3.1392 \(\int \frac{1}{x^{10} \sqrt{2+x^6}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0251987, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{x^6+2}}{18 x^3}-\frac{\sqrt{x^6+2}}{18 x^9} \]

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)

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Rubi in Sympy [A]  time = 3.22543, size = 26, normalized size = 0.79 \[ \frac{\sqrt{x^{6} + 2}}{18 x^{3}} - \frac{\sqrt{x^{6} + 2}}{18 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(x**6+2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0118733, 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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Maple [A]  time = 0.006, size = 18, normalized size = 0.6 \[{\frac{{x}^{6}-1}{18\,{x}^{9}}\sqrt{{x}^{6}+2}} \]

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 = 1.48146, size = 34, normalized size = 1.03 \[ \frac{\sqrt{x^{6} + 2}}{12 \, x^{3}} - \frac{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}{36 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^10),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 = 0.218311, size = 70, normalized size = 2.12 \[ \frac{3 \, x^{6} - 3 \, \sqrt{x^{6} + 2} x^{3} + 2}{18 \,{\left (2 \, x^{18} + 3 \, x^{12} -{\left (2 \, x^{15} + x^{9}\right )} \sqrt{x^{6} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^10),x, algorithm="fricas")

[Out]

1/18*(3*x^6 - 3*sqrt(x^6 + 2)*x^3 + 2)/(2*x^18 + 3*x^12 - (2*x^15 + x^9)*sqrt(x^
6 + 2))

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Sympy [A]  time = 5.2497, 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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GIAC/XCAS [A]  time = 0.229089, size = 43, normalized size = 1.3 \[ -\frac{{\left (\frac{2}{x^{6}} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{\frac{2}{x^{6}} + 1}}{36 \,{\rm sign}\left (x\right )} - \frac{1}{18} \,{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^10),x, algorithm="giac")

[Out]

-1/36*((2/x^6 + 1)^(3/2) - 3*sqrt(2/x^6 + 1))/sign(x) - 1/18*sign(x)