3.6.16 \(\int x^{10} (1+x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=83 \[ \frac {x^{22}}{22}+\frac {11 x^{21}}{21}+\frac {11 x^{20}}{4}+\frac {165 x^{19}}{19}+\frac {55 x^{18}}{3}+\frac {462 x^{17}}{17}+\frac {231 x^{16}}{8}+22 x^{15}+\frac {165 x^{14}}{14}+\frac {55 x^{13}}{13}+\frac {11 x^{12}}{12}+\frac {x^{11}}{11} \]

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Rubi [A]  time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^{22}}{22}+\frac {11 x^{21}}{21}+\frac {11 x^{20}}{4}+\frac {165 x^{19}}{19}+\frac {55 x^{18}}{3}+\frac {462 x^{17}}{17}+\frac {231 x^{16}}{8}+22 x^{15}+\frac {165 x^{14}}{14}+\frac {55 x^{13}}{13}+\frac {11 x^{12}}{12}+\frac {x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^10*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^11/11 + (11*x^12)/12 + (55*x^13)/13 + (165*x^14)/14 + 22*x^15 + (231*x^16)/8 + (462*x^17)/17 + (55*x^18)/3 +
(165*x^19)/19 + (11*x^20)/4 + (11*x^21)/21 + x^22/22

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^{10} (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^{10} (1+x)^{11} \, dx\\ &=\int \left (x^{10}+11 x^{11}+55 x^{12}+165 x^{13}+330 x^{14}+462 x^{15}+462 x^{16}+330 x^{17}+165 x^{18}+55 x^{19}+11 x^{20}+x^{21}\right ) \, dx\\ &=\frac {x^{11}}{11}+\frac {11 x^{12}}{12}+\frac {55 x^{13}}{13}+\frac {165 x^{14}}{14}+22 x^{15}+\frac {231 x^{16}}{8}+\frac {462 x^{17}}{17}+\frac {55 x^{18}}{3}+\frac {165 x^{19}}{19}+\frac {11 x^{20}}{4}+\frac {11 x^{21}}{21}+\frac {x^{22}}{22}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 83, normalized size = 1.00 \begin {gather*} \frac {x^{22}}{22}+\frac {11 x^{21}}{21}+\frac {11 x^{20}}{4}+\frac {165 x^{19}}{19}+\frac {55 x^{18}}{3}+\frac {462 x^{17}}{17}+\frac {231 x^{16}}{8}+22 x^{15}+\frac {165 x^{14}}{14}+\frac {55 x^{13}}{13}+\frac {11 x^{12}}{12}+\frac {x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^10*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^11/11 + (11*x^12)/12 + (55*x^13)/13 + (165*x^14)/14 + 22*x^15 + (231*x^16)/8 + (462*x^17)/17 + (55*x^18)/3 +
(165*x^19)/19 + (11*x^20)/4 + (11*x^21)/21 + x^22/22

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{10} (1+x) \left (1+2 x+x^2\right )^5 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^10*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

IntegrateAlgebraic[x^10*(1 + x)*(1 + 2*x + x^2)^5, x]

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fricas [A]  time = 0.36, size = 61, normalized size = 0.73 \begin {gather*} \frac {1}{22} x^{22} + \frac {11}{21} x^{21} + \frac {11}{4} x^{20} + \frac {165}{19} x^{19} + \frac {55}{3} x^{18} + \frac {462}{17} x^{17} + \frac {231}{8} x^{16} + 22 x^{15} + \frac {165}{14} x^{14} + \frac {55}{13} x^{13} + \frac {11}{12} x^{12} + \frac {1}{11} x^{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/22*x^22 + 11/21*x^21 + 11/4*x^20 + 165/19*x^19 + 55/3*x^18 + 462/17*x^17 + 231/8*x^16 + 22*x^15 + 165/14*x^1
4 + 55/13*x^13 + 11/12*x^12 + 1/11*x^11

