3.685 \(\int \frac{x^9}{2+3 x^4} \, dx\)

Optimal. Leaf size=38 \[ \frac{x^6}{18}-\frac{x^2}{9}+\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0513045, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{x^6}{18}-\frac{x^2}{9}+\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^9/(2 + 3*x^4),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{6}}{18} + \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{27} + \frac{\int ^{x^{2}} \left (- \frac{2}{9}\right )\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(3*x**4+2),x)

[Out]

x**6/18 + sqrt(6)*atan(sqrt(6)*x**2/2)/27 + Integral(-2/9, (x, x**2))/2

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Mathematica [A]  time = 0.0285102, size = 34, normalized size = 0.89 \[ \frac{1}{54} \left (3 x^6-6 x^2+2 \sqrt{6} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(2 + 3*x^4),x]

[Out]

(-6*x^2 + 3*x^6 + 2*Sqrt[6]*ArcTan[Sqrt[3/2]*x^2])/54

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Maple [A]  time = 0.005, size = 26, normalized size = 0.7 \[ -{\frac{{x}^{2}}{9}}+{\frac{{x}^{6}}{18}}+{\frac{\sqrt{6}}{27}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(3*x^4+2),x)

[Out]

-1/9*x^2+1/18*x^6+1/27*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.58673, size = 34, normalized size = 0.89 \[ \frac{1}{18} \, x^{6} - \frac{1}{9} \, x^{2} + \frac{1}{27} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(3*x^4 + 2),x, algorithm="maxima")

[Out]

1/18*x^6 - 1/9*x^2 + 1/27*sqrt(6)*arctan(1/2*sqrt(6)*x^2)

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Fricas [A]  time = 0.223301, size = 49, normalized size = 1.29 \[ \frac{1}{54} \, \sqrt{3}{\left (\sqrt{3}{\left (x^{6} - 2 \, x^{2}\right )} + 2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{3} \sqrt{2} x^{2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(3*x^4 + 2),x, algorithm="fricas")

[Out]

1/54*sqrt(3)*(sqrt(3)*(x^6 - 2*x^2) + 2*sqrt(2)*arctan(1/2*sqrt(3)*sqrt(2)*x^2))

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Sympy [A]  time = 0.202807, size = 27, normalized size = 0.71 \[ \frac{x^{6}}{18} - \frac{x^{2}}{9} + \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(3*x**4+2),x)

[Out]

x**6/18 - x**2/9 + sqrt(6)*atan(sqrt(6)*x**2/2)/27

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GIAC/XCAS [A]  time = 0.219429, size = 34, normalized size = 0.89 \[ \frac{1}{18} \, x^{6} - \frac{1}{9} \, x^{2} + \frac{1}{27} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(3*x^4 + 2),x, algorithm="giac")

[Out]

1/18*x^6 - 1/9*x^2 + 1/27*sqrt(6)*arctan(1/2*sqrt(6)*x^2)