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]

-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 = 0.01, 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 = {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 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^{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 = 0.00, 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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fricas [A]  time = 0.83, size = 22, normalized size = 0.67 \[ \frac {x^{9} + \sqrt {x^{6} + 2} {\left (x^{6} - 1\right )}}{18 \, x^{9}} \]

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

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,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is
real):Check [abs(x)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Val
ue

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maple [A]  time = 0.00, size = 18, normalized size = 0.55 \[ \frac {\sqrt {x^{6}+2}\, \left (x^{6}-1\right )}{18 x^{9}} \]

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 = 0.99, size = 25, normalized size = 0.76 \[ \frac {\sqrt {x^{6} + 2}}{12 \, x^{3}} - \frac {{\left (x^{6} + 2\right )}^{\frac {3}{2}}}{36 \, x^{9}} \]

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

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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sympy [A]  time = 1.98, size = 26, normalized size = 0.79 \[ \frac {\sqrt {1 + \frac {2}{x^{6}}}}{18} - \frac {\sqrt {1 + \frac {2}{x^{6}}}}{18 x^{6}} \]

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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