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

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

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Rubi [A]  time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 ) \end {gather*}

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 31

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

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]

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*}

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Mathematica [A]  time = 0.03, size = 49, normalized size = 0.96 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+x}{1+3 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + x)/(1 + 3*x + x^2),x]

[Out]

IntegrateAlgebraic[(3 + x)/(1 + 3*x + x^2), x]

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fricas [A]  time = 0.41, size = 49, normalized size = 0.96 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.15, size = 42, normalized size = 0.82 \begin {gather*} \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 {gather*}

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))

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maple [A]  time = 0.05, size = 29, normalized size = 0.57 \begin {gather*} -\frac {3 \sqrt {5}\, \arctanh \left (\frac {\left (2 x +3\right ) \sqrt {5}}{5}\right )}{5}+\frac {\ln \left (x^{2}+3 x +1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 39, normalized size = 0.76 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.12, size = 36, normalized size = 0.71 \begin {gather*} \ln \left (x-\frac {\sqrt {5}}{2}+\frac {3}{2}\right )\,\left (\frac {3\,\sqrt {5}}{10}+\frac {1}{2}\right )-\ln \left (x+\frac {\sqrt {5}}{2}+\frac {3}{2}\right )\,\left (\frac {3\,\sqrt {5}}{10}-\frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 3)/(3*x + x^2 + 1),x)

[Out]

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

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sympy [A]  time = 0.12, size = 49, normalized size = 0.96 \begin {gather*} \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 {gather*}

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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