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

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

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Rubi [A]  time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 31

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

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)]

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*}

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Mathematica [A]  time = 0.03, size = 47, normalized size = 1.00 \[ \frac {1}{12} \left (4-5 \sqrt {6}\right ) \log \left (\sqrt {6}-3 x\right )+\frac {1}{12} \left (4+5 \sqrt {6}\right ) \log \left (3 x+\sqrt {6}\right ) \]

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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IntegrateAlgebraic [A]  time = 0.04, size = 47, normalized size = 1.00 \[ \frac {1}{12} \left (4-5 \sqrt {6}\right ) \log \left (\sqrt {6}-3 x\right )+\frac {1}{12} \left (4+5 \sqrt {6}\right ) \log \left (3 x+\sqrt {6}\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-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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fricas [A]  time = 1.08, size = 40, normalized size = 0.85 \[ \frac {5}{12} \, \sqrt {6} \log \left (\frac {3 \, x^{2} + 2 \, \sqrt {6} x + 2}{3 \, x^{2} - 2}\right ) + \frac {1}{3} \, \log \left (3 \, x^{2} - 2\right ) \]

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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giac [A]  time = 0.96, size = 37, normalized size = 0.79 \[ \frac {1}{12} \, {\left (5 \, \sqrt {6} + 4\right )} \log \left ({\left | x + \frac {1}{3} \, \sqrt {6} \right |}\right ) - \frac {1}{12} \, {\left (5 \, \sqrt {6} - 4\right )} \log \left ({\left | x - \frac {1}{3} \, \sqrt {6} \right |}\right ) \]

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)))

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maple [A]  time = 0.29, size = 24, normalized size = 0.51

method	result	size
default	\(\frac {\ln \left (3 x^{2}-2\right )}{3}+\frac {5 \sqrt {6}\, \arctanh \left (\frac {x \sqrt {6}}{2}\right )}{6}\)	\(24\)
meijerg	\(\frac {5 \sqrt {6}\, \arctanh \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{6}+\frac {\ln \left (1-\frac {3 x^{2}}{2}\right )}{3}\)	\(27\)
risch	\(\frac {\ln \left (3 x +\sqrt {6}\right )}{3}+\frac {5 \ln \left (3 x +\sqrt {6}\right ) \sqrt {6}}{12}+\frac {\ln \left (3 x -\sqrt {6}\right )}{3}-\frac {5 \ln \left (3 x -\sqrt {6}\right ) \sqrt {6}}{12}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5+2*x)/(3*x^2-2),x,method=_RETURNVERBOSE)

[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 = 0.97, size = 36, normalized size = 0.77 \[ -\frac {5}{12} \, \sqrt {6} \log \left (\frac {3 \, x - \sqrt {6}}{3 \, x + \sqrt {6}}\right ) + \frac {1}{3} \, \log \left (3 \, x^{2} - 2\right ) \]

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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mupad [B]  time = 0.13, size = 47, normalized size = 1.00 \[ \frac {\ln \left (x-\frac {\sqrt {6}}{3}\right )}{3}+\frac {\ln \left (x+\frac {\sqrt {6}}{3}\right )}{3}-\frac {5\,\sqrt {6}\,\ln \left (x-\frac {\sqrt {6}}{3}\right )}{12}+\frac {5\,\sqrt {6}\,\ln \left (x+\frac {\sqrt {6}}{3}\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 5)/(3*x^2 - 2),x)

[Out]

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

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sympy [A]  time = 0.12, size = 42, normalized size = 0.89 \[ \left (\frac {1}{3} - \frac {5 \sqrt {6}}{12}\right ) \log {\left (x - \frac {\sqrt {6}}{3} \right )} + \left (\frac {1}{3} + \frac {5 \sqrt {6}}{12}\right ) \log {\left (x + \frac {\sqrt {6}}{3} \right )} \]

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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