3.840 \(\int \frac{x}{(x (2+x))^{3/2}} \, dx\)

Optimal. Leaf size=13 \[ \frac{x}{\sqrt{x^2+2 x}} \]

[Out]

x/Sqrt[2*x + x^2]

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Rubi [A]  time = 0.0252502, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x}{\sqrt{x^2+2 x}} \]

Antiderivative was successfully verified.

[In]  Int[x/(x*(2 + x))^(3/2),x]

[Out]

x/Sqrt[2*x + x^2]

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Rubi in Sympy [A]  time = 1.63904, size = 10, normalized size = 0.77 \[ \frac{x}{\sqrt{x^{2} + 2 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/sqrt(x**2 + 2*x)

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Mathematica [A]  time = 0.00911312, size = 11, normalized size = 0.85 \[ \frac{x}{\sqrt{x (x+2)}} \]

Antiderivative was successfully verified.

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

[Out]

x/Sqrt[x*(2 + x)]

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Maple [A]  time = 0.006, size = 15, normalized size = 1.2 \[{{x}^{2} \left ( 2+x \right ) \left ( x \left ( 2+x \right ) \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x*(2+x))^(3/2),x)

[Out]

x^2*(2+x)/(x*(2+x))^(3/2)

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Maxima [A]  time = 0.680424, size = 15, normalized size = 1.15 \[ \frac{x}{\sqrt{x^{2} + 2 \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/sqrt(x^2 + 2*x)

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Fricas [A]  time = 0.266967, size = 24, normalized size = 1.85 \[ \frac{2}{x - \sqrt{x^{2} + 2 \, x} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x + 2)*x)^(3/2),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. \[ \int \frac{x}{\left (x \left (x + 2\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x*(2+x))**(3/2),x)

[Out]

Integral(x/(x*(x + 2))**(3/2), x)

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GIAC/XCAS [A]  time = 0.261805, size = 22, normalized size = 1.69 \[ \frac{2}{x - \sqrt{{\left (x + 2\right )} x} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(x - sqrt((x + 2)*x) + 2)