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

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 = 0.003884, 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, Rules used = {636} \[ -\frac{2 (x+2)}{3 \sqrt{x^2+x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x/(1 + x + x^2)^(3/2),x]

[Out]

(-2*(2 + x))/(3*Sqrt[1 + x + x^2])

Rule 636

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

Rubi steps

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

Mathematica [A]  time = 0.0338795, 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 = 0.002, size = 14, normalized size = 0.8 \begin{align*} -{\frac{4+2\,x}{3}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \end{align*}

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 = 0.958795, size = 30, normalized size = 1.76 \begin{align*} -\frac{2 \, x}{3 \, \sqrt{x^{2} + x + 1}} - \frac{4}{3 \, \sqrt{x^{2} + x + 1}} \end{align*}

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 [B]  time = 2.05286, size = 84, normalized size = 4.94 \begin{align*} -\frac{2 \,{\left (x^{2} + \sqrt{x^{2} + x + 1}{\left (x + 2\right )} + x + 1\right )}}{3 \,{\left (x^{2} + x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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 [A]  time = 1.06772, size = 18, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (x + 2\right )}}{3 \, \sqrt{x^{2} + x + 1}} \end{align*}

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)