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

Optimal. Leaf size=30 \[ \frac{1}{3} \log \left (3 x^2+2\right )-\frac{5 \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]

[Out]

(-5*ArcTan[Sqrt[3/2]*x])/Sqrt[6] + Log[2 + 3*x^2]/3

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Rubi [A]  time = 0.0081567, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {635, 203, 260} \[ \frac{1}{3} \log \left (3 x^2+2\right )-\frac{5 \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Int[(-5 + 2*x)/(2 + 3*x^2),x]

[Out]

(-5*ArcTan[Sqrt[3/2]*x])/Sqrt[6] + Log[2 + 3*x^2]/3

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

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

Rubi steps

\begin{align*} \int \frac{-5+2 x}{2+3 x^2} \, dx &=2 \int \frac{x}{2+3 x^2} \, dx-5 \int \frac{1}{2+3 x^2} \, dx\\ &=-\frac{5 \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}}+\frac{1}{3} \log \left (2+3 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0131009, size = 30, normalized size = 1. \[ \frac{1}{3} \log \left (3 x^2+2\right )-\frac{5 \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 2*x)/(2 + 3*x^2),x]

[Out]

(-5*ArcTan[Sqrt[3/2]*x])/Sqrt[6] + Log[2 + 3*x^2]/3

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Maple [A]  time = 0.002, size = 24, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 3\,{x}^{2}+2 \right ) }{3}}-{\frac{5\,\sqrt{6}}{6}\arctan \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*arctan(1/2*x*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.40806, size = 31, normalized size = 1.03 \begin{align*} -\frac{5}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\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]

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

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Fricas [A]  time = 2.15712, size = 77, normalized size = 2.57 \begin{align*} -\frac{5}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\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]

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

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Sympy [A]  time = 0.104192, size = 27, normalized size = 0.9 \begin{align*} \frac{\log{\left (x^{2} + \frac{2}{3} \right )}}{3} - \frac{5 \sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 + 2/3)/3 - 5*sqrt(6)*atan(sqrt(6)*x/2)/6

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Giac [A]  time = 1.06772, size = 28, normalized size = 0.93 \begin{align*} -\frac{5}{6} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{1}{3} \, \log \left (x^{2} + \frac{2}{3}\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]

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