3.841 \(\int x^m (1+x) \left (1+2 x+x^2\right )^5 \, dx\)
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 = 0.0941608, 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
\[ \frac{x^{m+1}}{m+1}+\frac{11 x^{m+2}}{m+2}+\frac{55 x^{m+3}}{m+3}+\frac{165 x^{m+4}}{m+4}+\frac{330 x^{m+5}}{m+5}+\frac{462 x^{m+6}}{m+6}+\frac{462 x^{m+7}}{m+7}+\frac{330 x^{m+8}}{m+8}+\frac{165 x^{m+9}}{m+9}+\frac{55 x^{m+10}}{m+10}+\frac{11 x^{m+11}}{m+11}+\frac{x^{m+12}}{m+12} \]
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)
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Rubi in Sympy [A] time = 22.1115, size = 117, normalized size = 0.82 \[ \frac{x^{m + 1}}{m + 1} + \frac{11 x^{m + 2}}{m + 2} + \frac{55 x^{m + 3}}{m + 3} + \frac{165 x^{m + 4}}{m + 4} + \frac{330 x^{m + 5}}{m + 5} + \frac{462 x^{m + 6}}{m + 6} + \frac{462 x^{m + 7}}{m + 7} + \frac{330 x^{m + 8}}{m + 8} + \frac{165 x^{m + 9}}{m + 9} + \frac{55 x^{m + 10}}{m + 10} + \frac{11 x^{m + 11}}{m + 11} + \frac{x^{m + 12}}{m + 12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(1+x)*(x**2+2*x+1)**5,x)
[Out]
x**(m + 1)/(m + 1) + 11*x**(m + 2)/(m + 2) + 55*x**(m + 3)/(m + 3) + 165*x**(m +
4)/(m + 4) + 330*x**(m + 5)/(m + 5) + 462*x**(m + 6)/(m + 6) + 462*x**(m + 7)/(
m + 7) + 330*x**(m + 8)/(m + 8) + 165*x**(m + 9)/(m + 9) + 55*x**(m + 10)/(m + 1
0) + 11*x**(m + 11)/(m + 11) + x**(m + 12)/(m + 12)
_______________________________________________________________________________________
Mathematica [B] time = 0.407097, size = 357, normalized size = 2.5 \[ -\frac{x^m \left (-(m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (m+8) (m+9) (m+10) (m+11) (x+1)^{12}+m (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (m+8) (m+9) (m+10) (x+1)^{11}+11 m (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (m+8) (m+9) (x+1)^{10}+110 m (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (m+8) (x+1)^9+990 m (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (x+1)^8+7920 m (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (x+1)^7+55440 m (m+1) (m+2) (m+3) (m+4) (m+5) (x+1)^6+332640 m (m+1) (m+2) (m+3) (m+4) (x+1)^5+1663200 m (m+1) (m+2) (m+3) (x+1)^4+6652800 m (m+1) (m+2) (x+1)^3+19958400 m (m+1) (x+1)^2+39916800 m (x+1)+39916800\right )}{(m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (m+8) (m+9) (m+10) (m+11) (m+12)} \]
Antiderivative was successfully verified.
[In] Integrate[x^m*(1 + x)*(1 + 2*x + x^2)^5,x]
[Out]
-((x^m*(39916800 + 39916800*m*(1 + x) + 19958400*m*(1 + m)*(1 + x)^2 + 6652800*m
*(1 + m)*(2 + m)*(1 + x)^3 + 1663200*m*(1 + m)*(2 + m)*(3 + m)*(1 + x)^4 + 33264
0*m*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(1 + x)^5 + 55440*m*(1 + m)*(2 + m)*(3 + m)*
(4 + m)*(5 + m)*(1 + x)^6 + 7920*m*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 +
m)*(1 + x)^7 + 990*m*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(1
+ x)^8 + 110*m*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)*(
1 + x)^9 + 11*m*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)*
(9 + m)*(1 + x)^10 + m*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(
8 + m)*(9 + m)*(10 + m)*(1 + x)^11 - (1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6
+ m)*(7 + m)*(8 + m)*(9 + m)*(10 + m)*(11 + m)*(1 + x)^12))/((1 + m)*(2 + m)*(3
+ m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)*(9 + m)*(10 + m)*(11 + m)*(12 + m))
)
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Maple [B] time = 0.011, size = 1096, normalized size = 7.7 \[ \text{result too large to display} \]
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+523190
745*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+4125*m^10*x^2+399465*m^9*x^3+14523300*m^8*x^4+229661586*m^7*x^5+16870
68306*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+17470
6290*m^7*x^4+1822135392*m^6*x^5+8976008580*m^5*x^6+20948784780*m^4*x^7+223133394
00*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+9852674370*m^5*x^5+3237234
9240*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+2
634508800*x^9+53130*m^8+7130013*m^7*x+281209005*m^6*x^2+4343723835*m^5*x^3+29075
712600*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+714996
92880*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+109126448640*m^2*x^4+108
308914560*m*x^5+31614105600*x^6+38759930*m^5+1524718360*m^4*x+16822322440*m^3*x^
2+65582815320*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+71944
86816*m^2*x+24324220800*m*x^2+19758816000*x^3+924118272*m^2+6858181440*m*x+87816
96000*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] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 2*x + 1)^5*(x + 1)*x^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.290219, size = 1022, normalized size = 7.15 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 2*x + 1)^5*(x + 1)*x^m,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 + 2795331*m^6 + 1424159
0*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 + 152
70191*m^5 + 53368240*m^4 + 123524436*m^3 + 178770528*m^2 + 143854272*m + 4790016
0)*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 + 53
222400)*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 + 20257344
0*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 + 2344
34880*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 + 56511
9*m^7 + 4666158*m^6 + 26325599*m^5 + 100767754*m^4 + 254135820*m^3 + 397471608*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 + 496433
664*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 + 1414014888*m^3 + 1931559552*m^2 + 1486442880*m + 479001600)
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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/XCAS [A] time = 0.277181, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 + 2*x + 1)^5*(x + 1)*x^m,x, algorithm="giac")
[Out]
Done