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

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]

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+165/19*x^19+55/21*x^21+11/23*x^2
3+1/25*x^25

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Rubi [A]  time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {28, 270} \[ \frac {x^{25}}{25}+\frac {11 x^{23}}{23}+\frac {55 x^{21}}{21}+\frac {165 x^{19}}{19}+\frac {330 x^{17}}{17}+\frac {154 x^{15}}{5}+\frac {462 x^{13}}{13}+30 x^{11}+\frac {55 x^9}{3}+\frac {55 x^7}{7}+\frac {11 x^5}{5}+\frac {x^3}{3} \]

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

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^2 \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^2 \left (1+x^2\right )^{11} \, dx\\ &=\int \left (x^2+11 x^4+55 x^6+165 x^8+330 x^{10}+462 x^{12}+462 x^{14}+330 x^{16}+165 x^{18}+55 x^{20}+11 x^{22}+x^{24}\right ) \, dx\\ &=\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}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 83, normalized size = 1.00 \[ \frac {x^{25}}{25}+\frac {11 x^{23}}{23}+\frac {55 x^{21}}{21}+\frac {165 x^{19}}{19}+\frac {330 x^{17}}{17}+\frac {154 x^{15}}{5}+\frac {462 x^{13}}{13}+30 x^{11}+\frac {55 x^9}{3}+\frac {55 x^7}{7}+\frac {11 x^5}{5}+\frac {x^3}{3} \]

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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fricas [A]  time = 0.55, size = 61, normalized size = 0.73 \[ \frac {1}{25} x^{25} + \frac {11}{23} x^{23} + \frac {55}{21} x^{21} + \frac {165}{19} x^{19} + \frac {330}{17} x^{17} + \frac {154}{5} x^{15} + \frac {462}{13} x^{13} + 30 x^{11} + \frac {55}{3} x^{9} + \frac {55}{7} x^{7} + \frac {11}{5} x^{5} + \frac {1}{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^2+1)*(x^4+2*x^2+1)^5,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 + 462/13*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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giac [A]  time = 0.30, size = 61, normalized size = 0.73 \[ \frac {1}{25} \, x^{25} + \frac {11}{23} \, x^{23} + \frac {55}{21} \, x^{21} + \frac {165}{19} \, x^{19} + \frac {330}{17} \, x^{17} + \frac {154}{5} \, x^{15} + \frac {462}{13} \, x^{13} + 30 \, x^{11} + \frac {55}{3} \, x^{9} + \frac {55}{7} \, x^{7} + \frac {11}{5} \, x^{5} + \frac {1}{3} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^2+1)*(x^4+2*x^2+1)^5,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 + 462/13*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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maple [A]  time = 0.00, size = 62, normalized size = 0.75 \[ \frac {1}{25} x^{25}+\frac {11}{23} x^{23}+\frac {55}{21} x^{21}+\frac {165}{19} x^{19}+\frac {330}{17} x^{17}+\frac {154}{5} x^{15}+\frac {462}{13} x^{13}+30 x^{11}+\frac {55}{3} x^{9}+\frac {55}{7} x^{7}+\frac {11}{5} x^{5}+\frac {1}{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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+165/19*x^19+55/21*x^21+11/23*x^2
3+1/25*x^25

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maxima [A]  time = 0.88, size = 61, normalized size = 0.73 \[ \frac {1}{25} \, x^{25} + \frac {11}{23} \, x^{23} + \frac {55}{21} \, x^{21} + \frac {165}{19} \, x^{19} + \frac {330}{17} \, x^{17} + \frac {154}{5} \, x^{15} + \frac {462}{13} \, x^{13} + 30 \, x^{11} + \frac {55}{3} \, x^{9} + \frac {55}{7} \, x^{7} + \frac {11}{5} \, x^{5} + \frac {1}{3} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^2+1)*(x^4+2*x^2+1)^5,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 + 462/13*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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mupad [B]  time = 0.06, size = 61, normalized size = 0.73 \[ \frac {x^{25}}{25}+\frac {11\,x^{23}}{23}+\frac {55\,x^{21}}{21}+\frac {165\,x^{19}}{19}+\frac {330\,x^{17}}{17}+\frac {154\,x^{15}}{5}+\frac {462\,x^{13}}{13}+30\,x^{11}+\frac {55\,x^9}{3}+\frac {55\,x^7}{7}+\frac {11\,x^5}{5}+\frac {x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^2 + 1)*(2*x^2 + x^4 + 1)^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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sympy [A]  time = 0.07, size = 75, normalized size = 0.90 \[ \frac {x^{25}}{25} + \frac {11 x^{23}}{23} + \frac {55 x^{21}}{21} + \frac {165 x^{19}}{19} + \frac {330 x^{17}}{17} + \frac {154 x^{15}}{5} + \frac {462 x^{13}}{13} + 30 x^{11} + \frac {55 x^{9}}{3} + \frac {55 x^{7}}{7} + \frac {11 x^{5}}{5} + \frac {x^{3}}{3} \]

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 + 5
5*x**9/3 + 55*x**7/7 + 11*x**5/5 + x**3/3

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