3.590 \(\int x^9 (1+x) \left (1+2 x+x^2\right )^5 \, dx\)

Optimal. Leaf size=91 \[ \frac{1}{21} (x+1)^{21}-\frac{9}{20} (x+1)^{20}+\frac{36}{19} (x+1)^{19}-\frac{14}{3} (x+1)^{18}+\frac{126}{17} (x+1)^{17}-\frac{63}{8} (x+1)^{16}+\frac{28}{5} (x+1)^{15}-\frac{18}{7} (x+1)^{14}+\frac{9}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

[Out]

-(1 + x)^12/12 + (9*(1 + x)^13)/13 - (18*(1 + x)^14)/7 + (28*(1 + x)^15)/5 - (63
*(1 + x)^16)/8 + (126*(1 + x)^17)/17 - (14*(1 + x)^18)/3 + (36*(1 + x)^19)/19 -
(9*(1 + x)^20)/20 + (1 + x)^21/21

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Rubi [A]  time = 0.0657079, antiderivative size = 91, 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 \[ \frac{1}{21} (x+1)^{21}-\frac{9}{20} (x+1)^{20}+\frac{36}{19} (x+1)^{19}-\frac{14}{3} (x+1)^{18}+\frac{126}{17} (x+1)^{17}-\frac{63}{8} (x+1)^{16}+\frac{28}{5} (x+1)^{15}-\frac{18}{7} (x+1)^{14}+\frac{9}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]  Int[x^9*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

-(1 + x)^12/12 + (9*(1 + x)^13)/13 - (18*(1 + x)^14)/7 + (28*(1 + x)^15)/5 - (63
*(1 + x)^16)/8 + (126*(1 + x)^17)/17 - (14*(1 + x)^18)/3 + (36*(1 + x)^19)/19 -
(9*(1 + x)^20)/20 + (1 + x)^21/21

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Rubi in Sympy [A]  time = 12.8985, size = 73, normalized size = 0.8 \[ \frac{x^{21}}{21} + \frac{11 x^{20}}{20} + \frac{55 x^{19}}{19} + \frac{55 x^{18}}{6} + \frac{330 x^{17}}{17} + \frac{231 x^{16}}{8} + \frac{154 x^{15}}{5} + \frac{165 x^{14}}{7} + \frac{165 x^{13}}{13} + \frac{55 x^{12}}{12} + x^{11} + \frac{x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**21/21 + 11*x**20/20 + 55*x**19/19 + 55*x**18/6 + 330*x**17/17 + 231*x**16/8 +
154*x**15/5 + 165*x**14/7 + 165*x**13/13 + 55*x**12/12 + x**11 + x**10/10

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Mathematica [A]  time = 0.00276177, size = 81, normalized size = 0.89 \[ \frac{x^{21}}{21}+\frac{11 x^{20}}{20}+\frac{55 x^{19}}{19}+\frac{55 x^{18}}{6}+\frac{330 x^{17}}{17}+\frac{231 x^{16}}{8}+\frac{154 x^{15}}{5}+\frac{165 x^{14}}{7}+\frac{165 x^{13}}{13}+\frac{55 x^{12}}{12}+x^{11}+\frac{x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^10/10 + x^11 + (55*x^12)/12 + (165*x^13)/13 + (165*x^14)/7 + (154*x^15)/5 + (2
31*x^16)/8 + (330*x^17)/17 + (55*x^18)/6 + (55*x^19)/19 + (11*x^20)/20 + x^21/21

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Maple [A]  time = 0.002, size = 60, normalized size = 0.7 \[{\frac{{x}^{21}}{21}}+{\frac{11\,{x}^{20}}{20}}+{\frac{55\,{x}^{19}}{19}}+{\frac{55\,{x}^{18}}{6}}+{\frac{330\,{x}^{17}}{17}}+{\frac{231\,{x}^{16}}{8}}+{\frac{154\,{x}^{15}}{5}}+{\frac{165\,{x}^{14}}{7}}+{\frac{165\,{x}^{13}}{13}}+{\frac{55\,{x}^{12}}{12}}+{x}^{11}+{\frac{{x}^{10}}{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(1+x)*(x^2+2*x+1)^5,x)

