3.596 \(\int x^3 (1+x) \left (1+2 x+x^2\right )^5 \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{15} (x+1)^{15}-\frac{3}{14} (x+1)^{14}+\frac{3}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

[Out]

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

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Rubi [A]  time = 0.0407786, antiderivative size = 37, 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 \[ \frac{1}{15} (x+1)^{15}-\frac{3}{14} (x+1)^{14}+\frac{3}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.5921, size = 29, normalized size = 0.78 \[ \frac{\left (x + 1\right )^{15}}{15} - \frac{3 \left (x + 1\right )^{14}}{14} + \frac{3 \left (x + 1\right )^{13}}{13} - \frac{\left (x + 1\right )^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(1+x)*(x**2+2*x+1)**5,x)

[Out]

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

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Mathematica [B]  time = 0.00262322, size = 83, normalized size = 2.24 \[ \frac{x^{15}}{15}+\frac{11 x^{14}}{14}+\frac{55 x^{13}}{13}+\frac{55 x^{12}}{4}+30 x^{11}+\frac{231 x^{10}}{5}+\frac{154 x^9}{3}+\frac{165 x^8}{4}+\frac{165 x^7}{7}+\frac{55 x^6}{6}+\frac{11 x^5}{5}+\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^4/4 + (11*x^5)/5 + (55*x^6)/6 + (165*x^7)/7 + (165*x^8)/4 + (154*x^9)/3 + (231
*x^10)/5 + 30*x^11 + (55*x^12)/4 + (55*x^13)/13 + (11*x^14)/14 + x^15/15

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Maple [B]  time = 0.003, size = 62, normalized size = 1.7 \[{\frac{{x}^{15}}{15}}+{\frac{11\,{x}^{14}}{14}}+{\frac{55\,{x}^{13}}{13}}+{\frac{55\,{x}^{12}}{4}}+30\,{x}^{11}+{\frac{231\,{x}^{10}}{5}}+{\frac{154\,{x}^{9}}{3}}+{\frac{165\,{x}^{8}}{4}}+{\frac{165\,{x}^{7}}{7}}+{\frac{55\,{x}^{6}}{6}}+{\frac{11\,{x}^{5}}{5}}+{\frac{{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(1+x)*(x^2+2*x+1)^5,x)

[Out]

1/15*x^15+11/14*x^14+55/13*x^13+55/4*x^12+30*x^11+231/5*x^10+154/3*x^9+165/4*x^8
+165/7*x^7+55/6*x^6+11/5*x^5+1/4*x^4

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Maxima [A]  time = 0.687241, size = 82, normalized size = 2.22 \[ \frac{1}{15} \, x^{15} + \frac{11}{14} \, x^{14} + \frac{55}{13} \, x^{13} + \frac{55}{4} \, x^{12} + 30 \, x^{11} + \frac{231}{5} \, x^{10} + \frac{154}{3} \, x^{9} + \frac{165}{4} \, x^{8} + \frac{165}{7} \, x^{7} + \frac{55}{6} \, x^{6} + \frac{11}{5} \, x^{5} + \frac{1}{4} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^3,x, algorithm="maxima")

[Out]

1/15*x^15 + 11/14*x^14 + 55/13*x^13 + 55/4*x^12 + 30*x^11 + 231/5*x^10 + 154/3*x
^9 + 165/4*x^8 + 165/7*x^7 + 55/6*x^6 + 11/5*x^5 + 1/4*x^4

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Fricas [A]  time = 0.248315, size = 1, normalized size = 0.03 \[ \frac{1}{15} x^{15} + \frac{11}{14} x^{14} + \frac{55}{13} x^{13} + \frac{55}{4} x^{12} + 30 x^{11} + \frac{231}{5} x^{10} + \frac{154}{3} x^{9} + \frac{165}{4} x^{8} + \frac{165}{7} x^{7} + \frac{55}{6} x^{6} + \frac{11}{5} x^{5} + \frac{1}{4} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^3,x, algorithm="fricas")

[Out]

1/15*x^15 + 11/14*x^14 + 55/13*x^13 + 55/4*x^12 + 30*x^11 + 231/5*x^10 + 154/3*x
^9 + 165/4*x^8 + 165/7*x^7 + 55/6*x^6 + 11/5*x^5 + 1/4*x^4

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Sympy [A]  time = 0.104432, size = 75, normalized size = 2.03 \[ \frac{x^{15}}{15} + \frac{11 x^{14}}{14} + \frac{55 x^{13}}{13} + \frac{55 x^{12}}{4} + 30 x^{11} + \frac{231 x^{10}}{5} + \frac{154 x^{9}}{3} + \frac{165 x^{8}}{4} + \frac{165 x^{7}}{7} + \frac{55 x^{6}}{6} + \frac{11 x^{5}}{5} + \frac{x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**15/15 + 11*x**14/14 + 55*x**13/13 + 55*x**12/4 + 30*x**11 + 231*x**10/5 + 154
*x**9/3 + 165*x**8/4 + 165*x**7/7 + 55*x**6/6 + 11*x**5/5 + x**4/4

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GIAC/XCAS [A]  time = 0.273387, size = 82, normalized size = 2.22 \[ \frac{1}{15} \, x^{15} + \frac{11}{14} \, x^{14} + \frac{55}{13} \, x^{13} + \frac{55}{4} \, x^{12} + 30 \, x^{11} + \frac{231}{5} \, x^{10} + \frac{154}{3} \, x^{9} + \frac{165}{4} \, x^{8} + \frac{165}{7} \, x^{7} + \frac{55}{6} \, x^{6} + \frac{11}{5} \, x^{5} + \frac{1}{4} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)*x^3,x, algorithm="giac")

[Out]

1/15*x^15 + 11/14*x^14 + 55/13*x^13 + 55/4*x^12 + 30*x^11 + 231/5*x^10 + 154/3*x
^9 + 165/4*x^8 + 165/7*x^7 + 55/6*x^6 + 11/5*x^5 + 1/4*x^4