3.1394 \(\int \frac{1}{x^{10} \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=33 \[ \frac{\sqrt{x^6+2}}{18 x^3}-\frac{\sqrt{x^6+2}}{18 x^9} \]

[Out]

-Sqrt[2 + x^6]/(18*x^9) + Sqrt[2 + x^6]/(18*x^3)

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Rubi [A]  time = 0.0062895, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {271, 264} \[ \frac{\sqrt{x^6+2}}{18 x^3}-\frac{\sqrt{x^6+2}}{18 x^9} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^10*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(18*x^9) + Sqrt[2 + x^6]/(18*x^3)

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^{10} \sqrt{2+x^6}} \, dx &=-\frac{\sqrt{2+x^6}}{18 x^9}-\frac{1}{3} \int \frac{1}{x^4 \sqrt{2+x^6}} \, dx\\ &=-\frac{\sqrt{2+x^6}}{18 x^9}+\frac{\sqrt{2+x^6}}{18 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0042947, size = 21, normalized size = 0.64 \[ \frac{\left (x^6-1\right ) \sqrt{x^6+2}}{18 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^10*Sqrt[2 + x^6]),x]

[Out]

((-1 + x^6)*Sqrt[2 + x^6])/(18*x^9)

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Maple [A]  time = 0.002, size = 18, normalized size = 0.6 \begin{align*}{\frac{{x}^{6}-1}{18\,{x}^{9}}\sqrt{{x}^{6}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(x^6+2)^(1/2),x)

[Out]

1/18*(x^6+2)^(1/2)*(x^6-1)/x^9

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Maxima [A]  time = 0.974661, size = 34, normalized size = 1.03 \begin{align*} \frac{\sqrt{x^{6} + 2}}{12 \, x^{3}} - \frac{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}{36 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

1/12*sqrt(x^6 + 2)/x^3 - 1/36*(x^6 + 2)^(3/2)/x^9

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Fricas [A]  time = 1.64142, size = 57, normalized size = 1.73 \begin{align*} \frac{x^{9} + \sqrt{x^{6} + 2}{\left (x^{6} - 1\right )}}{18 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

1/18*(x^9 + sqrt(x^6 + 2)*(x^6 - 1))/x^9

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Sympy [A]  time = 1.21725, size = 26, normalized size = 0.79 \begin{align*} \frac{\sqrt{1 + \frac{2}{x^{6}}}}{18} - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{18 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(x**6+2)**(1/2),x)

[Out]

sqrt(1 + 2/x**6)/18 - sqrt(1 + 2/x**6)/(18*x**6)

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Giac [A]  time = 1.20563, size = 43, normalized size = 1.3 \begin{align*} -\frac{{\left (\frac{2}{x^{6}} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{\frac{2}{x^{6}} + 1}}{36 \, \mathrm{sgn}\left (x\right )} - \frac{1}{18} \, \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

-1/36*((2/x^6 + 1)^(3/2) - 3*sqrt(2/x^6 + 1))/sgn(x) - 1/18*sgn(x)