3.687 \(\int \frac{x^5}{2+3 x^4} \, dx\)

Optimal. Leaf size=29 \[ \frac{x^2}{6}-\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{3 \sqrt{6}} \]

[Out]

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

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Rubi [A]  time = 0.0376713, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{x^2}{6}-\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{3 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(2 + 3*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.47535, size = 22, normalized size = 0.76 \[ \frac{x^{2}}{6} - \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(3*x**4+2),x)

[Out]

x**2/6 - sqrt(6)*atan(sqrt(6)*x**2/2)/18

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Mathematica [A]  time = 0.0186189, size = 29, normalized size = 1. \[ \frac{x^2}{6}-\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{3 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(2 + 3*x^4),x]

[Out]

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

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Maple [A]  time = 0.003, size = 21, normalized size = 0.7 \[{\frac{{x}^{2}}{6}}-{\frac{\sqrt{6}}{18}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(3*x^4+2),x)

[Out]

1/6*x^2-1/18*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.5861, size = 27, normalized size = 0.93 \[ \frac{1}{6} \, x^{2} - \frac{1}{18} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(3*x^4 + 2),x, algorithm="maxima")

[Out]

1/6*x^2 - 1/18*sqrt(6)*arctan(1/2*sqrt(6)*x^2)

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Fricas [A]  time = 0.22674, size = 32, normalized size = 1.1 \[ \frac{1}{36} \, \sqrt{6}{\left (\sqrt{6} x^{2} - 2 \, \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(3*x^4 + 2),x, algorithm="fricas")

[Out]

1/36*sqrt(6)*(sqrt(6)*x^2 - 2*arctan(1/2*sqrt(6)*x^2))

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Sympy [A]  time = 0.200022, size = 22, normalized size = 0.76 \[ \frac{x^{2}}{6} - \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(3*x**4+2),x)

[Out]

x**2/6 - sqrt(6)*atan(sqrt(6)*x**2/2)/18

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GIAC/XCAS [A]  time = 0.219578, size = 27, normalized size = 0.93 \[ \frac{1}{6} \, x^{2} - \frac{1}{18} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(3*x^4 + 2),x, algorithm="giac")

[Out]

1/6*x^2 - 1/18*sqrt(6)*arctan(1/2*sqrt(6)*x^2)