3.591 \(\int x^8 (1+x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=80 \[ \frac{1}{20} (x+1)^{20}-\frac{8}{19} (x+1)^{19}+\frac{14}{9} (x+1)^{18}-\frac{56}{17} (x+1)^{17}+\frac{35}{8} (x+1)^{16}-\frac{56}{15} (x+1)^{15}+2 (x+1)^{14}-\frac{8}{13} (x+1)^{13}+\frac{1}{12} (x+1)^{12} \]

[Out]

(1 + x)^12/12 - (8*(1 + x)^13)/13 + 2*(1 + x)^14 - (56*(1 + x)^15)/15 + (35*(1 + x)^16)/8 - (56*(1 + x)^17)/17
+ (14*(1 + x)^18)/9 - (8*(1 + x)^19)/19 + (1 + x)^20/20

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Rubi [A]  time = 0.0249253, antiderivative size = 80, 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} \[ \frac{1}{20} (x+1)^{20}-\frac{8}{19} (x+1)^{19}+\frac{14}{9} (x+1)^{18}-\frac{56}{17} (x+1)^{17}+\frac{35}{8} (x+1)^{16}-\frac{56}{15} (x+1)^{15}+2 (x+1)^{14}-\frac{8}{13} (x+1)^{13}+\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

(1 + x)^12/12 - (8*(1 + x)^13)/13 + 2*(1 + x)^14 - (56*(1 + x)^15)/15 + (35*(1 + x)^16)/8 - (56*(1 + x)^17)/17
+ (14*(1 + x)^18)/9 - (8*(1 + x)^19)/19 + (1 + x)^20/20

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^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^8 (1+x)^{11} \, dx\\ &=\int \left ((1+x)^{11}-8 (1+x)^{12}+28 (1+x)^{13}-56 (1+x)^{14}+70 (1+x)^{15}-56 (1+x)^{16}+28 (1+x)^{17}-8 (1+x)^{18}+(1+x)^{19}\right ) \, dx\\ &=\frac{1}{12} (1+x)^{12}-\frac{8}{13} (1+x)^{13}+2 (1+x)^{14}-\frac{56}{15} (1+x)^{15}+\frac{35}{8} (1+x)^{16}-\frac{56}{17} (1+x)^{17}+\frac{14}{9} (1+x)^{18}-\frac{8}{19} (1+x)^{19}+\frac{1}{20} (1+x)^{20}\\ \end{align*}

Mathematica [A]  time = 0.0015597, size = 81, normalized size = 1.01 \[ \frac{x^{20}}{20}+\frac{11 x^{19}}{19}+\frac{55 x^{18}}{18}+\frac{165 x^{17}}{17}+\frac{165 x^{16}}{8}+\frac{154 x^{15}}{5}+33 x^{14}+\frac{330 x^{13}}{13}+\frac{55 x^{12}}{4}+5 x^{11}+\frac{11 x^{10}}{10}+\frac{x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^9/9 + (11*x^10)/10 + 5*x^11 + (55*x^12)/4 + (330*x^13)/13 + 33*x^14 + (154*x^15)/5 + (165*x^16)/8 + (165*x^1
7)/17 + (55*x^18)/18 + (11*x^19)/19 + x^20/20

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Maple [A]  time = 0.001, size = 62, normalized size = 0.8 \begin{align*}{\frac{{x}^{20}}{20}}+{\frac{11\,{x}^{19}}{19}}+{\frac{55\,{x}^{18}}{18}}+{\frac{165\,{x}^{17}}{17}}+{\frac{165\,{x}^{16}}{8}}+{\frac{154\,{x}^{15}}{5}}+33\,{x}^{14}+{\frac{330\,{x}^{13}}{13}}+{\frac{55\,{x}^{12}}{4}}+5\,{x}^{11}+{\frac{11\,{x}^{10}}{10}}+{\frac{{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(1+x)*(x^2+2*x+1)^5,x)

[Out]

1/20*x^20+11/19*x^19+55/18*x^18+165/17*x^17+165/8*x^16+154/5*x^15+33*x^14+330/13*x^13+55/4*x^12+5*x^11+11/10*x
^10+1/9*x^9

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Maxima [A]  time = 1.02427, size = 82, normalized size = 1.02 \begin{align*} \frac{1}{20} \, x^{20} + \frac{11}{19} \, x^{19} + \frac{55}{18} \, x^{18} + \frac{165}{17} \, x^{17} + \frac{165}{8} \, x^{16} + \frac{154}{5} \, x^{15} + 33 \, x^{14} + \frac{330}{13} \, x^{13} + \frac{55}{4} \, x^{12} + 5 \, x^{11} + \frac{11}{10} \, x^{10} + \frac{1}{9} \, x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/20*x^20 + 11/19*x^19 + 55/18*x^18 + 165/17*x^17 + 165/8*x^16 + 154/5*x^15 + 33*x^14 + 330/13*x^13 + 55/4*x^1
2 + 5*x^11 + 11/10*x^10 + 1/9*x^9

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Fricas [A]  time = 1.10644, size = 196, normalized size = 2.45 \begin{align*} \frac{1}{20} x^{20} + \frac{11}{19} x^{19} + \frac{55}{18} x^{18} + \frac{165}{17} x^{17} + \frac{165}{8} x^{16} + \frac{154}{5} x^{15} + 33 x^{14} + \frac{330}{13} x^{13} + \frac{55}{4} x^{12} + 5 x^{11} + \frac{11}{10} x^{10} + \frac{1}{9} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/20*x^20 + 11/19*x^19 + 55/18*x^18 + 165/17*x^17 + 165/8*x^16 + 154/5*x^15 + 33*x^14 + 330/13*x^13 + 55/4*x^1
2 + 5*x^11 + 11/10*x^10 + 1/9*x^9

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Sympy [A]  time = 0.074646, size = 73, normalized size = 0.91 \begin{align*} \frac{x^{20}}{20} + \frac{11 x^{19}}{19} + \frac{55 x^{18}}{18} + \frac{165 x^{17}}{17} + \frac{165 x^{16}}{8} + \frac{154 x^{15}}{5} + 33 x^{14} + \frac{330 x^{13}}{13} + \frac{55 x^{12}}{4} + 5 x^{11} + \frac{11 x^{10}}{10} + \frac{x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**20/20 + 11*x**19/19 + 55*x**18/18 + 165*x**17/17 + 165*x**16/8 + 154*x**15/5 + 33*x**14 + 330*x**13/13 + 55
*x**12/4 + 5*x**11 + 11*x**10/10 + x**9/9

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Giac [A]  time = 1.14219, size = 82, normalized size = 1.02 \begin{align*} \frac{1}{20} \, x^{20} + \frac{11}{19} \, x^{19} + \frac{55}{18} \, x^{18} + \frac{165}{17} \, x^{17} + \frac{165}{8} \, x^{16} + \frac{154}{5} \, x^{15} + 33 \, x^{14} + \frac{330}{13} \, x^{13} + \frac{55}{4} \, x^{12} + 5 \, x^{11} + \frac{11}{10} \, x^{10} + \frac{1}{9} \, x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/20*x^20 + 11/19*x^19 + 55/18*x^18 + 165/17*x^17 + 165/8*x^16 + 154/5*x^15 + 33*x^14 + 330/13*x^13 + 55/4*x^1
2 + 5*x^11 + 11/10*x^10 + 1/9*x^9