3.219 \(\int \frac{1+x}{2 x+x^2} \, dx\)

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

[Out]

Log[2*x + x^2]/2

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Rubi [A]  time = 0.00839859, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{1}{2} \log \left (x^2+2 x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)/(2*x + x^2),x]

[Out]

Log[2*x + x^2]/2

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Rubi in Sympy [A]  time = 2.28037, size = 8, normalized size = 0.67 \[ \frac{\log{\left (x^{2} + 2 x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + 2*x)/2

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Mathematica [A]  time = 0.00366637, size = 15, normalized size = 1.25 \[ \frac{\log (x)}{2}+\frac{1}{2} \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

Log[x]/2 + Log[2 + x]/2

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Maple [A]  time = 0.002, size = 9, normalized size = 0.8 \[{\frac{\ln \left ( x \left ( 2+x \right ) \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(x^2+2*x),x)

[Out]

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

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Maxima [A]  time = 0.727553, size = 14, normalized size = 1.17 \[ \frac{1}{2} \, \log \left (x^{2} + 2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(x^2 + 2*x)

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Fricas [A]  time = 0.214921, size = 14, normalized size = 1.17 \[ \frac{1}{2} \, \log \left (x^{2} + 2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(x^2 + 2*x)

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Sympy [A]  time = 0.142941, size = 8, normalized size = 0.67 \[ \frac{\log{\left (x^{2} + 2 x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)/(x**2+2*x),x)

[Out]

log(x**2 + 2*x)/2

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GIAC/XCAS [A]  time = 0.205228, size = 18, normalized size = 1.5 \[ \frac{1}{2} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(abs(x + 2)) + 1/2*ln(abs(x))