3.68 \(\int x^3 \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{26} \left (x^2+1\right )^{13}-\frac{1}{24} \left (x^2+1\right )^{12} \]

[Out]

-(1 + x^2)^12/24 + (1 + x^2)^13/26

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Rubi [A]  time = 0.0528938, antiderivative size = 23, 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 \[ \frac{1}{26} \left (x^2+1\right )^{13}-\frac{1}{24} \left (x^2+1\right )^{12} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

-(1 + x^2)^12/24 + (1 + x^2)^13/26

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Rubi in Sympy [A]  time = 9.5844, size = 15, normalized size = 0.65 \[ \frac{\left (x^{2} + 1\right )^{13}}{26} - \frac{\left (x^{2} + 1\right )^{12}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(x**2+1)*(x**4+2*x**2+1)**5,x)

[Out]

(x**2 + 1)**13/26 - (x**2 + 1)**12/24

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Mathematica [B]  time = 0.00263026, size = 83, normalized size = 3.61 \[ \frac{x^{26}}{26}+\frac{11 x^{24}}{24}+\frac{5 x^{22}}{2}+\frac{33 x^{20}}{4}+\frac{55 x^{18}}{3}+\frac{231 x^{16}}{8}+33 x^{14}+\frac{55 x^{12}}{2}+\frac{33 x^{10}}{2}+\frac{55 x^8}{8}+\frac{11 x^6}{6}+\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

x^4/4 + (11*x^6)/6 + (55*x^8)/8 + (33*x^10)/2 + (55*x^12)/2 + 33*x^14 + (231*x^1
6)/8 + (55*x^18)/3 + (33*x^20)/4 + (5*x^22)/2 + (11*x^24)/24 + x^26/26

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Maple [B]  time = 0.002, size = 62, normalized size = 2.7 \[{\frac{{x}^{26}}{26}}+{\frac{11\,{x}^{24}}{24}}+{\frac{5\,{x}^{22}}{2}}+{\frac{33\,{x}^{20}}{4}}+{\frac{55\,{x}^{18}}{3}}+{\frac{231\,{x}^{16}}{8}}+33\,{x}^{14}+{\frac{55\,{x}^{12}}{2}}+{\frac{33\,{x}^{10}}{2}}+{\frac{55\,{x}^{8}}{8}}+{\frac{11\,{x}^{6}}{6}}+{\frac{{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(x^2+1)*(x^4+2*x^2+1)^5,x)

[Out]

1/26*x^26+11/24*x^24+5/2*x^22+33/4*x^20+55/3*x^18+231/8*x^16+33*x^14+55/2*x^12+3
3/2*x^10+55/8*x^8+11/6*x^6+1/4*x^4

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Maxima [A]  time = 0.700554, size = 82, normalized size = 3.57 \[ \frac{1}{26} \, x^{26} + \frac{11}{24} \, x^{24} + \frac{5}{2} \, x^{22} + \frac{33}{4} \, x^{20} + \frac{55}{3} \, x^{18} + \frac{231}{8} \, x^{16} + 33 \, x^{14} + \frac{55}{2} \, x^{12} + \frac{33}{2} \, x^{10} + \frac{55}{8} \, x^{8} + \frac{11}{6} \, x^{6} + \frac{1}{4} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 2*x^2 + 1)^5*(x^2 + 1)*x^3,x, algorithm="maxima")

[Out]

1/26*x^26 + 11/24*x^24 + 5/2*x^22 + 33/4*x^20 + 55/3*x^18 + 231/8*x^16 + 33*x^14
+ 55/2*x^12 + 33/2*x^10 + 55/8*x^8 + 11/6*x^6 + 1/4*x^4

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Fricas [A]  time = 0.227383, size = 1, normalized size = 0.04 \[ \frac{1}{26} x^{26} + \frac{11}{24} x^{24} + \frac{5}{2} x^{22} + \frac{33}{4} x^{20} + \frac{55}{3} x^{18} + \frac{231}{8} x^{16} + 33 x^{14} + \frac{55}{2} x^{12} + \frac{33}{2} x^{10} + \frac{55}{8} x^{8} + \frac{11}{6} x^{6} + \frac{1}{4} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 2*x^2 + 1)^5*(x^2 + 1)*x^3,x, algorithm="fricas")

[Out]

1/26*x^26 + 11/24*x^24 + 5/2*x^22 + 33/4*x^20 + 55/3*x^18 + 231/8*x^16 + 33*x^14
+ 55/2*x^12 + 33/2*x^10 + 55/8*x^8 + 11/6*x^6 + 1/4*x^4

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Sympy [A]  time = 0.103432, size = 75, normalized size = 3.26 \[ \frac{x^{26}}{26} + \frac{11 x^{24}}{24} + \frac{5 x^{22}}{2} + \frac{33 x^{20}}{4} + \frac{55 x^{18}}{3} + \frac{231 x^{16}}{8} + 33 x^{14} + \frac{55 x^{12}}{2} + \frac{33 x^{10}}{2} + \frac{55 x^{8}}{8} + \frac{11 x^{6}}{6} + \frac{x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(x**2+1)*(x**4+2*x**2+1)**5,x)

[Out]

x**26/26 + 11*x**24/24 + 5*x**22/2 + 33*x**20/4 + 55*x**18/3 + 231*x**16/8 + 33*
x**14 + 55*x**12/2 + 33*x**10/2 + 55*x**8/8 + 11*x**6/6 + x**4/4

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GIAC/XCAS [A]  time = 0.267673, size = 82, normalized size = 3.57 \[ \frac{1}{26} \, x^{26} + \frac{11}{24} \, x^{24} + \frac{5}{2} \, x^{22} + \frac{33}{4} \, x^{20} + \frac{55}{3} \, x^{18} + \frac{231}{8} \, x^{16} + 33 \, x^{14} + \frac{55}{2} \, x^{12} + \frac{33}{2} \, x^{10} + \frac{55}{8} \, x^{8} + \frac{11}{6} \, x^{6} + \frac{1}{4} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 2*x^2 + 1)^5*(x^2 + 1)*x^3,x, algorithm="giac")

[Out]

1/26*x^26 + 11/24*x^24 + 5/2*x^22 + 33/4*x^20 + 55/3*x^18 + 231/8*x^16 + 33*x^14
+ 55/2*x^12 + 33/2*x^10 + 55/8*x^8 + 11/6*x^6 + 1/4*x^4