3.327 \(\int \frac{2+x}{2+x+x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{2} \log \left (x^2+x+2\right )+\frac{3 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{7}}\right )}{\sqrt{7}} \]

[Out]

(3*ArcTan[(1 + 2*x)/Sqrt[7]])/Sqrt[7] + Log[2 + x + x^2]/2

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Rubi [A]  time = 0.0145056, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {634, 618, 204, 628} \[ \frac{1}{2} \log \left (x^2+x+2\right )+\frac{3 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{7}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + x)/(2 + x + x^2),x]

[Out]

(3*ArcTan[(1 + 2*x)/Sqrt[7]])/Sqrt[7] + Log[2 + x + x^2]/2

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

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

Rule 204

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

Rule 628

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

Rubi steps

\begin{align*} \int \frac{2+x}{2+x+x^2} \, dx &=\frac{1}{2} \int \frac{1+2 x}{2+x+x^2} \, dx+\frac{3}{2} \int \frac{1}{2+x+x^2} \, dx\\ &=\frac{1}{2} \log \left (2+x+x^2\right )-3 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,1+2 x\right )\\ &=\frac{3 \tan ^{-1}\left (\frac{1+2 x}{\sqrt{7}}\right )}{\sqrt{7}}+\frac{1}{2} \log \left (2+x+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0098404, size = 31, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+x+2\right )+\frac{3 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{7}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x)/(2 + x + x^2),x]

[Out]

(3*ArcTan[(1 + 2*x)/Sqrt[7]])/Sqrt[7] + Log[2 + x + x^2]/2

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Maple [A]  time = 0.001, size = 27, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({x}^{2}+x+2 \right ) }{2}}+{\frac{3\,\sqrt{7}}{7}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{7}}{7}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(x^2+x+2)+3/7*arctan(1/7*(1+2*x)*7^(1/2))*7^(1/2)

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Maxima [A]  time = 1.40508, size = 35, normalized size = 1.13 \begin{align*} \frac{3}{7} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2} \, \log \left (x^{2} + x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*sqrt(7)*arctan(1/7*sqrt(7)*(2*x + 1)) + 1/2*log(x^2 + x + 2)

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Fricas [A]  time = 2.11347, size = 89, normalized size = 2.87 \begin{align*} \frac{3}{7} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2} \, \log \left (x^{2} + x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*sqrt(7)*arctan(1/7*sqrt(7)*(2*x + 1)) + 1/2*log(x^2 + x + 2)

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Sympy [A]  time = 0.099768, size = 36, normalized size = 1.16 \begin{align*} \frac{\log{\left (x^{2} + x + 2 \right )}}{2} + \frac{3 \sqrt{7} \operatorname{atan}{\left (\frac{2 \sqrt{7} x}{7} + \frac{\sqrt{7}}{7} \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 + x + 2)/2 + 3*sqrt(7)*atan(2*sqrt(7)*x/7 + sqrt(7)/7)/7

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Giac [A]  time = 1.05416, size = 35, normalized size = 1.13 \begin{align*} \frac{3}{7} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2} \, \log \left (x^{2} + x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*sqrt(7)*arctan(1/7*sqrt(7)*(2*x + 1)) + 1/2*log(x^2 + x + 2)