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

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

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Rubi [A]  time = 0.02, antiderivative size = 64, 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 {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} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 81, normalized size = 1.27 \begin {gather*} \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 {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^6 (1+x) \left (1+2 x+x^2\right )^5 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.35, size = 61, normalized size = 0.95 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.15, size = 61, normalized size = 0.95 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.04, size = 62, normalized size = 0.97 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x+1)*(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 = 0.66, size = 61, normalized size = 0.95 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.06, size = 61, normalized size = 0.95 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x + 1)*(2*x + x^2 + 1)^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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sympy [A]  time = 0.07, size = 73, normalized size = 1.14 \begin {gather*} \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 {gather*}

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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