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

Optimal. Leaf size=83 \[ \frac{x^{27}}{27}+\frac{11 x^{25}}{25}+\frac{55 x^{23}}{23}+\frac{55 x^{21}}{7}+\frac{330 x^{19}}{19}+\frac{462 x^{17}}{17}+\frac{154 x^{15}}{5}+\frac{330 x^{13}}{13}+15 x^{11}+\frac{55 x^9}{9}+\frac{11 x^7}{7}+\frac{x^5}{5} \]

[Out]

x^5/5 + (11*x^7)/7 + (55*x^9)/9 + 15*x^11 + (330*x^13)/13 + (154*x^15)/5 + (462*x^17)/17 + (330*x^19)/19 + (55
*x^21)/7 + (55*x^23)/23 + (11*x^25)/25 + x^27/27

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Rubi [A]  time = 0.0314287, 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, Rules used = {28, 270} \[ \frac{x^{27}}{27}+\frac{11 x^{25}}{25}+\frac{55 x^{23}}{23}+\frac{55 x^{21}}{7}+\frac{330 x^{19}}{19}+\frac{462 x^{17}}{17}+\frac{154 x^{15}}{5}+\frac{330 x^{13}}{13}+15 x^{11}+\frac{55 x^9}{9}+\frac{11 x^7}{7}+\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^5/5 + (11*x^7)/7 + (55*x^9)/9 + 15*x^11 + (330*x^13)/13 + (154*x^15)/5 + (462*x^17)/17 + (330*x^19)/19 + (55
*x^21)/7 + (55*x^23)/23 + (11*x^25)/25 + x^27/27

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^4 \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^4 \left (1+x^2\right )^{11} \, dx\\ &=\int \left (x^4+11 x^6+55 x^8+165 x^{10}+330 x^{12}+462 x^{14}+462 x^{16}+330 x^{18}+165 x^{20}+55 x^{22}+11 x^{24}+x^{26}\right ) \, dx\\ &=\frac{x^5}{5}+\frac{11 x^7}{7}+\frac{55 x^9}{9}+15 x^{11}+\frac{330 x^{13}}{13}+\frac{154 x^{15}}{5}+\frac{462 x^{17}}{17}+\frac{330 x^{19}}{19}+\frac{55 x^{21}}{7}+\frac{55 x^{23}}{23}+\frac{11 x^{25}}{25}+\frac{x^{27}}{27}\\ \end{align*}

Mathematica [A]  time = 0.0016816, size = 83, normalized size = 1. \[ \frac{x^{27}}{27}+\frac{11 x^{25}}{25}+\frac{55 x^{23}}{23}+\frac{55 x^{21}}{7}+\frac{330 x^{19}}{19}+\frac{462 x^{17}}{17}+\frac{154 x^{15}}{5}+\frac{330 x^{13}}{13}+15 x^{11}+\frac{55 x^9}{9}+\frac{11 x^7}{7}+\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^5/5 + (11*x^7)/7 + (55*x^9)/9 + 15*x^11 + (330*x^13)/13 + (154*x^15)/5 + (462*x^17)/17 + (330*x^19)/19 + (55
*x^21)/7 + (55*x^23)/23 + (11*x^25)/25 + x^27/27

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Maple [A]  time = 0.001, size = 62, normalized size = 0.8 \begin{align*}{\frac{{x}^{5}}{5}}+{\frac{11\,{x}^{7}}{7}}+{\frac{55\,{x}^{9}}{9}}+15\,{x}^{11}+{\frac{330\,{x}^{13}}{13}}+{\frac{154\,{x}^{15}}{5}}+{\frac{462\,{x}^{17}}{17}}+{\frac{330\,{x}^{19}}{19}}+{\frac{55\,{x}^{21}}{7}}+{\frac{55\,{x}^{23}}{23}}+{\frac{11\,{x}^{25}}{25}}+{\frac{{x}^{27}}{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5+11/7*x^7+55/9*x^9+15*x^11+330/13*x^13+154/5*x^15+462/17*x^17+330/19*x^19+55/7*x^21+55/23*x^23+11/25*x^
25+1/27*x^27

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Maxima [A]  time = 0.964315, size = 82, normalized size = 0.99 \begin{align*} \frac{1}{27} \, x^{27} + \frac{11}{25} \, x^{25} + \frac{55}{23} \, x^{23} + \frac{55}{7} \, x^{21} + \frac{330}{19} \, x^{19} + \frac{462}{17} \, x^{17} + \frac{154}{5} \, x^{15} + \frac{330}{13} \, x^{13} + 15 \, x^{11} + \frac{55}{9} \, x^{9} + \frac{11}{7} \, x^{7} + \frac{1}{5} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*x^27 + 11/25*x^25 + 55/23*x^23 + 55/7*x^21 + 330/19*x^19 + 462/17*x^17 + 154/5*x^15 + 330/13*x^13 + 15*x^
11 + 55/9*x^9 + 11/7*x^7 + 1/5*x^5

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Fricas [A]  time = 1.25451, size = 197, normalized size = 2.37 \begin{align*} \frac{1}{27} x^{27} + \frac{11}{25} x^{25} + \frac{55}{23} x^{23} + \frac{55}{7} x^{21} + \frac{330}{19} x^{19} + \frac{462}{17} x^{17} + \frac{154}{5} x^{15} + \frac{330}{13} x^{13} + 15 x^{11} + \frac{55}{9} x^{9} + \frac{11}{7} x^{7} + \frac{1}{5} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*x^27 + 11/25*x^25 + 55/23*x^23 + 55/7*x^21 + 330/19*x^19 + 462/17*x^17 + 154/5*x^15 + 330/13*x^13 + 15*x^
11 + 55/9*x^9 + 11/7*x^7 + 1/5*x^5

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Sympy [A]  time = 0.069771, size = 75, normalized size = 0.9 \begin{align*} \frac{x^{27}}{27} + \frac{11 x^{25}}{25} + \frac{55 x^{23}}{23} + \frac{55 x^{21}}{7} + \frac{330 x^{19}}{19} + \frac{462 x^{17}}{17} + \frac{154 x^{15}}{5} + \frac{330 x^{13}}{13} + 15 x^{11} + \frac{55 x^{9}}{9} + \frac{11 x^{7}}{7} + \frac{x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(x**2+1)*(x**4+2*x**2+1)**5,x)

[Out]

x**27/27 + 11*x**25/25 + 55*x**23/23 + 55*x**21/7 + 330*x**19/19 + 462*x**17/17 + 154*x**15/5 + 330*x**13/13 +
15*x**11 + 55*x**9/9 + 11*x**7/7 + x**5/5

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Giac [A]  time = 1.10929, size = 82, normalized size = 0.99 \begin{align*} \frac{1}{27} \, x^{27} + \frac{11}{25} \, x^{25} + \frac{55}{23} \, x^{23} + \frac{55}{7} \, x^{21} + \frac{330}{19} \, x^{19} + \frac{462}{17} \, x^{17} + \frac{154}{5} \, x^{15} + \frac{330}{13} \, x^{13} + 15 \, x^{11} + \frac{55}{9} \, x^{9} + \frac{11}{7} \, x^{7} + \frac{1}{5} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*x^27 + 11/25*x^25 + 55/23*x^23 + 55/7*x^21 + 330/19*x^19 + 462/17*x^17 + 154/5*x^15 + 330/13*x^13 + 15*x^
11 + 55/9*x^9 + 11/7*x^7 + 1/5*x^5