3.269 \(\int \frac{x}{\left (1+x+x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=17 \[ -\frac{2 (x+2)}{3 \sqrt{x^2+x+1}} \]

[Out]

(-2*(2 + x))/(3*Sqrt[1 + x + x^2])

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Rubi [A]  time = 0.0123177, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 (x+2)}{3 \sqrt{x^2+x+1}} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + x + x^2)^(3/2),x]

[Out]

(-2*(2 + x))/(3*Sqrt[1 + x + x^2])

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Rubi in Sympy [A]  time = 1.37436, size = 17, normalized size = 1. \[ - \frac{2 x + 4}{3 \sqrt{x^{2} + x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 4)/(3*sqrt(x**2 + x + 1))

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Mathematica [A]  time = 0.0111808, size = 17, normalized size = 1. \[ -\frac{2 (x+2)}{3 \sqrt{x^2+x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(2 + x))/(3*Sqrt[1 + x + x^2])

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Maple [A]  time = 0.006, size = 14, normalized size = 0.8 \[ -{\frac{2\,x+4}{3}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^2+x+1)^(3/2),x)

[Out]

-2/3*(2+x)/(x^2+x+1)^(1/2)

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Maxima [A]  time = 1.38131, size = 30, normalized size = 1.76 \[ -\frac{2 \, x}{3 \, \sqrt{x^{2} + x + 1}} - \frac{4}{3 \, \sqrt{x^{2} + x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*x/sqrt(x^2 + x + 1) - 4/3/sqrt(x^2 + x + 1)

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Fricas [A]  time = 0.200396, size = 55, normalized size = 3.24 \[ \frac{2 \,{\left (x - \sqrt{x^{2} + x + 1}\right )}}{2 \, x^{2} - \sqrt{x^{2} + x + 1}{\left (2 \, x + 1\right )} + 2 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (x^{2} + x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(x**2 + x + 1)**(3/2), x)

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GIAC/XCAS [A]  time = 0.201742, size = 18, normalized size = 1.06 \[ -\frac{2 \,{\left (x + 2\right )}}{3 \, \sqrt{x^{2} + x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(x + 2)/sqrt(x^2 + x + 1)