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

Optimal. Leaf size=51 \[ \frac{1}{10} \left (5+3 \sqrt{5}\right ) \log \left (2 x-\sqrt{5}+3\right )+\frac{1}{10} \left (5-3 \sqrt{5}\right ) \log \left (2 x+\sqrt{5}+3\right ) \]

[Out]

((5 + 3*Sqrt[5])*Log[3 - Sqrt[5] + 2*x])/10 + ((5 - 3*Sqrt[5])*Log[3 + Sqrt[5] + 2*x])/10

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Rubi [A]  time = 0.0186711, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {632, 31} \[ \frac{1}{10} \left (5+3 \sqrt{5}\right ) \log \left (2 x-\sqrt{5}+3\right )+\frac{1}{10} \left (5-3 \sqrt{5}\right ) \log \left (2 x+\sqrt{5}+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[(3 + x)/(1 + 3*x + x^2),x]

[Out]

((5 + 3*Sqrt[5])*Log[3 - Sqrt[5] + 2*x])/10 + ((5 - 3*Sqrt[5])*Log[3 + Sqrt[5] + 2*x])/10

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{3+x}{1+3 x+x^2} \, dx &=-\left (\frac{1}{10} \left (-5+3 \sqrt{5}\right ) \int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x} \, dx\right )+\frac{1}{10} \left (5+3 \sqrt{5}\right ) \int \frac{1}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x} \, dx\\ &=\frac{1}{10} \left (5+3 \sqrt{5}\right ) \log \left (3-\sqrt{5}+2 x\right )+\frac{1}{10} \left (5-3 \sqrt{5}\right ) \log \left (3+\sqrt{5}+2 x\right )\\ \end{align*}

Mathematica [A]  time = 0.0250529, size = 49, normalized size = 0.96 \[ \frac{1}{10} \left (5+3 \sqrt{5}\right ) \log \left (-2 x+\sqrt{5}-3\right )+\frac{1}{10} \left (5-3 \sqrt{5}\right ) \log \left (2 x+\sqrt{5}+3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + x)/(1 + 3*x + x^2),x]

[Out]

((5 + 3*Sqrt[5])*Log[-3 + Sqrt[5] - 2*x])/10 + ((5 - 3*Sqrt[5])*Log[3 + Sqrt[5] + 2*x])/10

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Maple [A]  time = 0.003, size = 29, normalized size = 0.6 \begin{align*}{\frac{\ln \left ({x}^{2}+3\,x+1 \right ) }{2}}-{\frac{3\,\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 3+2\,x \right ) \sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(x^2+3*x+1)-3/5*5^(1/2)*arctanh(1/5*(3+2*x)*5^(1/2))

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Maxima [A]  time = 1.53988, size = 53, normalized size = 1.04 \begin{align*} \frac{3}{10} \, \sqrt{5} \log \left (\frac{2 \, x - \sqrt{5} + 3}{2 \, x + \sqrt{5} + 3}\right ) + \frac{1}{2} \, \log \left (x^{2} + 3 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*sqrt(5)*log((2*x - sqrt(5) + 3)/(2*x + sqrt(5) + 3)) + 1/2*log(x^2 + 3*x + 1)

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Fricas [A]  time = 1.62943, size = 132, normalized size = 2.59 \begin{align*} \frac{3}{10} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5}{\left (2 \, x + 3\right )} + 6 \, x + 7}{x^{2} + 3 \, x + 1}\right ) + \frac{1}{2} \, \log \left (x^{2} + 3 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*sqrt(5)*log((2*x^2 - sqrt(5)*(2*x + 3) + 6*x + 7)/(x^2 + 3*x + 1)) + 1/2*log(x^2 + 3*x + 1)

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Sympy [A]  time = 0.111567, size = 49, normalized size = 0.96 \begin{align*} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \log{\left (x - \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} + \left (\frac{1}{2} - \frac{3 \sqrt{5}}{10}\right ) \log{\left (x + \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(x**2+3*x+1),x)

[Out]

(1/2 + 3*sqrt(5)/10)*log(x - sqrt(5)/2 + 3/2) + (1/2 - 3*sqrt(5)/10)*log(x + sqrt(5)/2 + 3/2)

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Giac [A]  time = 1.19062, size = 57, normalized size = 1.12 \begin{align*} \frac{3}{10} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x - \sqrt{5} + 3 \right |}}{{\left | 2 \, x + \sqrt{5} + 3 \right |}}\right ) + \frac{1}{2} \, \log \left ({\left | x^{2} + 3 \, x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*sqrt(5)*log(abs(2*x - sqrt(5) + 3)/abs(2*x + sqrt(5) + 3)) + 1/2*log(abs(x^2 + 3*x + 1))