3.1416 \(\int \frac{1}{x^4 \left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{x^3}{6 \sqrt{x^6+2}}-\frac{1}{6 \sqrt{x^6+2} x^3} \]

[Out]

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

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Rubi [A]  time = 0.0243014, 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 \[ -\frac{x^3}{6 \sqrt{x^6+2}}-\frac{1}{6 \sqrt{x^6+2} x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(2 + x^6)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.20135, size = 29, normalized size = 0.88 \[ - \frac{x^{3}}{6 \sqrt{x^{6} + 2}} - \frac{1}{6 x^{3} \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(x**6+2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.01267, size = 21, normalized size = 0.64 \[ -\frac{x^6+1}{6 x^3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(2 + x^6)^(3/2)),x]

[Out]

-(1 + x^6)/(6*x^3*Sqrt[2 + x^6])

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Maple [A]  time = 0.006, size = 18, normalized size = 0.6 \[ -{\frac{{x}^{6}+1}{6\,{x}^{3}}{\frac{1}{\sqrt{{x}^{6}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^6+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44011, size = 34, normalized size = 1.03 \[ -\frac{x^{3}}{12 \, \sqrt{x^{6} + 2}} - \frac{\sqrt{x^{6} + 2}}{12 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.218174, size = 39, normalized size = 1.18 \[ \frac{1}{6 \,{\left (x^{12} + 2 \, x^{6} -{\left (x^{9} + x^{3}\right )} \sqrt{x^{6} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.82349, size = 31, normalized size = 0.94 \[ - \frac{1}{6 \sqrt{1 + \frac{2}{x^{6}}}} - \frac{1}{6 x^{6} \sqrt{1 + \frac{2}{x^{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(x**6+2)**(3/2),x)

[Out]

-1/(6*sqrt(1 + 2/x**6)) - 1/(6*x**6*sqrt(1 + 2/x**6))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{6} + 2\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^4),x, algorithm="giac")

[Out]

integrate(1/((x^6 + 2)^(3/2)*x^4), x)