3.14.94 \(\int \frac {1}{x^{10} \sqrt {2+x^6}} \, dx\) [1394]

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

[Out]

-1/18*(x^6+2)^(1/2)/x^9+1/18*(x^6+2)^(1/2)/x^3

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Rubi [A]
time = 0.00, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {277, 270} \begin {gather*} \frac {\sqrt {x^6+2}}{18 x^3}-\frac {\sqrt {x^6+2}}{18 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^10*Sqrt[2 + x^6]),x]

[Out]

-1/18*Sqrt[2 + x^6]/x^9 + Sqrt[2 + x^6]/(18*x^3)

Rule 270

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 277

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^{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*}

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Mathematica [A]
time = 0.12, size = 21, normalized size = 0.64 \begin {gather*} \frac {\left (-1+x^6\right ) \sqrt {2+x^6}}{18 x^9} \end {gather*}

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 = 0.18, size = 18, normalized size = 0.55

method	result	size
gosper	\(\frac {\sqrt {x^{6}+2}\, \left (x^{6}-1\right )}{18 x^{9}}\)	\(18\)
risch	\(\frac {x^{12}+x^{6}-2}{18 x^{9} \sqrt {x^{6}+2}}\)	\(21\)
meijerg	\(-\frac {\sqrt {2}\, \left (-x^{6}+1\right ) \sqrt {1+\frac {x^{6}}{2}}}{18 x^{9}}\)	\(25\)
trager	\(\frac {\left (x -1\right ) \left (x^{5}+x^{4}+x^{3}+x^{2}+x +1\right ) \sqrt {x^{6}+2}}{18 x^{9}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 25, normalized size = 0.76 \begin {gather*} \frac {\sqrt {x^{6} + 2}}{12 \, x^{3}} - \frac {{\left (x^{6} + 2\right )}^{\frac {3}{2}}}{36 \, x^{9}} \end {gather*}

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 = 0.37, size = 22, normalized size = 0.67 \begin {gather*} \frac {x^{9} + \sqrt {x^{6} + 2} {\left (x^{6} - 1\right )}}{18 \, x^{9}} \end {gather*}

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 = 0.54, size = 26, normalized size = 0.79 \begin {gather*} \frac {\sqrt {1 + \frac {2}{x^{6}}}}{18} - \frac {\sqrt {1 + \frac {2}{x^{6}}}}{18 x^{6}} \end {gather*}

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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [B]
time = 1.17, size = 17, normalized size = 0.52 \begin {gather*} \frac {\left (x^6-1\right )\,\sqrt {x^6+2}}{18\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^10*(x^6 + 2)^(1/2)),x)

[Out]

((x^6 - 1)*(x^6 + 2)^(1/2))/(18*x^9)

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