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

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 = 0.0019644, 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, Rules used = {613} \[ \frac{2 (2 x+1)}{3 \sqrt{x^2+x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x + x^2)^(-3/2),x]

[Out]

(2*(1 + 2*x))/(3*Sqrt[1 + x + x^2])

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (1+x+x^2\right )^{3/2}} \, dx &=\frac{2 (1+2 x)}{3 \sqrt{1+x+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0041424, 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 = 0.004, size = 16, normalized size = 0.8 \begin{align*}{\frac{2+4\,x}{3}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \end{align*}

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 = 0.942825, size = 30, normalized size = 1.58 \begin{align*} \frac{4 \, x}{3 \, \sqrt{x^{2} + x + 1}} + \frac{2}{3 \, \sqrt{x^{2} + x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 [B]  time = 2.24154, size = 90, normalized size = 4.74 \begin{align*} \frac{2 \,{\left (2 \, x^{2} + \sqrt{x^{2} + x + 1}{\left (2 \, x + 1\right )} + 2 \, x + 2\right )}}{3 \,{\left (x^{2} + x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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 [A]  time = 1.0733, size = 20, normalized size = 1.05 \begin{align*} \frac{2 \,{\left (2 \, x + 1\right )}}{3 \, \sqrt{x^{2} + x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(2*x + 1)/sqrt(x^2 + x + 1)