3.292 \(\int \frac{5+2 x-x^2+x^3}{1+x+x^2} \, dx\)

Optimal. Leaf size=41 \[ \frac{x^2}{2}+\frac{3}{2} \log \left (x^2+x+1\right )-2 x+\frac{11 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-2*x + x^2/2 + (11*ArcTan[(1 + 2*x)/Sqrt[3]])/Sqrt[3] + (3*Log[1 + x + x^2])/2

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Rubi [A]  time = 0.0629781, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{x^2}{2}+\frac{3}{2} \log \left (x^2+x+1\right )-2 x+\frac{11 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(5 + 2*x - x^2 + x^3)/(1 + x + x^2),x]

[Out]

-2*x + x^2/2 + (11*ArcTan[(1 + 2*x)/Sqrt[3]])/Sqrt[3] + (3*Log[1 + x + x^2])/2

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Rubi in Sympy [A]  time = 22.9296, size = 42, normalized size = 1.02 \[ \frac{x^{2}}{2} - 2 x + \frac{3 \log{\left (x^{2} + x + 1 \right )}}{2} + \frac{11 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3-x**2+2*x+5)/(x**2+x+1),x)

[Out]

x**2/2 - 2*x + 3*log(x**2 + x + 1)/2 + 11*sqrt(3)*atan(sqrt(3)*(2*x/3 + 1/3))/3

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Mathematica [A]  time = 0.0247558, size = 41, normalized size = 1. \[ \frac{x^2}{2}+\frac{3}{2} \log \left (x^2+x+1\right )-2 x+\frac{11 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 + 2*x - x^2 + x^3)/(1 + x + x^2),x]

[Out]

-2*x + x^2/2 + (11*ArcTan[(1 + 2*x)/Sqrt[3]])/Sqrt[3] + (3*Log[1 + x + x^2])/2

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Maple [A]  time = 0.004, size = 35, normalized size = 0.9 \[ -2\,x+{\frac{{x}^{2}}{2}}+{\frac{3\,\ln \left ({x}^{2}+x+1 \right ) }{2}}+{\frac{11\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3-x^2+2*x+5)/(x^2+x+1),x)

[Out]

-2*x+1/2*x^2+3/2*ln(x^2+x+1)+11/3*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)

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Maxima [A]  time = 0.881089, size = 46, normalized size = 1.12 \[ \frac{1}{2} \, x^{2} + \frac{11}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 2 \, x + \frac{3}{2} \, \log \left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - x^2 + 2*x + 5)/(x^2 + x + 1),x, algorithm="maxima")

[Out]

1/2*x^2 + 11/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 2*x + 3/2*log(x^2 + x + 1
)

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Fricas [A]  time = 0.249227, size = 57, normalized size = 1.39 \[ \frac{1}{6} \, \sqrt{3}{\left (\sqrt{3}{\left (x^{2} - 4 \, x\right )} + 3 \, \sqrt{3} \log \left (x^{2} + x + 1\right ) + 22 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - x^2 + 2*x + 5)/(x^2 + x + 1),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*(sqrt(3)*(x^2 - 4*x) + 3*sqrt(3)*log(x^2 + x + 1) + 22*arctan(1/3*sq
rt(3)*(2*x + 1)))

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Sympy [A]  time = 0.230536, size = 46, normalized size = 1.12 \[ \frac{x^{2}}{2} - 2 x + \frac{3 \log{\left (x^{2} + x + 1 \right )}}{2} + \frac{11 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3-x**2+2*x+5)/(x**2+x+1),x)

[Out]

x**2/2 - 2*x + 3*log(x**2 + x + 1)/2 + 11*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3
)/3

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GIAC/XCAS [A]  time = 0.262447, size = 46, normalized size = 1.12 \[ \frac{1}{2} \, x^{2} + \frac{11}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 2 \, x + \frac{3}{2} \,{\rm ln}\left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - x^2 + 2*x + 5)/(x^2 + x + 1),x, algorithm="giac")

[Out]

1/2*x^2 + 11/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 2*x + 3/2*ln(x^2 + x + 1)