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

Optimal. Leaf size=49 \[ -\frac{\sqrt{x^2+2}}{15 x}+\frac{\sqrt{x^2+2}}{15 x^3}-\frac{\sqrt{x^2+2}}{10 x^5} \]

[Out]

-Sqrt[2 + x^2]/(10*x^5) + Sqrt[2 + x^2]/(15*x^3) - Sqrt[2 + x^2]/(15*x)

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Rubi [A]  time = 0.0110739, 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} \[ -\frac{\sqrt{x^2+2}}{15 x}+\frac{\sqrt{x^2+2}}{15 x^3}-\frac{\sqrt{x^2+2}}{10 x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*Sqrt[2 + x^2]),x]

[Out]

-Sqrt[2 + x^2]/(10*x^5) + Sqrt[2 + x^2]/(15*x^3) - Sqrt[2 + x^2]/(15*x)

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^6 \sqrt{2+x^2}} \, dx &=-\frac{\sqrt{2+x^2}}{10 x^5}-\frac{2}{5} \int \frac{1}{x^4 \sqrt{2+x^2}} \, dx\\ &=-\frac{\sqrt{2+x^2}}{10 x^5}+\frac{\sqrt{2+x^2}}{15 x^3}+\frac{2}{15} \int \frac{1}{x^2 \sqrt{2+x^2}} \, dx\\ &=-\frac{\sqrt{2+x^2}}{10 x^5}+\frac{\sqrt{2+x^2}}{15 x^3}-\frac{\sqrt{2+x^2}}{15 x}\\ \end{align*}

Mathematica [A]  time = 0.006025, size = 28, normalized size = 0.57 \[ -\frac{\sqrt{x^2+2} \left (2 x^4-2 x^2+3\right )}{30 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*Sqrt[2 + x^2]),x]

[Out]

-(Sqrt[2 + x^2]*(3 - 2*x^2 + 2*x^4))/(30*x^5)

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{2\,{x}^{4}-2\,{x}^{2}+3}{30\,{x}^{5}}\sqrt{{x}^{2}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(x^2+2)^(1/2)*(2*x^4-2*x^2+3)/x^5

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Maxima [A]  time = 1.46857, size = 50, normalized size = 1.02 \begin{align*} -\frac{\sqrt{x^{2} + 2}}{15 \, x} + \frac{\sqrt{x^{2} + 2}}{15 \, x^{3}} - \frac{\sqrt{x^{2} + 2}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*sqrt(x^2 + 2)/x + 1/15*sqrt(x^2 + 2)/x^3 - 1/10*sqrt(x^2 + 2)/x^5

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Fricas [A]  time = 1.8217, size = 74, normalized size = 1.51 \begin{align*} -\frac{2 \, x^{5} +{\left (2 \, x^{4} - 2 \, x^{2} + 3\right )} \sqrt{x^{2} + 2}}{30 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(2*x^5 + (2*x^4 - 2*x^2 + 3)*sqrt(x^2 + 2))/x^5

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Sympy [A]  time = 4.00259, size = 41, normalized size = 0.84 \begin{align*} - \frac{\sqrt{1 + \frac{2}{x^{2}}}}{15} + \frac{\sqrt{1 + \frac{2}{x^{2}}}}{15 x^{2}} - \frac{\sqrt{1 + \frac{2}{x^{2}}}}{10 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 + 2/x**2)/15 + sqrt(1 + 2/x**2)/(15*x**2) - sqrt(1 + 2/x**2)/(10*x**4)

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Giac [A]  time = 1.07276, size = 69, normalized size = 1.41 \begin{align*} \frac{32 \,{\left (5 \,{\left (x - \sqrt{x^{2} + 2}\right )}^{4} - 5 \,{\left (x - \sqrt{x^{2} + 2}\right )}^{2} + 2\right )}}{15 \,{\left ({\left (x - \sqrt{x^{2} + 2}\right )}^{2} - 2\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

32/15*(5*(x - sqrt(x^2 + 2))^4 - 5*(x - sqrt(x^2 + 2))^2 + 2)/((x - sqrt(x^2 + 2))^2 - 2)^5