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

Optimal. Leaf size=34 \[ \frac{1}{28} \left (x^2+1\right )^{14}-\frac{1}{13} \left (x^2+1\right )^{13}+\frac{1}{24} \left (x^2+1\right )^{12} \]

[Out]

(1 + x^2)^12/24 - (1 + x^2)^13/13 + (1 + x^2)^14/28

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Rubi [A]  time = 0.0938088, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{28} \left (x^2+1\right )^{14}-\frac{1}{13} \left (x^2+1\right )^{13}+\frac{1}{24} \left (x^2+1\right )^{12} \]

Antiderivative was successfully verified.

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

[Out]

(1 + x^2)^12/24 - (1 + x^2)^13/13 + (1 + x^2)^14/28

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Rubi in Sympy [A]  time = 10.4658, size = 24, normalized size = 0.71 \[ \frac{\left (x^{2} + 1\right )^{14}}{28} - \frac{\left (x^{2} + 1\right )^{13}}{13} + \frac{\left (x^{2} + 1\right )^{12}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x**2 + 1)**14/28 - (x**2 + 1)**13/13 + (x**2 + 1)**12/24

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Mathematica [B]  time = 0.00350701, size = 85, normalized size = 2.5 \[ \frac{x^{28}}{28}+\frac{11 x^{26}}{26}+\frac{55 x^{24}}{24}+\frac{15 x^{22}}{2}+\frac{33 x^{20}}{2}+\frac{77 x^{18}}{3}+\frac{231 x^{16}}{8}+\frac{165 x^{14}}{7}+\frac{55 x^{12}}{4}+\frac{11 x^{10}}{2}+\frac{11 x^8}{8}+\frac{x^6}{6} \]

Antiderivative was successfully verified.

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

[Out]

x^6/6 + (11*x^8)/8 + (11*x^10)/2 + (55*x^12)/4 + (165*x^14)/7 + (231*x^16)/8 + (
77*x^18)/3 + (33*x^20)/2 + (15*x^22)/2 + (55*x^24)/24 + (11*x^26)/26 + x^28/28

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Maple [B]  time = 0.002, size = 62, normalized size = 1.8 \[{\frac{{x}^{28}}{28}}+{\frac{11\,{x}^{26}}{26}}+{\frac{55\,{x}^{24}}{24}}+{\frac{15\,{x}^{22}}{2}}+{\frac{33\,{x}^{20}}{2}}+{\frac{77\,{x}^{18}}{3}}+{\frac{231\,{x}^{16}}{8}}+{\frac{165\,{x}^{14}}{7}}+{\frac{55\,{x}^{12}}{4}}+{\frac{11\,{x}^{10}}{2}}+{\frac{11\,{x}^{8}}{8}}+{\frac{{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/28*x^28+11/26*x^26+55/24*x^24+15/2*x^22+33/2*x^20+77/3*x^18+231/8*x^16+165/7*x
^14+55/4*x^12+11/2*x^10+11/8*x^8+1/6*x^6

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Maxima [A]  time = 0.699125, size = 82, normalized size = 2.41 \[ \frac{1}{28} \, x^{28} + \frac{11}{26} \, x^{26} + \frac{55}{24} \, x^{24} + \frac{15}{2} \, x^{22} + \frac{33}{2} \, x^{20} + \frac{77}{3} \, x^{18} + \frac{231}{8} \, x^{16} + \frac{165}{7} \, x^{14} + \frac{55}{4} \, x^{12} + \frac{11}{2} \, x^{10} + \frac{11}{8} \, x^{8} + \frac{1}{6} \, x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/28*x^28 + 11/26*x^26 + 55/24*x^24 + 15/2*x^22 + 33/2*x^20 + 77/3*x^18 + 231/8*
x^16 + 165/7*x^14 + 55/4*x^12 + 11/2*x^10 + 11/8*x^8 + 1/6*x^6

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Fricas [A]  time = 0.258133, size = 1, normalized size = 0.03 \[ \frac{1}{28} x^{28} + \frac{11}{26} x^{26} + \frac{55}{24} x^{24} + \frac{15}{2} x^{22} + \frac{33}{2} x^{20} + \frac{77}{3} x^{18} + \frac{231}{8} x^{16} + \frac{165}{7} x^{14} + \frac{55}{4} x^{12} + \frac{11}{2} x^{10} + \frac{11}{8} x^{8} + \frac{1}{6} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/28*x^28 + 11/26*x^26 + 55/24*x^24 + 15/2*x^22 + 33/2*x^20 + 77/3*x^18 + 231/8*
x^16 + 165/7*x^14 + 55/4*x^12 + 11/2*x^10 + 11/8*x^8 + 1/6*x^6

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Sympy [A]  time = 0.108114, size = 76, normalized size = 2.24 \[ \frac{x^{28}}{28} + \frac{11 x^{26}}{26} + \frac{55 x^{24}}{24} + \frac{15 x^{22}}{2} + \frac{33 x^{20}}{2} + \frac{77 x^{18}}{3} + \frac{231 x^{16}}{8} + \frac{165 x^{14}}{7} + \frac{55 x^{12}}{4} + \frac{11 x^{10}}{2} + \frac{11 x^{8}}{8} + \frac{x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**28/28 + 11*x**26/26 + 55*x**24/24 + 15*x**22/2 + 33*x**20/2 + 77*x**18/3 + 23
1*x**16/8 + 165*x**14/7 + 55*x**12/4 + 11*x**10/2 + 11*x**8/8 + x**6/6

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GIAC/XCAS [A]  time = 0.268875, size = 82, normalized size = 2.41 \[ \frac{1}{28} \, x^{28} + \frac{11}{26} \, x^{26} + \frac{55}{24} \, x^{24} + \frac{15}{2} \, x^{22} + \frac{33}{2} \, x^{20} + \frac{77}{3} \, x^{18} + \frac{231}{8} \, x^{16} + \frac{165}{7} \, x^{14} + \frac{55}{4} \, x^{12} + \frac{11}{2} \, x^{10} + \frac{11}{8} \, x^{8} + \frac{1}{6} \, x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/28*x^28 + 11/26*x^26 + 55/24*x^24 + 15/2*x^22 + 33/2*x^20 + 77/3*x^18 + 231/8*
x^16 + 165/7*x^14 + 55/4*x^12 + 11/2*x^10 + 11/8*x^8 + 1/6*x^6