3.199 \(\int \frac {x}{3+6 x+2 x^2} \, dx\)

Optimal. Leaf size=49 \[ \frac {1}{4} \left (1-\sqrt {3}\right ) \log \left (2 x-\sqrt {3}+3\right )+\frac {1}{4} \left (1+\sqrt {3}\right ) \log \left (2 x+\sqrt {3}+3\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 49, 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} \[ \frac {1}{4} \left (1-\sqrt {3}\right ) \log \left (2 x-\sqrt {3}+3\right )+\frac {1}{4} \left (1+\sqrt {3}\right ) \log \left (2 x+\sqrt {3}+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(3 + 6*x + 2*x^2),x]

[Out]

((1 - Sqrt[3])*Log[3 - Sqrt[3] + 2*x])/4 + ((1 + Sqrt[3])*Log[3 + Sqrt[3] + 2*x])/4

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 {x}{3+6 x+2 x^2} \, dx &=\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {1}{3-\sqrt {3}+2 x} \, dx+\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {1}{3+\sqrt {3}+2 x} \, dx\\ &=\frac {1}{4} \left (1-\sqrt {3}\right ) \log \left (3-\sqrt {3}+2 x\right )+\frac {1}{4} \left (1+\sqrt {3}\right ) \log \left (3+\sqrt {3}+2 x\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 44, normalized size = 0.90 \[ \frac {1}{4} \left (\left (1+\sqrt {3}\right ) \log \left (2 x+\sqrt {3}+3\right )-\left (\sqrt {3}-1\right ) \log \left (-2 x+\sqrt {3}-3\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(3 + 6*x + 2*x^2),x]

[Out]

(-((-1 + Sqrt[3])*Log[-3 + Sqrt[3] - 2*x]) + (1 + Sqrt[3])*Log[3 + Sqrt[3] + 2*x])/4

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IntegrateAlgebraic [A]  time = 0.03, size = 47, normalized size = 0.96 \[ \frac {1}{4} \left (1-\sqrt {3}\right ) \log \left (-2 x+\sqrt {3}-3\right )+\frac {1}{4} \left (1+\sqrt {3}\right ) \log \left (2 x+\sqrt {3}+3\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/(3 + 6*x + 2*x^2),x]

[Out]

((1 - Sqrt[3])*Log[-3 + Sqrt[3] - 2*x])/4 + ((1 + Sqrt[3])*Log[3 + Sqrt[3] + 2*x])/4

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fricas [A]  time = 1.25, size = 52, normalized size = 1.06 \[ \frac {1}{4} \, \sqrt {3} \log \left (\frac {2 \, x^{2} + \sqrt {3} {\left (2 \, x + 3\right )} + 6 \, x + 6}{2 \, x^{2} + 6 \, x + 3}\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} + 6 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2*x^2+6*x+3),x, algorithm="fricas")

[Out]

1/4*sqrt(3)*log((2*x^2 + sqrt(3)*(2*x + 3) + 6*x + 6)/(2*x^2 + 6*x + 3)) + 1/4*log(2*x^2 + 6*x + 3)

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giac [A]  time = 0.96, size = 46, normalized size = 0.94 \[ -\frac {1}{4} \, \sqrt {3} \log \left (\frac {{\left | 4 \, x - 2 \, \sqrt {3} + 6 \right |}}{{\left | 4 \, x + 2 \, \sqrt {3} + 6 \right |}}\right ) + \frac {1}{4} \, \log \left ({\left | 2 \, x^{2} + 6 \, x + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2*x^2+6*x+3),x, algorithm="giac")

[Out]

-1/4*sqrt(3)*log(abs(4*x - 2*sqrt(3) + 6)/abs(4*x + 2*sqrt(3) + 6)) + 1/4*log(abs(2*x^2 + 6*x + 3))

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maple [A]  time = 0.29, size = 31, normalized size = 0.63

method	result	size
default	\(\frac {\ln \left (2 x^{2}+6 x +3\right )}{4}+\frac {\sqrt {3}\, \arctanh \left (\frac {\left (4 x +6\right ) \sqrt {3}}{6}\right )}{2}\)	\(31\)
risch	\(\frac {\ln \left (3+2 x +\sqrt {3}\right )}{4}+\frac {\ln \left (3+2 x +\sqrt {3}\right ) \sqrt {3}}{4}+\frac {\ln \left (3+2 x -\sqrt {3}\right )}{4}-\frac {\ln \left (3+2 x -\sqrt {3}\right ) \sqrt {3}}{4}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(2*x^2+6*x+3),x,method=_RETURNVERBOSE)

[Out]

1/4*ln(2*x^2+6*x+3)+1/2*3^(1/2)*arctanh(1/6*(4*x+6)*3^(1/2))

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maxima [A]  time = 1.38, size = 41, normalized size = 0.84 \[ -\frac {1}{4} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3} + 3}{2 \, x + \sqrt {3} + 3}\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} + 6 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2*x^2+6*x+3),x, algorithm="maxima")

[Out]

-1/4*sqrt(3)*log((2*x - sqrt(3) + 3)/(2*x + sqrt(3) + 3)) + 1/4*log(2*x^2 + 6*x + 3)

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mupad [B]  time = 0.17, size = 36, normalized size = 0.73 \[ \ln \left (x+\frac {\sqrt {3}}{2}+\frac {3}{2}\right )\,\left (\frac {\sqrt {3}}{4}+\frac {1}{4}\right )-\ln \left (x-\frac {\sqrt {3}}{2}+\frac {3}{2}\right )\,\left (\frac {\sqrt {3}}{4}-\frac {1}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(6*x + 2*x^2 + 3),x)

[Out]

log(x + 3^(1/2)/2 + 3/2)*(3^(1/2)/4 + 1/4) - log(x - 3^(1/2)/2 + 3/2)*(3^(1/2)/4 - 1/4)

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sympy [A]  time = 0.12, size = 46, normalized size = 0.94 \[ \left (\frac {1}{4} - \frac {\sqrt {3}}{4}\right ) \log {\left (x - \frac {\sqrt {3}}{2} + \frac {3}{2} \right )} + \left (\frac {1}{4} + \frac {\sqrt {3}}{4}\right ) \log {\left (x + \frac {\sqrt {3}}{2} + \frac {3}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2*x**2+6*x+3),x)

[Out]

(1/4 - sqrt(3)/4)*log(x - sqrt(3)/2 + 3/2) + (1/4 + sqrt(3)/4)*log(x + sqrt(3)/2 + 3/2)

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