3.593 \(\int x^6 (1+x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=64 \[ \frac{1}{18} (x+1)^{18}-\frac{6}{17} (x+1)^{17}+\frac{15}{16} (x+1)^{16}-\frac{4}{3} (x+1)^{15}+\frac{15}{14} (x+1)^{14}-\frac{6}{13} (x+1)^{13}+\frac{1}{12} (x+1)^{12} \]

[Out]

(1 + x)^12/12 - (6*(1 + x)^13)/13 + (15*(1 + x)^14)/14 - (4*(1 + x)^15)/3 + (15*(1 + x)^16)/16 - (6*(1 + x)^17
)/17 + (1 + x)^18/18

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Rubi [A]  time = 0.0210965, antiderivative size = 64, 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} \[ \frac{1}{18} (x+1)^{18}-\frac{6}{17} (x+1)^{17}+\frac{15}{16} (x+1)^{16}-\frac{4}{3} (x+1)^{15}+\frac{15}{14} (x+1)^{14}-\frac{6}{13} (x+1)^{13}+\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

(1 + x)^12/12 - (6*(1 + x)^13)/13 + (15*(1 + x)^14)/14 - (4*(1 + x)^15)/3 + (15*(1 + x)^16)/16 - (6*(1 + x)^17
)/17 + (1 + x)^18/18

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^6 (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^6 (1+x)^{11} \, dx\\ &=\int \left ((1+x)^{11}-6 (1+x)^{12}+15 (1+x)^{13}-20 (1+x)^{14}+15 (1+x)^{15}-6 (1+x)^{16}+(1+x)^{17}\right ) \, dx\\ &=\frac{1}{12} (1+x)^{12}-\frac{6}{13} (1+x)^{13}+\frac{15}{14} (1+x)^{14}-\frac{4}{3} (1+x)^{15}+\frac{15}{16} (1+x)^{16}-\frac{6}{17} (1+x)^{17}+\frac{1}{18} (1+x)^{18}\\ \end{align*}

Mathematica [A]  time = 0.001677, size = 81, normalized size = 1.27 \[ \frac{x^{18}}{18}+\frac{11 x^{17}}{17}+\frac{55 x^{16}}{16}+11 x^{15}+\frac{165 x^{14}}{7}+\frac{462 x^{13}}{13}+\frac{77 x^{12}}{2}+30 x^{11}+\frac{33 x^{10}}{2}+\frac{55 x^9}{9}+\frac{11 x^8}{8}+\frac{x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^7/7 + (11*x^8)/8 + (55*x^9)/9 + (33*x^10)/2 + 30*x^11 + (77*x^12)/2 + (462*x^13)/13 + (165*x^14)/7 + 11*x^15
+ (55*x^16)/16 + (11*x^17)/17 + x^18/18

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Maple [A]  time = 0., size = 62, normalized size = 1. \begin{align*}{\frac{{x}^{18}}{18}}+{\frac{11\,{x}^{17}}{17}}+{\frac{55\,{x}^{16}}{16}}+11\,{x}^{15}+{\frac{165\,{x}^{14}}{7}}+{\frac{462\,{x}^{13}}{13}}+{\frac{77\,{x}^{12}}{2}}+30\,{x}^{11}+{\frac{33\,{x}^{10}}{2}}+{\frac{55\,{x}^{9}}{9}}+{\frac{11\,{x}^{8}}{8}}+{\frac{{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(1+x)*(x^2+2*x+1)^5,x)

[Out]

1/18*x^18+11/17*x^17+55/16*x^16+11*x^15+165/7*x^14+462/13*x^13+77/2*x^12+30*x^11+33/2*x^10+55/9*x^9+11/8*x^8+1
/7*x^7

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Maxima [A]  time = 0.985321, size = 82, normalized size = 1.28 \begin{align*} \frac{1}{18} \, x^{18} + \frac{11}{17} \, x^{17} + \frac{55}{16} \, x^{16} + 11 \, x^{15} + \frac{165}{7} \, x^{14} + \frac{462}{13} \, x^{13} + \frac{77}{2} \, x^{12} + 30 \, x^{11} + \frac{33}{2} \, x^{10} + \frac{55}{9} \, x^{9} + \frac{11}{8} \, x^{8} + \frac{1}{7} \, x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/18*x^18 + 11/17*x^17 + 55/16*x^16 + 11*x^15 + 165/7*x^14 + 462/13*x^13 + 77/2*x^12 + 30*x^11 + 33/2*x^10 + 5
5/9*x^9 + 11/8*x^8 + 1/7*x^7

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Fricas [A]  time = 1.07157, size = 189, normalized size = 2.95 \begin{align*} \frac{1}{18} x^{18} + \frac{11}{17} x^{17} + \frac{55}{16} x^{16} + 11 x^{15} + \frac{165}{7} x^{14} + \frac{462}{13} x^{13} + \frac{77}{2} x^{12} + 30 x^{11} + \frac{33}{2} x^{10} + \frac{55}{9} x^{9} + \frac{11}{8} x^{8} + \frac{1}{7} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/18*x^18 + 11/17*x^17 + 55/16*x^16 + 11*x^15 + 165/7*x^14 + 462/13*x^13 + 77/2*x^12 + 30*x^11 + 33/2*x^10 + 5
5/9*x^9 + 11/8*x^8 + 1/7*x^7

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Sympy [A]  time = 0.064727, size = 73, normalized size = 1.14 \begin{align*} \frac{x^{18}}{18} + \frac{11 x^{17}}{17} + \frac{55 x^{16}}{16} + 11 x^{15} + \frac{165 x^{14}}{7} + \frac{462 x^{13}}{13} + \frac{77 x^{12}}{2} + 30 x^{11} + \frac{33 x^{10}}{2} + \frac{55 x^{9}}{9} + \frac{11 x^{8}}{8} + \frac{x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**18/18 + 11*x**17/17 + 55*x**16/16 + 11*x**15 + 165*x**14/7 + 462*x**13/13 + 77*x**12/2 + 30*x**11 + 33*x**1
0/2 + 55*x**9/9 + 11*x**8/8 + x**7/7

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Giac [A]  time = 1.1773, size = 82, normalized size = 1.28 \begin{align*} \frac{1}{18} \, x^{18} + \frac{11}{17} \, x^{17} + \frac{55}{16} \, x^{16} + 11 \, x^{15} + \frac{165}{7} \, x^{14} + \frac{462}{13} \, x^{13} + \frac{77}{2} \, x^{12} + 30 \, x^{11} + \frac{33}{2} \, x^{10} + \frac{55}{9} \, x^{9} + \frac{11}{8} \, x^{8} + \frac{1}{7} \, x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/18*x^18 + 11/17*x^17 + 55/16*x^16 + 11*x^15 + 165/7*x^14 + 462/13*x^13 + 77/2*x^12 + 30*x^11 + 33/2*x^10 + 5
5/9*x^9 + 11/8*x^8 + 1/7*x^7