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

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

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Rubi [A]  time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, 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]

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

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IntegrateAlgebraic [A]  time = 0.03, 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]

IntegrateAlgebraic[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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fricas [A]  time = 0.64, size = 31, normalized size = 0.63 \[ -\frac {2 \, x^{5} + {\left (2 \, x^{4} - 2 \, x^{2} + 3\right )} \sqrt {x^{2} + 2}}{30 \, x^{5}} \]

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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giac [A]  time = 0.63, size = 51, normalized size = 1.04 \[ \frac {32 \, {\left (5 \, {\left (x - \sqrt {x^{2} + 2}\right )}^{4} - 5 \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2} + 2\right )}}{15 \, {\left ({\left (x - \sqrt {x^{2} + 2}\right )}^{2} - 2\right )}^{5}} \]

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

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maple [A]  time = 0.32, size = 25, normalized size = 0.51

method	result	size
gosper	\(-\frac {\sqrt {x^{2}+2}\, \left (2 x^{4}-2 x^{2}+3\right )}{30 x^{5}}\)	\(25\)
trager	\(-\frac {\sqrt {x^{2}+2}\, \left (2 x^{4}-2 x^{2}+3\right )}{30 x^{5}}\)	\(25\)
meijerg	\(-\frac {\sqrt {2}\, \left (\frac {2}{3} x^{4}-\frac {2}{3} x^{2}+1\right ) \sqrt {1+\frac {x^{2}}{2}}}{10 x^{5}}\)	\(30\)
risch	\(-\frac {2 x^{6}+2 x^{4}-x^{2}+6}{30 x^{5} \sqrt {x^{2}+2}}\)	\(30\)
default	\(-\frac {\sqrt {x^{2}+2}}{10 x^{5}}+\frac {\sqrt {x^{2}+2}}{15 x^{3}}-\frac {\sqrt {x^{2}+2}}{15 x}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 1.09, size = 37, normalized size = 0.76 \[ -\frac {\sqrt {x^{2} + 2}}{15 \, x} + \frac {\sqrt {x^{2} + 2}}{15 \, x^{3}} - \frac {\sqrt {x^{2} + 2}}{10 \, x^{5}} \]

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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mupad [B]  time = 0.03, size = 25, normalized size = 0.51 \[ -\sqrt {x^2+2}\,\left (\frac {1}{15\,x}-\frac {1}{15\,x^3}+\frac {1}{10\,x^5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(x^2 + 2)^(1/2)),x)

[Out]

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

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sympy [A]  time = 2.49, size = 41, normalized size = 0.84 \[ - \frac {\sqrt {1 + \frac {2}{x^{2}}}}{15} + \frac {\sqrt {1 + \frac {2}{x^{2}}}}{15 x^{2}} - \frac {\sqrt {1 + \frac {2}{x^{2}}}}{10 x^{4}} \]

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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