3.1395 \(\int \frac{1}{x^{16} \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\sqrt{x^6+2}}{45 x^3}+\frac{\sqrt{x^6+2}}{45 x^9}-\frac{\sqrt{x^6+2}}{30 x^{15}} \]

[Out]

-Sqrt[2 + x^6]/(30*x^15) + Sqrt[2 + x^6]/(45*x^9) - Sqrt[2 + x^6]/(45*x^3)

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Rubi [A]  time = 0.0100154, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {271, 264} \[ -\frac{\sqrt{x^6+2}}{45 x^3}+\frac{\sqrt{x^6+2}}{45 x^9}-\frac{\sqrt{x^6+2}}{30 x^{15}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^16*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(30*x^15) + Sqrt[2 + x^6]/(45*x^9) - Sqrt[2 + x^6]/(45*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^{16} \sqrt{2+x^6}} \, dx &=-\frac{\sqrt{2+x^6}}{30 x^{15}}-\frac{2}{5} \int \frac{1}{x^{10} \sqrt{2+x^6}} \, dx\\ &=-\frac{\sqrt{2+x^6}}{30 x^{15}}+\frac{\sqrt{2+x^6}}{45 x^9}+\frac{2}{15} \int \frac{1}{x^4 \sqrt{2+x^6}} \, dx\\ &=-\frac{\sqrt{2+x^6}}{30 x^{15}}+\frac{\sqrt{2+x^6}}{45 x^9}-\frac{\sqrt{2+x^6}}{45 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0062481, size = 28, normalized size = 0.57 \[ -\frac{\sqrt{x^6+2} \left (2 x^{12}-2 x^6+3\right )}{90 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^16*Sqrt[2 + x^6]),x]

[Out]

-(Sqrt[2 + x^6]*(3 - 2*x^6 + 2*x^12))/(90*x^15)

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{2\,{x}^{12}-2\,{x}^{6}+3}{90\,{x}^{15}}\sqrt{{x}^{6}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^16/(x^6+2)^(1/2),x)

[Out]

-1/90*(x^6+2)^(1/2)*(2*x^12-2*x^6+3)/x^15

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Maxima [A]  time = 0.998992, size = 50, normalized size = 1.02 \begin{align*} -\frac{\sqrt{x^{6} + 2}}{24 \, x^{3}} + \frac{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}{36 \, x^{9}} - \frac{{\left (x^{6} + 2\right )}^{\frac{5}{2}}}{120 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

-1/24*sqrt(x^6 + 2)/x^3 + 1/36*(x^6 + 2)^(3/2)/x^9 - 1/120*(x^6 + 2)^(5/2)/x^15

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Fricas [A]  time = 1.48217, size = 78, normalized size = 1.59 \begin{align*} -\frac{2 \, x^{15} +{\left (2 \, x^{12} - 2 \, x^{6} + 3\right )} \sqrt{x^{6} + 2}}{90 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

-1/90*(2*x^15 + (2*x^12 - 2*x^6 + 3)*sqrt(x^6 + 2))/x^15

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Sympy [A]  time = 3.61322, size = 41, normalized size = 0.84 \begin{align*} - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{45} + \frac{\sqrt{1 + \frac{2}{x^{6}}}}{45 x^{6}} - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{30 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**16/(x**6+2)**(1/2),x)

[Out]

-sqrt(1 + 2/x**6)/45 + sqrt(1 + 2/x**6)/(45*x**6) - sqrt(1 + 2/x**6)/(30*x**12)

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Giac [A]  time = 1.63931, size = 61, normalized size = 1.24 \begin{align*} -\frac{3 \,{\left (\frac{2}{x^{6}} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (\frac{2}{x^{6}} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{\frac{2}{x^{6}} + 1}}{360 \, \mathrm{sgn}\left (x\right )} + \frac{1}{45} \, \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

-1/360*(3*(2/x^6 + 1)^(5/2) - 10*(2/x^6 + 1)^(3/2) + 15*sqrt(2/x^6 + 1))/sgn(x) + 1/45*sgn(x)