3.153 \(\int \frac{x}{\sqrt{1+x+x^2}} \, dx\)

Optimal. Leaf size=27 \[ \sqrt{x^2+x+1}-\frac{1}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

[Out]

Sqrt[1 + x + x^2] - ArcSinh[(1 + 2*x)/Sqrt[3]]/2

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Rubi [A]  time = 0.0109699, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {640, 619, 215} \[ \sqrt{x^2+x+1}-\frac{1}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[1 + x + x^2],x]

[Out]

Sqrt[1 + x + x^2] - ArcSinh[(1 + 2*x)/Sqrt[3]]/2

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0038627, size = 27, normalized size = 1. \[ \sqrt{x^2+x+1}-\frac{1}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[1 + x + x^2],x]

[Out]

Sqrt[1 + x + x^2] - ArcSinh[(1 + 2*x)/Sqrt[3]]/2

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Maple [A]  time = 0.004, size = 21, normalized size = 0.8 \begin{align*} \sqrt{{x}^{2}+x+1}-{\frac{1}{2}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2+x+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.39499, size = 30, normalized size = 1.11 \begin{align*} \sqrt{x^{2} + x + 1} - \frac{1}{2} \, \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.00863, size = 82, normalized size = 3.04 \begin{align*} \sqrt{x^{2} + x + 1} + \frac{1}{2} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2+x+1)**(1/2),x)

[Out]

Integral(x/sqrt(x**2 + x + 1), x)

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Giac [A]  time = 1.08945, size = 36, normalized size = 1.33 \begin{align*} \sqrt{x^{2} + x + 1} + \frac{1}{2} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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