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

Optimal. Leaf size=71 \[ \frac{2}{3} \log \left (x^2+x+1\right )-\frac{1}{6} \log \left (2 x^2-x+2\right )-\frac{1}{3} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{8 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

-(Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/3 + (8*ArcTan[(1 + 2*x)/Sqrt[3]])/(3*Sqr
t[3]) + (2*Log[1 + x + x^2])/3 - Log[2 - x + 2*x^2]/6

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Rubi [A]  time = 0.166109, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{2}{3} \log \left (x^2+x+1\right )-\frac{1}{6} \log \left (2 x^2-x+2\right )-\frac{1}{3} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{8 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/3 + (8*ArcTan[(1 + 2*x)/Sqrt[3]])/(3*Sqr
t[3]) + (2*Log[1 + x + x^2])/3 - Log[2 - x + 2*x^2]/6

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.0284967, size = 65, normalized size = 0.92 \[ \frac{1}{18} \left (12 \log \left (x^2+x+1\right )-3 \log \left (2 x^2-x+2\right )+16 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )+2 \sqrt{15} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{15}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(16*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + 2*Sqrt[15]*ArcTan[(-1 + 4*x)/Sqrt[15]] +
12*Log[1 + x + x^2] - 3*Log[2 - x + 2*x^2])/18

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Maple [A]  time = 0.006, size = 56, normalized size = 0.8 \[{\frac{2\,\ln \left ({x}^{2}+x+1 \right ) }{3}}+{\frac{8\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 2\,{x}^{2}-x+2 \right ) }{6}}+{\frac{\sqrt{15}}{9}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{15}}{15}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*ln(x^2+x+1)+8/9*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/6*ln(2*x^2-x+2)+1/9*15
^(1/2)*arctan(1/15*(-1+4*x)*15^(1/2))

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Maxima [A]  time = 0.895591, size = 74, normalized size = 1.04 \[ \frac{1}{9} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) + \frac{8}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{2}{3} \, \log \left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) + 8/9*sqrt(3)*arctan(1/3*sqrt(3)*(2
*x + 1)) - 1/6*log(2*x^2 - x + 2) + 2/3*log(x^2 + x + 1)

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Fricas [A]  time = 0.275955, size = 89, normalized size = 1.25 \[ \frac{1}{18} \, \sqrt{3}{\left (2 \, \sqrt{5} \arctan \left (\frac{1}{15} \, \sqrt{5} \sqrt{3}{\left (4 \, x - 1\right )}\right ) - \sqrt{3} \log \left (2 \, x^{2} - x + 2\right ) + 4 \, \sqrt{3} \log \left (x^{2} + x + 1\right ) + 16 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(2*sqrt(5)*arctan(1/15*sqrt(5)*sqrt(3)*(4*x - 1)) - sqrt(3)*log(2*x
^2 - x + 2) + 4*sqrt(3)*log(x^2 + x + 1) + 16*arctan(1/3*sqrt(3)*(2*x + 1)))

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Sympy [A]  time = 0.663386, size = 75, normalized size = 1.06 \[ - \frac{\log{\left (x^{2} - \frac{x}{2} + 1 \right )}}{6} + \frac{2 \log{\left (x^{2} + x + 1 \right )}}{3} + \frac{\sqrt{15} \operatorname{atan}{\left (\frac{4 \sqrt{15} x}{15} - \frac{\sqrt{15}}{15} \right )}}{9} + \frac{8 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**2 - x/2 + 1)/6 + 2*log(x**2 + x + 1)/3 + sqrt(15)*atan(4*sqrt(15)*x/15 -
sqrt(15)/15)/9 + 8*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/9

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GIAC/XCAS [A]  time = 0.263081, size = 74, normalized size = 1.04 \[ \frac{1}{9} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) + \frac{8}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \,{\rm ln}\left (2 \, x^{2} - x + 2\right ) + \frac{2}{3} \,{\rm ln}\left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) + 8/9*sqrt(3)*arctan(1/3*sqrt(3)*(2
*x + 1)) - 1/6*ln(2*x^2 - x + 2) + 2/3*ln(x^2 + x + 1)