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

Optimal. Leaf size=17 \[ \frac{\sqrt{x^2+2 x}}{x+2} \]

[Out]

Sqrt[2*x + x^2]/(2 + x)

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Rubi [A]  time = 0.0236695, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{\sqrt{x^2+2 x}}{x+2} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[2*x + x^2]/(2 + x)

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Rubi in Sympy [A]  time = 4.16075, size = 12, normalized size = 0.71 \[ \frac{\sqrt{x^{2} + 2 x}}{x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x**2 + 2*x)/(x + 2)

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Mathematica [A]  time = 0.011474, size = 11, normalized size = 0.65 \[ \frac{x}{\sqrt{x (x+2)}} \]

Antiderivative was successfully verified.

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

[Out]

x/Sqrt[x*(2 + x)]

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Maple [A]  time = 0.004, size = 12, normalized size = 0.7 \[{x{\frac{1}{\sqrt{{x}^{2}+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2+x)/(x^2+2*x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.688519, size = 20, normalized size = 1.18 \[ \frac{\sqrt{x^{2} + 2 \, x}}{x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 + 2*x)/(x + 2)

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Fricas [A]  time = 0.21037, size = 24, normalized size = 1.41 \[ \frac{2}{x - \sqrt{x^{2} + 2 \, x} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(x - sqrt(x^2 + 2*x) + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x \left (x + 2\right )} \left (x + 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(x*(x + 2))*(x + 2)), x)

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GIAC/XCAS [A]  time = 0.206661, size = 24, normalized size = 1.41 \[ \frac{2}{x - \sqrt{x^{2} + 2 \, x} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(x - sqrt(x^2 + 2*x) + 2)