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

Optimal. Leaf size=83 \[ \frac{x^{25}}{25}+\frac{11 x^{23}}{23}+\frac{55 x^{21}}{21}+\frac{165 x^{19}}{19}+\frac{330 x^{17}}{17}+\frac{154 x^{15}}{5}+\frac{462 x^{13}}{13}+30 x^{11}+\frac{55 x^9}{3}+\frac{55 x^7}{7}+\frac{11 x^5}{5}+\frac{x^3}{3} \]

[Out]

x^3/3 + (11*x^5)/5 + (55*x^7)/7 + (55*x^9)/3 + 30*x^11 + (462*x^13)/13 + (154*x^
15)/5 + (330*x^17)/17 + (165*x^19)/19 + (55*x^21)/21 + (11*x^23)/23 + x^25/25

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Rubi [A]  time = 0.0593104, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{x^{25}}{25}+\frac{11 x^{23}}{23}+\frac{55 x^{21}}{21}+\frac{165 x^{19}}{19}+\frac{330 x^{17}}{17}+\frac{154 x^{15}}{5}+\frac{462 x^{13}}{13}+30 x^{11}+\frac{55 x^9}{3}+\frac{55 x^7}{7}+\frac{11 x^5}{5}+\frac{x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

x^3/3 + (11*x^5)/5 + (55*x^7)/7 + (55*x^9)/3 + 30*x^11 + (462*x^13)/13 + (154*x^
15)/5 + (330*x^17)/17 + (165*x^19)/19 + (55*x^21)/21 + (11*x^23)/23 + x^25/25

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Rubi in Sympy [A]  time = 11.2157, size = 75, normalized size = 0.9 \[ \frac{x^{25}}{25} + \frac{11 x^{23}}{23} + \frac{55 x^{21}}{21} + \frac{165 x^{19}}{19} + \frac{330 x^{17}}{17} + \frac{154 x^{15}}{5} + \frac{462 x^{13}}{13} + 30 x^{11} + \frac{55 x^{9}}{3} + \frac{55 x^{7}}{7} + \frac{11 x^{5}}{5} + \frac{x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**25/25 + 11*x**23/23 + 55*x**21/21 + 165*x**19/19 + 330*x**17/17 + 154*x**15/5
+ 462*x**13/13 + 30*x**11 + 55*x**9/3 + 55*x**7/7 + 11*x**5/5 + x**3/3

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Mathematica [A]  time = 0.00232436, size = 83, normalized size = 1. \[ \frac{x^{25}}{25}+\frac{11 x^{23}}{23}+\frac{55 x^{21}}{21}+\frac{165 x^{19}}{19}+\frac{330 x^{17}}{17}+\frac{154 x^{15}}{5}+\frac{462 x^{13}}{13}+30 x^{11}+\frac{55 x^9}{3}+\frac{55 x^7}{7}+\frac{11 x^5}{5}+\frac{x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

x^3/3 + (11*x^5)/5 + (55*x^7)/7 + (55*x^9)/3 + 30*x^11 + (462*x^13)/13 + (154*x^
15)/5 + (330*x^17)/17 + (165*x^19)/19 + (55*x^21)/21 + (11*x^23)/23 + x^25/25

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Maple [A]  time = 0.002, size = 62, normalized size = 0.8 \[{\frac{{x}^{3}}{3}}+{\frac{11\,{x}^{5}}{5}}+{\frac{55\,{x}^{7}}{7}}+{\frac{55\,{x}^{9}}{3}}+30\,{x}^{11}+{\frac{462\,{x}^{13}}{13}}+{\frac{154\,{x}^{15}}{5}}+{\frac{330\,{x}^{17}}{17}}+{\frac{165\,{x}^{19}}{19}}+{\frac{55\,{x}^{21}}{21}}+{\frac{11\,{x}^{23}}{23}}+{\frac{{x}^{25}}{25}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3+11/5*x^5+55/7*x^7+55/3*x^9+30*x^11+462/13*x^13+154/5*x^15+330/17*x^17+16
5/19*x^19+55/21*x^21+11/23*x^23+1/25*x^25

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Maxima [A]  time = 0.696201, size = 82, normalized size = 0.99 \[ \frac{1}{25} \, x^{25} + \frac{11}{23} \, x^{23} + \frac{55}{21} \, x^{21} + \frac{165}{19} \, x^{19} + \frac{330}{17} \, x^{17} + \frac{154}{5} \, x^{15} + \frac{462}{13} \, x^{13} + 30 \, x^{11} + \frac{55}{3} \, x^{9} + \frac{55}{7} \, x^{7} + \frac{11}{5} \, x^{5} + \frac{1}{3} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*x^25 + 11/23*x^23 + 55/21*x^21 + 165/19*x^19 + 330/17*x^17 + 154/5*x^15 + 4
62/13*x^13 + 30*x^11 + 55/3*x^9 + 55/7*x^7 + 11/5*x^5 + 1/3*x^3

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Fricas [A]  time = 0.226396, size = 1, normalized size = 0.01 \[ \frac{1}{25} x^{25} + \frac{11}{23} x^{23} + \frac{55}{21} x^{21} + \frac{165}{19} x^{19} + \frac{330}{17} x^{17} + \frac{154}{5} x^{15} + \frac{462}{13} x^{13} + 30 x^{11} + \frac{55}{3} x^{9} + \frac{55}{7} x^{7} + \frac{11}{5} x^{5} + \frac{1}{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*x^25 + 11/23*x^23 + 55/21*x^21 + 165/19*x^19 + 330/17*x^17 + 154/5*x^15 + 4
62/13*x^13 + 30*x^11 + 55/3*x^9 + 55/7*x^7 + 11/5*x^5 + 1/3*x^3

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Sympy [A]  time = 0.100533, size = 75, normalized size = 0.9 \[ \frac{x^{25}}{25} + \frac{11 x^{23}}{23} + \frac{55 x^{21}}{21} + \frac{165 x^{19}}{19} + \frac{330 x^{17}}{17} + \frac{154 x^{15}}{5} + \frac{462 x^{13}}{13} + 30 x^{11} + \frac{55 x^{9}}{3} + \frac{55 x^{7}}{7} + \frac{11 x^{5}}{5} + \frac{x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**25/25 + 11*x**23/23 + 55*x**21/21 + 165*x**19/19 + 330*x**17/17 + 154*x**15/5
+ 462*x**13/13 + 30*x**11 + 55*x**9/3 + 55*x**7/7 + 11*x**5/5 + x**3/3

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GIAC/XCAS [A]  time = 0.267529, size = 82, normalized size = 0.99 \[ \frac{1}{25} \, x^{25} + \frac{11}{23} \, x^{23} + \frac{55}{21} \, x^{21} + \frac{165}{19} \, x^{19} + \frac{330}{17} \, x^{17} + \frac{154}{5} \, x^{15} + \frac{462}{13} \, x^{13} + 30 \, x^{11} + \frac{55}{3} \, x^{9} + \frac{55}{7} \, x^{7} + \frac{11}{5} \, x^{5} + \frac{1}{3} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*x^25 + 11/23*x^23 + 55/21*x^21 + 165/19*x^19 + 330/17*x^17 + 154/5*x^15 + 4
62/13*x^13 + 30*x^11 + 55/3*x^9 + 55/7*x^7 + 11/5*x^5 + 1/3*x^3