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giac [A]  time = 0.17, size = 61, normalized size = 0.73 \begin {gather*} \frac {1}{22} \, x^{22} + \frac {11}{21} \, x^{21} + \frac {11}{4} \, x^{20} + \frac {165}{19} \, x^{19} + \frac {55}{3} \, x^{18} + \frac {462}{17} \, x^{17} + \frac {231}{8} \, x^{16} + 22 \, x^{15} + \frac {165}{14} \, x^{14} + \frac {55}{13} \, x^{13} + \frac {11}{12} \, x^{12} + \frac {1}{11} \, x^{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/22*x^22 + 11/21*x^21 + 11/4*x^20 + 165/19*x^19 + 55/3*x^18 + 462/17*x^17 + 231/8*x^16 + 22*x^15 + 165/14*x^1
4 + 55/13*x^13 + 11/12*x^12 + 1/11*x^11

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maple [A]  time = 0.04, size = 62, normalized size = 0.75 \begin {gather*} \frac {1}{22} x^{22}+\frac {11}{21} x^{21}+\frac {11}{4} x^{20}+\frac {165}{19} x^{19}+\frac {55}{3} x^{18}+\frac {462}{17} x^{17}+\frac {231}{8} x^{16}+22 x^{15}+\frac {165}{14} x^{14}+\frac {55}{13} x^{13}+\frac {11}{12} x^{12}+\frac {1}{11} x^{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10*(x+1)*(x^2+2*x+1)^5,x)

[Out]

1/11*x^11+11/12*x^12+55/13*x^13+165/14*x^14+22*x^15+231/8*x^16+462/17*x^17+55/3*x^18+165/19*x^19+11/4*x^20+11/
21*x^21+1/22*x^22

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maxima [A]  time = 0.60, size = 61, normalized size = 0.73 \begin {gather*} \frac {1}{22} \, x^{22} + \frac {11}{21} \, x^{21} + \frac {11}{4} \, x^{20} + \frac {165}{19} \, x^{19} + \frac {55}{3} \, x^{18} + \frac {462}{17} \, x^{17} + \frac {231}{8} \, x^{16} + 22 \, x^{15} + \frac {165}{14} \, x^{14} + \frac {55}{13} \, x^{13} + \frac {11}{12} \, x^{12} + \frac {1}{11} \, x^{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/22*x^22 + 11/21*x^21 + 11/4*x^20 + 165/19*x^19 + 55/3*x^18 + 462/17*x^17 + 231/8*x^16 + 22*x^15 + 165/14*x^1
4 + 55/13*x^13 + 11/12*x^12 + 1/11*x^11

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mupad [B]  time = 0.06, size = 61, normalized size = 0.73 \begin {gather*} \frac {x^{22}}{22}+\frac {11\,x^{21}}{21}+\frac {11\,x^{20}}{4}+\frac {165\,x^{19}}{19}+\frac {55\,x^{18}}{3}+\frac {462\,x^{17}}{17}+\frac {231\,x^{16}}{8}+22\,x^{15}+\frac {165\,x^{14}}{14}+\frac {55\,x^{13}}{13}+\frac {11\,x^{12}}{12}+\frac {x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10*(x + 1)*(2*x + x^2 + 1)^5,x)

[Out]

x^11/11 + (11*x^12)/12 + (55*x^13)/13 + (165*x^14)/14 + 22*x^15 + (231*x^16)/8 + (462*x^17)/17 + (55*x^18)/3 +
(165*x^19)/19 + (11*x^20)/4 + (11*x^21)/21 + x^22/22

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sympy [A]  time = 0.07, size = 75, normalized size = 0.90 \begin {gather*} \frac {x^{22}}{22} + \frac {11 x^{21}}{21} + \frac {11 x^{20}}{4} + \frac {165 x^{19}}{19} + \frac {55 x^{18}}{3} + \frac {462 x^{17}}{17} + \frac {231 x^{16}}{8} + 22 x^{15} + \frac {165 x^{14}}{14} + \frac {55 x^{13}}{13} + \frac {11 x^{12}}{12} + \frac {x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**22/22 + 11*x**21/21 + 11*x**20/4 + 165*x**19/19 + 55*x**18/3 + 462*x**17/17 + 231*x**16/8 + 22*x**15 + 165*
x**14/14 + 55*x**13/13 + 11*x**12/12 + x**11/11

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