3.1415 \(\int \frac{x^2}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ \frac{x^3}{6 \sqrt{x^6+2}} \]

[Out]

x^3/(6*Sqrt[2 + x^6])

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Rubi [A]  time = 0.0136617, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^3}{6 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 + x^6)^(3/2),x]

[Out]

x^3/(6*Sqrt[2 + x^6])

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Rubi in Sympy [A]  time = 2.28054, size = 12, normalized size = 0.75 \[ \frac{x^{3}}{6 \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/(6*sqrt(x**6 + 2))

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Mathematica [A]  time = 0.00915727, size = 16, normalized size = 1. \[ \frac{x^3}{6 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

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

[Out]

x^3/(6*Sqrt[2 + x^6])

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Maple [A]  time = 0.005, size = 13, normalized size = 0.8 \[{\frac{{x}^{3}}{6}{\frac{1}{\sqrt{{x}^{6}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^6+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.43471, size = 16, normalized size = 1. \[ \frac{x^{3}}{6 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^3/sqrt(x^6 + 2)

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Fricas [A]  time = 0.225076, size = 28, normalized size = 1.75 \[ \frac{1}{3 \,{\left (x^{6} - \sqrt{x^{6} + 2} x^{3} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/(x^6 - sqrt(x^6 + 2)*x^3 + 2)

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Sympy [A]  time = 1.62997, size = 12, normalized size = 0.75 \[ \frac{x^{3}}{6 \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/(6*sqrt(x**6 + 2))

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GIAC/XCAS [A]  time = 0.228558, size = 16, normalized size = 1. \[ \frac{x^{3}}{6 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^3/sqrt(x^6 + 2)