3.592 \(\int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx\)

Optimal. Leaf size=73 \[ \frac{1}{19} (x+1)^{19}-\frac{7}{18} (x+1)^{18}+\frac{21}{17} (x+1)^{17}-\frac{35}{16} (x+1)^{16}+\frac{7}{3} (x+1)^{15}-\frac{3}{2} (x+1)^{14}+\frac{7}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

[Out]

-(1 + x)^12/12 + (7*(1 + x)^13)/13 - (3*(1 + x)^14)/2 + (7*(1 + x)^15)/3 - (35*(
1 + x)^16)/16 + (21*(1 + x)^17)/17 - (7*(1 + x)^18)/18 + (1 + x)^19/19

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Rubi [A]  time = 0.055527, antiderivative size = 73, 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 \[ \frac{1}{19} (x+1)^{19}-\frac{7}{18} (x+1)^{18}+\frac{21}{17} (x+1)^{17}-\frac{35}{16} (x+1)^{16}+\frac{7}{3} (x+1)^{15}-\frac{3}{2} (x+1)^{14}+\frac{7}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

-(1 + x)^12/12 + (7*(1 + x)^13)/13 - (3*(1 + x)^14)/2 + (7*(1 + x)^15)/3 - (35*(
1 + x)^16)/16 + (21*(1 + x)^17)/17 - (7*(1 + x)^18)/18 + (1 + x)^19/19

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Rubi in Sympy [A]  time = 12.7948, size = 71, normalized size = 0.97 \[ \frac{x^{19}}{19} + \frac{11 x^{18}}{18} + \frac{55 x^{17}}{17} + \frac{165 x^{16}}{16} + 22 x^{15} + 33 x^{14} + \frac{462 x^{13}}{13} + \frac{55 x^{12}}{2} + 15 x^{11} + \frac{11 x^{10}}{2} + \frac{11 x^{9}}{9} + \frac{x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**19/19 + 11*x**18/18 + 55*x**17/17 + 165*x**16/16 + 22*x**15 + 33*x**14 + 462*
x**13/13 + 55*x**12/2 + 15*x**11 + 11*x**10/2 + 11*x**9/9 + x**8/8

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Mathematica [A]  time = 0.00272626, size = 79, normalized size = 1.08 \[ \frac{x^{19}}{19}+\frac{11 x^{18}}{18}+\frac{55 x^{17}}{17}+\frac{165 x^{16}}{16}+22 x^{15}+33 x^{14}+\frac{462 x^{13}}{13}+\frac{55 x^{12}}{2}+15 x^{11}+\frac{11 x^{10}}{2}+\frac{11 x^9}{9}+\frac{x^8}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^8/8 + (11*x^9)/9 + (11*x^10)/2 + 15*x^11 + (55*x^12)/2 + (462*x^13)/13 + 33*x^
14 + 22*x^15 + (165*x^16)/16 + (55*x^17)/17 + (11*x^18)/18 + x^19/19

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Maple [A]  time = 0.003, size = 62, normalized size = 0.9 \[{\frac{{x}^{19}}{19}}+{\frac{11\,{x}^{18}}{18}}+{\frac{55\,{x}^{17}}{17}}+{\frac{165\,{x}^{16}}{16}}+22\,{x}^{15}+33\,{x}^{14}+{\frac{462\,{x}^{13}}{13}}+{\frac{55\,{x}^{12}}{2}}+15\,{x}^{11}+{\frac{11\,{x}^{10}}{2}}+{\frac{11\,{x}^{9}}{9}}+{\frac{{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(1+x)*(x^2+2*x+1)^5,x)

[Out]

1/19*x^19+11/18*x^18+55/17*x^17+165/16*x^16+22*x^15+33*x^14+462/13*x^13+55/2*x^1
2+15*x^11+11/2*x^10+11/9*x^9+1/8*x^8

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Maxima [A]  time = 0.68914, size = 82, normalized size = 1.12 \[ \frac{1}{19} \, x^{19} + \frac{11}{18} \, x^{18} + \frac{55}{17} \, x^{17} + \frac{165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac{462}{13} \, x^{13} + \frac{55}{2} \, x^{12} + 15 \, x^{11} + \frac{11}{2} \, x^{10} + \frac{11}{9} \, x^{9} + \frac{1}{8} \, x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^7,x, algorithm="maxima")

[Out]

1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 462/13*x
^13 + 55/2*x^12 + 15*x^11 + 11/2*x^10 + 11/9*x^9 + 1/8*x^8

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Fricas [A]  time = 0.255453, size = 1, normalized size = 0.01 \[ \frac{1}{19} x^{19} + \frac{11}{18} x^{18} + \frac{55}{17} x^{17} + \frac{165}{16} x^{16} + 22 x^{15} + 33 x^{14} + \frac{462}{13} x^{13} + \frac{55}{2} x^{12} + 15 x^{11} + \frac{11}{2} x^{10} + \frac{11}{9} x^{9} + \frac{1}{8} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^7,x, algorithm="fricas")

[Out]

1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 462/13*x
^13 + 55/2*x^12 + 15*x^11 + 11/2*x^10 + 11/9*x^9 + 1/8*x^8

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Sympy [A]  time = 0.11915, size = 71, normalized size = 0.97 \[ \frac{x^{19}}{19} + \frac{11 x^{18}}{18} + \frac{55 x^{17}}{17} + \frac{165 x^{16}}{16} + 22 x^{15} + 33 x^{14} + \frac{462 x^{13}}{13} + \frac{55 x^{12}}{2} + 15 x^{11} + \frac{11 x^{10}}{2} + \frac{11 x^{9}}{9} + \frac{x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**19/19 + 11*x**18/18 + 55*x**17/17 + 165*x**16/16 + 22*x**15 + 33*x**14 + 462*
x**13/13 + 55*x**12/2 + 15*x**11 + 11*x**10/2 + 11*x**9/9 + x**8/8

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GIAC/XCAS [A]  time = 0.267192, size = 82, normalized size = 1.12 \[ \frac{1}{19} \, x^{19} + \frac{11}{18} \, x^{18} + \frac{55}{17} \, x^{17} + \frac{165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac{462}{13} \, x^{13} + \frac{55}{2} \, x^{12} + 15 \, x^{11} + \frac{11}{2} \, x^{10} + \frac{11}{9} \, x^{9} + \frac{1}{8} \, x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^7,x, algorithm="giac")

[Out]

1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 462/13*x
^13 + 55/2*x^12 + 15*x^11 + 11/2*x^10 + 11/9*x^9 + 1/8*x^8