3.27 \(\int \frac{-5+2 x}{-2+3 x^2} \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{12} \left (4-5 \sqrt{6}\right ) \log \left (\sqrt{6}-3 x\right )+\frac{1}{12} \left (4+5 \sqrt{6}\right ) \log \left (3 x+\sqrt{6}\right ) \]

[Out]

((4 - 5*Sqrt[6])*Log[Sqrt[6] - 3*x])/12 + ((4 + 5*Sqrt[6])*Log[Sqrt[6] + 3*x])/12

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Rubi [A]  time = 0.0227613, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {633, 31} \[ \frac{1}{12} \left (4-5 \sqrt{6}\right ) \log \left (\sqrt{6}-3 x\right )+\frac{1}{12} \left (4+5 \sqrt{6}\right ) \log \left (3 x+\sqrt{6}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-5 + 2*x)/(-2 + 3*x^2),x]

[Out]

((4 - 5*Sqrt[6])*Log[Sqrt[6] - 3*x])/12 + ((4 + 5*Sqrt[6])*Log[Sqrt[6] + 3*x])/12

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(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{-5+2 x}{-2+3 x^2} \, dx &=\frac{1}{4} \left (4-5 \sqrt{6}\right ) \int \frac{1}{-\sqrt{6}+3 x} \, dx+\frac{1}{4} \left (4+5 \sqrt{6}\right ) \int \frac{1}{\sqrt{6}+3 x} \, dx\\ &=\frac{1}{12} \left (4-5 \sqrt{6}\right ) \log \left (\sqrt{6}-3 x\right )+\frac{1}{12} \left (4+5 \sqrt{6}\right ) \log \left (\sqrt{6}+3 x\right )\\ \end{align*}

Mathematica [A]  time = 0.027503, size = 47, normalized size = 1. \[ \frac{1}{12} \left (4-5 \sqrt{6}\right ) \log \left (\sqrt{6}-3 x\right )+\frac{1}{12} \left (4+5 \sqrt{6}\right ) \log \left (3 x+\sqrt{6}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 2*x)/(-2 + 3*x^2),x]

[Out]

((4 - 5*Sqrt[6])*Log[Sqrt[6] - 3*x])/12 + ((4 + 5*Sqrt[6])*Log[Sqrt[6] + 3*x])/12

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Maple [A]  time = 0.002, size = 24, normalized size = 0.5 \begin{align*}{\frac{\ln \left ( 3\,{x}^{2}-2 \right ) }{3}}+{\frac{5\,\sqrt{6}}{6}{\it Artanh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x-5)/(3*x^2-2),x)

[Out]

1/3*ln(3*x^2-2)+5/6*6^(1/2)*arctanh(1/2*x*6^(1/2))

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Maxima [A]  time = 1.40894, size = 49, normalized size = 1.04 \begin{align*} -\frac{5}{12} \, \sqrt{6} \log \left (\frac{3 \, x - \sqrt{6}}{3 \, x + \sqrt{6}}\right ) + \frac{1}{3} \, \log \left (3 \, x^{2} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+2*x)/(3*x^2-2),x, algorithm="maxima")

[Out]

-5/12*sqrt(6)*log((3*x - sqrt(6))/(3*x + sqrt(6))) + 1/3*log(3*x^2 - 2)

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Fricas [A]  time = 2.03127, size = 105, normalized size = 2.23 \begin{align*} \frac{5}{12} \, \sqrt{6} \log \left (\frac{3 \, x^{2} + 2 \, \sqrt{6} x + 2}{3 \, x^{2} - 2}\right ) + \frac{1}{3} \, \log \left (3 \, x^{2} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+2*x)/(3*x^2-2),x, algorithm="fricas")

[Out]

5/12*sqrt(6)*log((3*x^2 + 2*sqrt(6)*x + 2)/(3*x^2 - 2)) + 1/3*log(3*x^2 - 2)

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Sympy [A]  time = 0.106597, size = 42, normalized size = 0.89 \begin{align*} \left (\frac{1}{3} - \frac{5 \sqrt{6}}{12}\right ) \log{\left (x - \frac{\sqrt{6}}{3} \right )} + \left (\frac{1}{3} + \frac{5 \sqrt{6}}{12}\right ) \log{\left (x + \frac{\sqrt{6}}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+2*x)/(3*x**2-2),x)

[Out]

(1/3 - 5*sqrt(6)/12)*log(x - sqrt(6)/3) + (1/3 + 5*sqrt(6)/12)*log(x + sqrt(6)/3)

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Giac [A]  time = 1.07114, size = 50, normalized size = 1.06 \begin{align*} \frac{1}{12} \,{\left (5 \, \sqrt{6} + 4\right )} \log \left ({\left | x + \frac{1}{3} \, \sqrt{6} \right |}\right ) - \frac{1}{12} \,{\left (5 \, \sqrt{6} - 4\right )} \log \left ({\left | x - \frac{1}{3} \, \sqrt{6} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5+2*x)/(3*x^2-2),x, algorithm="giac")

[Out]

1/12*(5*sqrt(6) + 4)*log(abs(x + 1/3*sqrt(6))) - 1/12*(5*sqrt(6) - 4)*log(abs(x - 1/3*sqrt(6)))