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

Optimal. Leaf size=36 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

[Out]

(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) + (d*Log[2 + 3*x^4])/12

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Rubi [A]  time = 0.0744684, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(b*x + d*x^3)/(2 + 3*x^4),x]

[Out]

(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) + (d*Log[2 + 3*x^4])/12

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Rubi in Sympy [A]  time = 9.55335, size = 31, normalized size = 0.86 \[ \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} + \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(6)*b*atan(sqrt(6)*x**2/2)/12 + d*log(3*x**4 + 2)/12

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Mathematica [C]  time = 0.0570111, size = 65, normalized size = 1.81 \[ \frac{1}{24} \left (2 d+i \sqrt{6} b\right ) \log \left (\sqrt{6}-3 i x^2\right )+\frac{1}{24} \left (2 d-i \sqrt{6} b\right ) \log \left (\sqrt{6}+3 i x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + d*x^3)/(2 + 3*x^4),x]

[Out]

((I*Sqrt[6]*b + 2*d)*Log[Sqrt[6] - (3*I)*x^2])/24 + (((-I)*Sqrt[6]*b + 2*d)*Log[
Sqrt[6] + (3*I)*x^2])/24

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Maple [A]  time = 0.004, size = 28, normalized size = 0.8 \[{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}}+{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+b*x)/(3*x^4+2),x)

[Out]

1/12*d*ln(3*x^4+2)+1/12*b*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.52483, size = 153, normalized size = 4.25 \[ -\frac{1}{12} \, \sqrt{3} \sqrt{2} b \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{12} \, \sqrt{3} \sqrt{2} b \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{12} \, d \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{12} \, d \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(3)*sqrt(2)*b*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4
))) + 1/12*sqrt(3)*sqrt(2)*b*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2
^(3/4))) + 1/12*d*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/12*d*log(sq
rt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2))

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Fricas [A]  time = 0.228033, size = 42, normalized size = 1.17 \[ \frac{1}{72} \, \sqrt{6}{\left (\sqrt{6} d \log \left (3 \, x^{4} + 2\right ) + 6 \, b \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/72*sqrt(6)*(sqrt(6)*d*log(3*x^4 + 2) + 6*b*arctan(1/2*sqrt(6)*x^2))

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Sympy [A]  time = 0.497149, size = 53, normalized size = 1.47 \[ \left (- \frac{\sqrt{6} i b}{24} + \frac{d}{12}\right ) \log{\left (x^{2} - \frac{\sqrt{6} i}{3} \right )} + \left (\frac{\sqrt{6} i b}{24} + \frac{d}{12}\right ) \log{\left (x^{2} + \frac{\sqrt{6} i}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-sqrt(6)*I*b/24 + d/12)*log(x**2 - sqrt(6)*I/3) + (sqrt(6)*I*b/24 + d/12)*log(x
**2 + sqrt(6)*I/3)

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GIAC/XCAS [A]  time = 0.221105, size = 126, normalized size = 3.5 \[ -\frac{1}{12} \, \sqrt{6} b \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{12} \, \sqrt{6} b \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{12} \, d{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{12} \, d{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(6)*b*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/
12*sqrt(6)*b*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/12*
d*ln(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) + 1/12*d*ln(x^2 - sqrt(2)*(2/3)^(1
/4)*x + sqrt(2/3))