[Out]

1/21*x^21+11/20*x^20+55/19*x^19+55/6*x^18+330/17*x^17+231/8*x^16+154/5*x^15+165/
7*x^14+165/13*x^13+55/12*x^12+x^11+1/10*x^10

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Maxima [A]  time = 0.686104, size = 80, normalized size = 0.88 \[ \frac{1}{21} \, x^{21} + \frac{11}{20} \, x^{20} + \frac{55}{19} \, x^{19} + \frac{55}{6} \, x^{18} + \frac{330}{17} \, x^{17} + \frac{231}{8} \, x^{16} + \frac{154}{5} \, x^{15} + \frac{165}{7} \, x^{14} + \frac{165}{13} \, x^{13} + \frac{55}{12} \, x^{12} + x^{11} + \frac{1}{10} \, x^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^9,x, algorithm="maxima")

[Out]

1/21*x^21 + 11/20*x^20 + 55/19*x^19 + 55/6*x^18 + 330/17*x^17 + 231/8*x^16 + 154
/5*x^15 + 165/7*x^14 + 165/13*x^13 + 55/12*x^12 + x^11 + 1/10*x^10

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Fricas [A]  time = 0.246148, size = 1, normalized size = 0.01 \[ \frac{1}{21} x^{21} + \frac{11}{20} x^{20} + \frac{55}{19} x^{19} + \frac{55}{6} x^{18} + \frac{330}{17} x^{17} + \frac{231}{8} x^{16} + \frac{154}{5} x^{15} + \frac{165}{7} x^{14} + \frac{165}{13} x^{13} + \frac{55}{12} x^{12} + x^{11} + \frac{1}{10} x^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^9,x, algorithm="fricas")

[Out]

1/21*x^21 + 11/20*x^20 + 55/19*x^19 + 55/6*x^18 + 330/17*x^17 + 231/8*x^16 + 154
/5*x^15 + 165/7*x^14 + 165/13*x^13 + 55/12*x^12 + x^11 + 1/10*x^10

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Sympy [A]  time = 0.110547, size = 73, normalized size = 0.8 \[ \frac{x^{21}}{21} + \frac{11 x^{20}}{20} + \frac{55 x^{19}}{19} + \frac{55 x^{18}}{6} + \frac{330 x^{17}}{17} + \frac{231 x^{16}}{8} + \frac{154 x^{15}}{5} + \frac{165 x^{14}}{7} + \frac{165 x^{13}}{13} + \frac{55 x^{12}}{12} + x^{11} + \frac{x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**21/21 + 11*x**20/20 + 55*x**19/19 + 55*x**18/6 + 330*x**17/17 + 231*x**16/8 +
154*x**15/5 + 165*x**14/7 + 165*x**13/13 + 55*x**12/12 + x**11 + x**10/10

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GIAC/XCAS [A]  time = 0.271894, size = 80, normalized size = 0.88 \[ \frac{1}{21} \, x^{21} + \frac{11}{20} \, x^{20} + \frac{55}{19} \, x^{19} + \frac{55}{6} \, x^{18} + \frac{330}{17} \, x^{17} + \frac{231}{8} \, x^{16} + \frac{154}{5} \, x^{15} + \frac{165}{7} \, x^{14} + \frac{165}{13} \, x^{13} + \frac{55}{12} \, x^{12} + x^{11} + \frac{1}{10} \, x^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^9,x, algorithm="giac")

[Out]

1/21*x^21 + 11/20*x^20 + 55/19*x^19 + 55/6*x^18 + 330/17*x^17 + 231/8*x^16 + 154
/5*x^15 + 165/7*x^14 + 165/13*x^13 + 55/12*x^12 + x^11 + 1/10*x^10