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

Optimal. Leaf size=19 \[ \frac{2 (2 x+1)}{3 \sqrt{x^2+x+1}} \]

[Out]

(2*(1 + 2*x))/(3*Sqrt[1 + x + x^2])

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Rubi [A]  time = 0.0070777, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 (2 x+1)}{3 \sqrt{x^2+x+1}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x + x^2)^(-3/2),x]

[Out]

(2*(1 + 2*x))/(3*Sqrt[1 + x + x^2])

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Rubi in Sympy [A]  time = 0.563205, size = 15, normalized size = 0.79 \[ \frac{4 x + 2}{3 \sqrt{x^{2} + x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4*x + 2)/(3*sqrt(x**2 + x + 1))

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Mathematica [A]  time = 0.010649, size = 19, normalized size = 1. \[ \frac{2 (2 x+1)}{3 \sqrt{x^2+x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(1 + 2*x))/(3*Sqrt[1 + x + x^2])

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \[{\frac{4\,x+2}{3}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+x+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.39681, size = 30, normalized size = 1.58 \[ \frac{4 \, x}{3 \, \sqrt{x^{2} + x + 1}} + \frac{2}{3 \, \sqrt{x^{2} + x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.201689, size = 39, normalized size = 2.05 \[ \frac{2}{2 \, x^{2} - \sqrt{x^{2} + x + 1}{\left (2 \, x + 1\right )} + 2 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x**2 + x + 1)**(-3/2), x)

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GIAC/XCAS [A]  time = 0.2029, size = 20, normalized size = 1.05 \[ \frac{2 \,{\left (2 \, x + 1\right )}}{3 \, \sqrt{x^{2} + x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(2*x + 1)/sqrt(x^2 + x + 1)