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

Optimal. Leaf size=12 \[ x^2+\log \left (x^2+x+1\right )+x \]

[Out]

x + x^2 + Log[1 + x + x^2]

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Rubi [A]  time = 0.0236842, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ x^2+\log \left (x^2+x+1\right )+x \]

Antiderivative was successfully verified.

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

[Out]

x + x^2 + Log[1 + x + x^2]

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Rubi in Sympy [A]  time = 21.6663, size = 12, normalized size = 1. \[ x^{2} + x + \log{\left (x^{2} + x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2 + x + log(x**2 + x + 1)

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Mathematica [A]  time = 0.00860242, size = 12, normalized size = 1. \[ x^2+\log \left (x^2+x+1\right )+x \]

Antiderivative was successfully verified.

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

[Out]

x + x^2 + Log[1 + x + x^2]

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Maple [A]  time = 0.003, size = 13, normalized size = 1.1 \[ x+{x}^{2}+\ln \left ({x}^{2}+x+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x+x^2+ln(x^2+x+1)

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Maxima [A]  time = 0.809502, size = 16, normalized size = 1.33 \[ x^{2} + x + \log \left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2 + x + log(x^2 + x + 1)

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Fricas [A]  time = 0.246551, size = 16, normalized size = 1.33 \[ x^{2} + x + \log \left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2 + x + log(x^2 + x + 1)

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Sympy [A]  time = 0.148496, size = 12, normalized size = 1. \[ x^{2} + x + \log{\left (x^{2} + x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2 + x + log(x**2 + x + 1)

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GIAC/XCAS [A]  time = 0.26417, size = 16, normalized size = 1.33 \[ x^{2} + x +{\rm ln}\left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2 + x + ln(x^2 + x + 1)