916800*x^12*x^m + 77*m^10*x*x^m + 28215*
m^9*x^2*x^m + 2656170*m^8*x^3*x^m + 93244635*m^7*x^4*x^m + 1412257770*m^6*x^5*x^m + 9852674370*m^5*x^6*x^m + 3
2372349240*m^4*x^7*x^m + 49287977640*m^3*x^8*x^m + 32492401920*m^2*x^9*x^m + 7911984960*m*x^10*x^m + 479001600
*x^11*x^m + 2640*m^9*x*x^m + 557040*m^8*x^2*x^m + 33251955*m^7*x^3*x^m + 769916070*m^6*x^4*x^m + 7785487380*m^
5*x^5*x^m + 36088363080*m^4*x^6*x^m + 77023113552*m^3*x^7*x^m + 72321091920*m^2*x^8*x^m + 26275708800*m*x^9*x^
m + 2634508800*x^10*x^m + 53130*m^8*x*x^m + 7130013*m^7*x^2*x^m + 281209005*m^6*x^3*x^m + 4343723835*m^5*x^4*x
^m + 29075712600*m^4*x^5*x^m + 87099379752*m^3*x^6*x^m + 114113083392*m^2*x^7*x^m + 58845916800*m*x^8*x^m + 87
81696000*x^9*x^m + 696333*m^7*x*x^m + 61932948*m^6*x^2*x^m + 1630835690*m^5*x^3*x^m + 16626679410*m^4*x^4*x^m
+ 71499692880*m^3*x^5*x^m + 130678599744*m^2*x^6*x^m + 93588929280*m*x^7*x^m + 19758816000*x^8*x^m + 6230301*m
^6*x*x^m + 371026645*m^5*x^2*x^m + 6441351180*m^4*x^3*x^m + 41932410300*m^3*x^4*x^m + 109126448640*m^2*x^5*x^m
+ 108308914560*m*x^6*x^m + 31614105600*x^7*x^m + 38759930*m^5*x*x^m + 1524718360*m^4*x^2*x^m + 16822322440*m^
3*x^3*x^m + 65582815320*m^2*x^4*x^m + 91782408960*m*x^5*x^m + 36883123200*x^6*x^m + 167310220*m^4*x*x^m + 4179
838476*m^3*x^2*x^m + 27303851520*m^2*x^3*x^m + 56376064800*m*x^4*x^m + 31614105600*x^5*x^m + 489896616*m^3*x*x
^m + 7194486816*m^2*x^2*x^m + 24324220800*m*x^3*x^m + 19758816000*x^4*x^m + 924118272*m^2*x*x^m + 6858181440*m
*x^2*x^m + 8781696000*x^3*x^m + 1007441280*m*x*x^m + 2634508800*x^2*x^m + 479001600*x*x^m)/(m^12 + 78*m^11 + 2
717*m^10 + 55770*m^9 + 749463*m^8 + 6926634*m^7 + 44990231*m^6 + 206070150*m^5 + 657206836*m^4 + 1414014888*m^
3 + 1931559552*m^2 + 1486442880*m + 479001600)