3.2164 \(\int \frac{2+2 x+x^2}{2+x} \, dx\)

Optimal. Leaf size=14 \[ \frac{x^2}{2}+2 \log (x+2) \]

[Out]

x^2/2 + 2*Log[2 + x]

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Rubi [A]  time = 0.0246137, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{x^2}{2}+2 \log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[(2 + 2*x + x^2)/(2 + x),x]

[Out]

x^2/2 + 2*Log[2 + x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 \log{\left (x + 2 \right )} + \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(x + 2) + Integral(x, x)

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Mathematica [A]  time = 0.00669884, size = 15, normalized size = 1.07 \[ \frac{1}{2} \left (x^2+4 \log (x+2)-4\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-4 + x^2 + 4*Log[2 + x])/2

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Maple [A]  time = 0.005, size = 13, normalized size = 0.9 \[{\frac{{x}^{2}}{2}}+2\,\ln \left ( 2+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+2*x+2)/(2+x),x)

[Out]

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

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Maxima [A]  time = 0.805176, size = 16, normalized size = 1.14 \[ \frac{1}{2} \, x^{2} + 2 \, \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + 2*log(x + 2)

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Fricas [A]  time = 0.214002, size = 16, normalized size = 1.14 \[ \frac{1}{2} \, x^{2} + 2 \, \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + 2*log(x + 2)

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Sympy [A]  time = 0.125825, size = 10, normalized size = 0.71 \[ \frac{x^{2}}{2} + 2 \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+2*x+2)/(2+x),x)

[Out]

x**2/2 + 2*log(x + 2)

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GIAC/XCAS [A]  time = 0.203123, size = 18, normalized size = 1.29 \[ \frac{1}{2} \, x^{2} + 2 \,{\rm ln}\left ({\left | x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + 2*ln(abs(x + 2))