3.1417 \(\int \frac{1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{1}{18 \sqrt{x^6+2} x^9}+\frac{x^3}{9 \sqrt{x^6+2}}+\frac{1}{9 \sqrt{x^6+2} x^3} \]

[Out]

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

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Rubi [A]  time = 0.036567, 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 \[ -\frac{1}{18 \sqrt{x^6+2} x^9}+\frac{x^3}{9 \sqrt{x^6+2}}+\frac{1}{9 \sqrt{x^6+2} x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^10*(2 + x^6)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.34828, size = 42, normalized size = 0.86 \[ \frac{x^{3}}{9 \sqrt{x^{6} + 2}} + \frac{1}{9 x^{3} \sqrt{x^{6} + 2}} - \frac{1}{18 x^{9} \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(x**6+2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0165642, size = 28, normalized size = 0.57 \[ \frac{2 x^{12}+2 x^6-1}{18 x^9 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^10*(2 + x^6)^(3/2)),x]

[Out]

(-1 + 2*x^6 + 2*x^12)/(18*x^9*Sqrt[2 + x^6])

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Maple [A]  time = 0.007, size = 25, normalized size = 0.5 \[{\frac{2\,{x}^{12}+2\,{x}^{6}-1}{18\,{x}^{9}}{\frac{1}{\sqrt{{x}^{6}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(x^6+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.43894, size = 50, normalized size = 1.02 \[ \frac{x^{3}}{24 \, \sqrt{x^{6} + 2}} + \frac{\sqrt{x^{6} + 2}}{12 \, x^{3}} - \frac{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}{72 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^10),x, algorithm="maxima")

[Out]

1/24*x^3/sqrt(x^6 + 2) + 1/12*sqrt(x^6 + 2)/x^3 - 1/72*(x^6 + 2)^(3/2)/x^9

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Fricas [A]  time = 0.225412, size = 84, normalized size = 1.71 \[ \frac{2 \, x^{6} - 2 \, \sqrt{x^{6} + 2} x^{3} + 1}{18 \,{\left (2 \, x^{24} + 6 \, x^{18} + 4 \, x^{12} -{\left (2 \, x^{21} + 4 \, x^{15} + x^{9}\right )} \sqrt{x^{6} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^10),x, algorithm="fricas")

[Out]

1/18*(2*x^6 - 2*sqrt(x^6 + 2)*x^3 + 1)/(2*x^24 + 6*x^18 + 4*x^12 - (2*x^21 + 4*x
^15 + x^9)*sqrt(x^6 + 2))

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Sympy [A]  time = 8.89478, size = 70, normalized size = 1.43 \[ \frac{2 x^{12} \sqrt{1 + \frac{2}{x^{6}}}}{18 x^{12} + 36 x^{6}} + \frac{2 x^{6} \sqrt{1 + \frac{2}{x^{6}}}}{18 x^{12} + 36 x^{6}} - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{18 x^{12} + 36 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**10/(x**6+2)**(3/2),x)

[Out]

2*x**12*sqrt(1 + 2/x**6)/(18*x**12 + 36*x**6) + 2*x**6*sqrt(1 + 2/x**6)/(18*x**1
2 + 36*x**6) - sqrt(1 + 2/x**6)/(18*x**12 + 36*x**6)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^10),x, algorithm="giac")

[Out]

integrate(1/((x^6 + 2)^(3/2)*x^10), x)