3.595 \(\int x^4 (1+x) \left (1+2 x+x^2\right )^5 \, dx\)

Optimal. Leaf size=46 \[ \frac{1}{16} (x+1)^{16}-\frac{4}{15} (x+1)^{15}+\frac{3}{7} (x+1)^{14}-\frac{4}{13} (x+1)^{13}+\frac{1}{12} (x+1)^{12} \]

[Out]

(1 + x)^12/12 - (4*(1 + x)^13)/13 + (3*(1 + x)^14)/7 - (4*(1 + x)^15)/15 + (1 +
x)^16/16

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Rubi [A]  time = 0.0446687, antiderivative size = 46, 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 \[ \frac{1}{16} (x+1)^{16}-\frac{4}{15} (x+1)^{15}+\frac{3}{7} (x+1)^{14}-\frac{4}{13} (x+1)^{13}+\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

(1 + x)^12/12 - (4*(1 + x)^13)/13 + (3*(1 + x)^14)/7 - (4*(1 + x)^15)/15 + (1 +
x)^16/16

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Rubi in Sympy [A]  time = 11.3113, size = 37, normalized size = 0.8 \[ \frac{\left (x + 1\right )^{16}}{16} - \frac{4 \left (x + 1\right )^{15}}{15} + \frac{3 \left (x + 1\right )^{14}}{7} - \frac{4 \left (x + 1\right )^{13}}{13} + \frac{\left (x + 1\right )^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(1+x)*(x**2+2*x+1)**5,x)

[Out]

(x + 1)**16/16 - 4*(x + 1)**15/15 + 3*(x + 1)**14/7 - 4*(x + 1)**13/13 + (x + 1)
**12/12

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Mathematica [A]  time = 0.00246579, size = 83, normalized size = 1.8 \[ \frac{x^{16}}{16}+\frac{11 x^{15}}{15}+\frac{55 x^{14}}{14}+\frac{165 x^{13}}{13}+\frac{55 x^{12}}{2}+42 x^{11}+\frac{231 x^{10}}{5}+\frac{110 x^9}{3}+\frac{165 x^8}{8}+\frac{55 x^7}{7}+\frac{11 x^6}{6}+\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^5/5 + (11*x^6)/6 + (55*x^7)/7 + (165*x^8)/8 + (110*x^9)/3 + (231*x^10)/5 + 42*
x^11 + (55*x^12)/2 + (165*x^13)/13 + (55*x^14)/14 + (11*x^15)/15 + x^16/16

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Maple [A]  time = 0.002, size = 62, normalized size = 1.4 \[{\frac{{x}^{16}}{16}}+{\frac{11\,{x}^{15}}{15}}+{\frac{55\,{x}^{14}}{14}}+{\frac{165\,{x}^{13}}{13}}+{\frac{55\,{x}^{12}}{2}}+42\,{x}^{11}+{\frac{231\,{x}^{10}}{5}}+{\frac{110\,{x}^{9}}{3}}+{\frac{165\,{x}^{8}}{8}}+{\frac{55\,{x}^{7}}{7}}+{\frac{11\,{x}^{6}}{6}}+{\frac{{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(1+x)*(x^2+2*x+1)^5,x)

[Out]

1/16*x^16+11/15*x^15+55/14*x^14+165/13*x^13+55/2*x^12+42*x^11+231/5*x^10+110/3*x
^9+165/8*x^8+55/7*x^7+11/6*x^6+1/5*x^5

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Maxima [A]  time = 0.682295, size = 82, normalized size = 1.78 \[ \frac{1}{16} \, x^{16} + \frac{11}{15} \, x^{15} + \frac{55}{14} \, x^{14} + \frac{165}{13} \, x^{13} + \frac{55}{2} \, x^{12} + 42 \, x^{11} + \frac{231}{5} \, x^{10} + \frac{110}{3} \, x^{9} + \frac{165}{8} \, x^{8} + \frac{55}{7} \, x^{7} + \frac{11}{6} \, x^{6} + \frac{1}{5} \, x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^4,x, algorithm="maxima")

[Out]

1/16*x^16 + 11/15*x^15 + 55/14*x^14 + 165/13*x^13 + 55/2*x^12 + 42*x^11 + 231/5*
x^10 + 110/3*x^9 + 165/8*x^8 + 55/7*x^7 + 11/6*x^6 + 1/5*x^5

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Fricas [A]  time = 0.25929, size = 1, normalized size = 0.02 \[ \frac{1}{16} x^{16} + \frac{11}{15} x^{15} + \frac{55}{14} x^{14} + \frac{165}{13} x^{13} + \frac{55}{2} x^{12} + 42 x^{11} + \frac{231}{5} x^{10} + \frac{110}{3} x^{9} + \frac{165}{8} x^{8} + \frac{55}{7} x^{7} + \frac{11}{6} x^{6} + \frac{1}{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^4,x, algorithm="fricas")

[Out]

1/16*x^16 + 11/15*x^15 + 55/14*x^14 + 165/13*x^13 + 55/2*x^12 + 42*x^11 + 231/5*
x^10 + 110/3*x^9 + 165/8*x^8 + 55/7*x^7 + 11/6*x^6 + 1/5*x^5

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Sympy [A]  time = 0.104109, size = 75, normalized size = 1.63 \[ \frac{x^{16}}{16} + \frac{11 x^{15}}{15} + \frac{55 x^{14}}{14} + \frac{165 x^{13}}{13} + \frac{55 x^{12}}{2} + 42 x^{11} + \frac{231 x^{10}}{5} + \frac{110 x^{9}}{3} + \frac{165 x^{8}}{8} + \frac{55 x^{7}}{7} + \frac{11 x^{6}}{6} + \frac{x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**16/16 + 11*x**15/15 + 55*x**14/14 + 165*x**13/13 + 55*x**12/2 + 42*x**11 + 23
1*x**10/5 + 110*x**9/3 + 165*x**8/8 + 55*x**7/7 + 11*x**6/6 + x**5/5

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GIAC/XCAS [A]  time = 0.276003, size = 82, normalized size = 1.78 \[ \frac{1}{16} \, x^{16} + \frac{11}{15} \, x^{15} + \frac{55}{14} \, x^{14} + \frac{165}{13} \, x^{13} + \frac{55}{2} \, x^{12} + 42 \, x^{11} + \frac{231}{5} \, x^{10} + \frac{110}{3} \, x^{9} + \frac{165}{8} \, x^{8} + \frac{55}{7} \, x^{7} + \frac{11}{6} \, x^{6} + \frac{1}{5} \, x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^4,x, algorithm="giac")

[Out]

1/16*x^16 + 11/15*x^15 + 55/14*x^14 + 165/13*x^13 + 55/2*x^12 + 42*x^11 + 231/5*
x^10 + 110/3*x^9 + 165/8*x^8 + 55/7*x^7 + 11/6*x^6 + 1/5*x^5