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

Optimal. Leaf size=73 \[ \frac{x^{23}}{23}+\frac{11 x^{21}}{21}+\frac{55 x^{19}}{19}+\frac{165 x^{17}}{17}+22 x^{15}+\frac{462 x^{13}}{13}+42 x^{11}+\frac{110 x^9}{3}+\frac{165 x^7}{7}+11 x^5+\frac{11 x^3}{3}+x \]

[Out]

x + (11*x^3)/3 + 11*x^5 + (165*x^7)/7 + (110*x^9)/3 + 42*x^11 + (462*x^13)/13 +
22*x^15 + (165*x^17)/17 + (55*x^19)/19 + (11*x^21)/21 + x^23/23

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Rubi [A]  time = 0.0446194, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x^{23}}{23}+\frac{11 x^{21}}{21}+\frac{55 x^{19}}{19}+\frac{165 x^{17}}{17}+22 x^{15}+\frac{462 x^{13}}{13}+42 x^{11}+\frac{110 x^9}{3}+\frac{165 x^7}{7}+11 x^5+\frac{11 x^3}{3}+x \]

Antiderivative was successfully verified.

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

[Out]

x + (11*x^3)/3 + 11*x^5 + (165*x^7)/7 + (110*x^9)/3 + 42*x^11 + (462*x^13)/13 +
22*x^15 + (165*x^17)/17 + (55*x^19)/19 + (11*x^21)/21 + x^23/23

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Rubi in Sympy [A]  time = 9.60154, size = 68, normalized size = 0.93 \[ \frac{x^{23}}{23} + \frac{11 x^{21}}{21} + \frac{55 x^{19}}{19} + \frac{165 x^{17}}{17} + 22 x^{15} + \frac{462 x^{13}}{13} + 42 x^{11} + \frac{110 x^{9}}{3} + \frac{165 x^{7}}{7} + 11 x^{5} + \frac{11 x^{3}}{3} + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**23/23 + 11*x**21/21 + 55*x**19/19 + 165*x**17/17 + 22*x**15 + 462*x**13/13 +
42*x**11 + 110*x**9/3 + 165*x**7/7 + 11*x**5 + 11*x**3/3 + x

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Mathematica [A]  time = 0.0011577, size = 73, normalized size = 1. \[ \frac{x^{23}}{23}+\frac{11 x^{21}}{21}+\frac{55 x^{19}}{19}+\frac{165 x^{17}}{17}+22 x^{15}+\frac{462 x^{13}}{13}+42 x^{11}+\frac{110 x^9}{3}+\frac{165 x^7}{7}+11 x^5+\frac{11 x^3}{3}+x \]

Antiderivative was successfully verified.

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

[Out]

x + (11*x^3)/3 + 11*x^5 + (165*x^7)/7 + (110*x^9)/3 + 42*x^11 + (462*x^13)/13 +
22*x^15 + (165*x^17)/17 + (55*x^19)/19 + (11*x^21)/21 + x^23/23

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Maple [A]  time = 0.002, size = 58, normalized size = 0.8 \[ x+{\frac{11\,{x}^{3}}{3}}+11\,{x}^{5}+{\frac{165\,{x}^{7}}{7}}+{\frac{110\,{x}^{9}}{3}}+42\,{x}^{11}+{\frac{462\,{x}^{13}}{13}}+22\,{x}^{15}+{\frac{165\,{x}^{17}}{17}}+{\frac{55\,{x}^{19}}{19}}+{\frac{11\,{x}^{21}}{21}}+{\frac{{x}^{23}}{23}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)*(x^4+2*x^2+1)^5,x)

[Out]

x+11/3*x^3+11*x^5+165/7*x^7+110/3*x^9+42*x^11+462/13*x^13+22*x^15+165/17*x^17+55
/19*x^19+11/21*x^21+1/23*x^23

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Maxima [A]  time = 0.699206, size = 77, normalized size = 1.05 \[ \frac{1}{23} \, x^{23} + \frac{11}{21} \, x^{21} + \frac{55}{19} \, x^{19} + \frac{165}{17} \, x^{17} + 22 \, x^{15} + \frac{462}{13} \, x^{13} + 42 \, x^{11} + \frac{110}{3} \, x^{9} + \frac{165}{7} \, x^{7} + 11 \, x^{5} + \frac{11}{3} \, x^{3} + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/23*x^23 + 11/21*x^21 + 55/19*x^19 + 165/17*x^17 + 22*x^15 + 462/13*x^13 + 42*x
^11 + 110/3*x^9 + 165/7*x^7 + 11*x^5 + 11/3*x^3 + x

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Fricas [A]  time = 0.256262, size = 1, normalized size = 0.01 \[ \frac{1}{23} x^{23} + \frac{11}{21} x^{21} + \frac{55}{19} x^{19} + \frac{165}{17} x^{17} + 22 x^{15} + \frac{462}{13} x^{13} + 42 x^{11} + \frac{110}{3} x^{9} + \frac{165}{7} x^{7} + 11 x^{5} + \frac{11}{3} x^{3} + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/23*x^23 + 11/21*x^21 + 55/19*x^19 + 165/17*x^17 + 22*x^15 + 462/13*x^13 + 42*x
^11 + 110/3*x^9 + 165/7*x^7 + 11*x^5 + 11/3*x^3 + x

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Sympy [A]  time = 0.109544, size = 68, normalized size = 0.93 \[ \frac{x^{23}}{23} + \frac{11 x^{21}}{21} + \frac{55 x^{19}}{19} + \frac{165 x^{17}}{17} + 22 x^{15} + \frac{462 x^{13}}{13} + 42 x^{11} + \frac{110 x^{9}}{3} + \frac{165 x^{7}}{7} + 11 x^{5} + \frac{11 x^{3}}{3} + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**23/23 + 11*x**21/21 + 55*x**19/19 + 165*x**17/17 + 22*x**15 + 462*x**13/13 +
42*x**11 + 110*x**9/3 + 165*x**7/7 + 11*x**5 + 11*x**3/3 + x

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GIAC/XCAS [A]  time = 0.261, size = 77, normalized size = 1.05 \[ \frac{1}{23} \, x^{23} + \frac{11}{21} \, x^{21} + \frac{55}{19} \, x^{19} + \frac{165}{17} \, x^{17} + 22 \, x^{15} + \frac{462}{13} \, x^{13} + 42 \, x^{11} + \frac{110}{3} \, x^{9} + \frac{165}{7} \, x^{7} + 11 \, x^{5} + \frac{11}{3} \, x^{3} + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/23*x^23 + 11/21*x^21 + 55/19*x^19 + 165/17*x^17 + 22*x^15 + 462/13*x^13 + 42*x
^11 + 110/3*x^9 + 165/7*x^7 + 11*x^5 + 11/3*x^3 + x