3.153 \(\int \frac{b x}{2+3 x^4} \, dx\)

Optimal. Leaf size=22 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

[Out]

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

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Rubi [A]  time = 0.0362842, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x)/(2 + 3*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.87435, size = 19, normalized size = 0.86 \[ \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(6)*b*atan(sqrt(6)*x**2/2)/12

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Mathematica [A]  time = 0.0182195, size = 22, normalized size = 1. \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x)/(2 + 3*x^4),x]

[Out]

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

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Maple [A]  time = 0.002, size = 16, normalized size = 0.7 \[{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(b*x/(3*x^4+2),x)

[Out]

1/12*b*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.51463, size = 20, normalized size = 0.91 \[ \frac{1}{12} \, \sqrt{6} b \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(6)*b*arctan(1/2*sqrt(6)*x^2)

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Fricas [A]  time = 0.219502, size = 20, normalized size = 0.91 \[ \frac{1}{12} \, \sqrt{6} b \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(6)*b*arctan(1/2*sqrt(6)*x^2)

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Sympy [A]  time = 0.103153, size = 19, normalized size = 0.86 \[ \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(b*x/(3*x**4+2),x)

[Out]

sqrt(6)*b*atan(sqrt(6)*x**2/2)/12

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GIAC/XCAS [A]  time = 0.213107, size = 20, normalized size = 0.91 \[ \frac{1}{12} \, \sqrt{6} b \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(6)*b*arctan(1/2*sqrt(6)*x^2)