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

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 = 0.0264048, 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, Rules used = {27, 43} \[ \frac{1}{21} (x+1)^{21}-\frac{9}{20} (x+1)^{20}+\frac{36}{19} (x+1)^{19}-\frac{14}{3} (x+1)^{18}+\frac{126}{17} (x+1)^{17}-\frac{63}{8} (x+1)^{16}+\frac{28}{5} (x+1)^{15}-\frac{18}{7} (x+1)^{14}+\frac{9}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

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

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^9 (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^9 (1+x)^{11} \, dx\\ &=\int \left (-(1+x)^{11}+9 (1+x)^{12}-36 (1+x)^{13}+84 (1+x)^{14}-126 (1+x)^{15}+126 (1+x)^{16}-84 (1+x)^{17}+36 (1+x)^{18}-9 (1+x)^{19}+(1+x)^{20}\right ) \, dx\\ &=-\frac{1}{12} (1+x)^{12}+\frac{9}{13} (1+x)^{13}-\frac{18}{7} (1+x)^{14}+\frac{28}{5} (1+x)^{15}-\frac{63}{8} (1+x)^{16}+\frac{126}{17} (1+x)^{17}-\frac{14}{3} (1+x)^{18}+\frac{36}{19} (1+x)^{19}-\frac{9}{20} (1+x)^{20}+\frac{1}{21} (1+x)^{21}\\ \end{align*}

Mathematica [A]  time = 0.0016308, 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 + (231*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 = 0.001, size = 60, normalized size = 0.7 \begin{align*}{\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}} \end{align*}

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 = 0.958375, size = 80, normalized size = 0.88 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(1+x)*(x^2+2*x+1)^5,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 = 1.05527, size = 200, normalized size = 2.2 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(1+x)*(x^2+2*x+1)^5,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 = 0.08129, size = 73, normalized size = 0.8 \begin{align*} \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} \end{align*}

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 + 1
65*x**13/13 + 55*x**12/12 + x**11 + x**10/10

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Giac [A]  time = 1.15265, size = 80, normalized size = 0.88 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(1+x)*(x^2+2*x+1)^5,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