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

Optimal. Leaf size=11 \[ \frac{1}{24} \left (x^2+1\right )^{12} \]

[Out]

(1 + x^2)^12/24

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Rubi [A]  time = 0.0023995, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {28, 261} \[ \frac{1}{24} \left (x^2+1\right )^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x^2)^12/24

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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x \left (1+x^2\right )^{11} \, dx\\ &=\frac{1}{24} \left (1+x^2\right )^{12}\\ \end{align*}

Mathematica [A]  time = 0.0017517, size = 11, normalized size = 1. \[ \frac{1}{24} \left (x^2+1\right )^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x^2)^12/24

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Maple [B]  time = 0.001, size = 62, normalized size = 5.6 \begin{align*}{\frac{{x}^{24}}{24}}+{\frac{{x}^{22}}{2}}+{\frac{11\,{x}^{20}}{4}}+{\frac{55\,{x}^{18}}{6}}+{\frac{165\,{x}^{16}}{8}}+33\,{x}^{14}+{\frac{77\,{x}^{12}}{2}}+33\,{x}^{10}+{\frac{165\,{x}^{8}}{8}}+{\frac{55\,{x}^{6}}{6}}+{\frac{11\,{x}^{4}}{4}}+{\frac{{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*x^24+1/2*x^22+11/4*x^20+55/6*x^18+165/8*x^16+33*x^14+77/2*x^12+33*x^10+165/8*x^8+55/6*x^6+11/4*x^4+1/2*x^
2

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Maxima [B]  time = 0.981964, size = 82, normalized size = 7.45 \begin{align*} \frac{1}{24} \, x^{24} + \frac{1}{2} \, x^{22} + \frac{11}{4} \, x^{20} + \frac{55}{6} \, x^{18} + \frac{165}{8} \, x^{16} + 33 \, x^{14} + \frac{77}{2} \, x^{12} + 33 \, x^{10} + \frac{165}{8} \, x^{8} + \frac{55}{6} \, x^{6} + \frac{11}{4} \, x^{4} + \frac{1}{2} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*x^24 + 1/2*x^22 + 11/4*x^20 + 55/6*x^18 + 165/8*x^16 + 33*x^14 + 77/2*x^12 + 33*x^10 + 165/8*x^8 + 55/6*x
^6 + 11/4*x^4 + 1/2*x^2

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Fricas [B]  time = 1.46461, size = 182, normalized size = 16.55 \begin{align*} \frac{1}{24} x^{24} + \frac{1}{2} x^{22} + \frac{11}{4} x^{20} + \frac{55}{6} x^{18} + \frac{165}{8} x^{16} + 33 x^{14} + \frac{77}{2} x^{12} + 33 x^{10} + \frac{165}{8} x^{8} + \frac{55}{6} x^{6} + \frac{11}{4} x^{4} + \frac{1}{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*x^24 + 1/2*x^22 + 11/4*x^20 + 55/6*x^18 + 165/8*x^16 + 33*x^14 + 77/2*x^12 + 33*x^10 + 165/8*x^8 + 55/6*x
^6 + 11/4*x^4 + 1/2*x^2

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Sympy [B]  time = 0.068124, size = 71, normalized size = 6.45 \begin{align*} \frac{x^{24}}{24} + \frac{x^{22}}{2} + \frac{11 x^{20}}{4} + \frac{55 x^{18}}{6} + \frac{165 x^{16}}{8} + 33 x^{14} + \frac{77 x^{12}}{2} + 33 x^{10} + \frac{165 x^{8}}{8} + \frac{55 x^{6}}{6} + \frac{11 x^{4}}{4} + \frac{x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**24/24 + x**22/2 + 11*x**20/4 + 55*x**18/6 + 165*x**16/8 + 33*x**14 + 77*x**12/2 + 33*x**10 + 165*x**8/8 + 5
5*x**6/6 + 11*x**4/4 + x**2/2

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Giac [B]  time = 1.11146, size = 82, normalized size = 7.45 \begin{align*} \frac{1}{24} \, x^{24} + \frac{1}{2} \, x^{22} + \frac{11}{4} \, x^{20} + \frac{55}{6} \, x^{18} + \frac{165}{8} \, x^{16} + 33 \, x^{14} + \frac{77}{2} \, x^{12} + 33 \, x^{10} + \frac{165}{8} \, x^{8} + \frac{55}{6} \, x^{6} + \frac{11}{4} \, x^{4} + \frac{1}{2} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*x^24 + 1/2*x^22 + 11/4*x^20 + 55/6*x^18 + 165/8*x^16 + 33*x^14 + 77/2*x^12 + 33*x^10 + 165/8*x^8 + 55/6*x
^6 + 11/4*x^4 + 1/2*x^2