3.2404 \(\int \frac{x}{\sqrt{-2+4 x+3 x^2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{1}{3} \sqrt{3 x^2+4 x-2}-\frac{2 \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]

[Out]

Sqrt[-2 + 4*x + 3*x^2]/3 - (2*ArcTanh[(2 + 3*x)/(Sqrt[3]*Sqrt[-2 + 4*x + 3*x^2])
])/(3*Sqrt[3])

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Rubi [A]  time = 0.0447973, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{3} \sqrt{3 x^2+4 x-2}-\frac{2 \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[-2 + 4*x + 3*x^2],x]

[Out]

Sqrt[-2 + 4*x + 3*x^2]/3 - (2*ArcTanh[(2 + 3*x)/(Sqrt[3]*Sqrt[-2 + 4*x + 3*x^2])
])/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 4.59689, size = 49, normalized size = 0.91 \[ \frac{\sqrt{3 x^{2} + 4 x - 2}}{3} - \frac{2 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 4\right )}{6 \sqrt{3 x^{2} + 4 x - 2}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(3*x**2+4*x-2)**(1/2),x)

[Out]

sqrt(3*x**2 + 4*x - 2)/3 - 2*sqrt(3)*atanh(sqrt(3)*(6*x + 4)/(6*sqrt(3*x**2 + 4*
x - 2)))/9

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Mathematica [A]  time = 0.0370476, size = 48, normalized size = 0.89 \[ \frac{1}{9} \left (3 \sqrt{3 x^2+4 x-2}-2 \sqrt{3} \log \left (\sqrt{9 x^2+12 x-6}+3 x+2\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[-2 + 4*x + 3*x^2],x]

[Out]

(3*Sqrt[-2 + 4*x + 3*x^2] - 2*Sqrt[3]*Log[2 + 3*x + Sqrt[-6 + 12*x + 9*x^2]])/9

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Maple [A]  time = 0.008, size = 45, normalized size = 0.8 \[{\frac{1}{3}\sqrt{3\,{x}^{2}+4\,x-2}}-{\frac{2\,\sqrt{3}}{9}\ln \left ({\frac{ \left ( 2+3\,x \right ) \sqrt{3}}{3}}+\sqrt{3\,{x}^{2}+4\,x-2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(3*x^2+4*x-2)^(1/2),x)

[Out]

1/3*(3*x^2+4*x-2)^(1/2)-2/9*ln(1/3*(2+3*x)*3^(1/2)+(3*x^2+4*x-2)^(1/2))*3^(1/2)

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Maxima [A]  time = 0.74928, size = 58, normalized size = 1.07 \[ -\frac{2}{9} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 4 \, x - 2} + 6 \, x + 4\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(3*x^2 + 4*x - 2),x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 4*x - 2) + 6*x + 4) + 1/3*sqrt(3*x^2 + 4
*x - 2)

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Fricas [A]  time = 0.225885, size = 77, normalized size = 1.43 \[ \frac{1}{9} \, \sqrt{3}{\left (\sqrt{3} \sqrt{3 \, x^{2} + 4 \, x - 2} + \log \left (\sqrt{3}{\left (9 \, x^{2} + 12 \, x - 1\right )} - 3 \, \sqrt{3 \, x^{2} + 4 \, x - 2}{\left (3 \, x + 2\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(3*x^2 + 4*x - 2),x, algorithm="fricas")

[Out]

1/9*sqrt(3)*(sqrt(3)*sqrt(3*x^2 + 4*x - 2) + log(sqrt(3)*(9*x^2 + 12*x - 1) - 3*
sqrt(3*x^2 + 4*x - 2)*(3*x + 2)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{3 x^{2} + 4 x - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(3*x**2+4*x-2)**(1/2),x)

[Out]

Integral(x/sqrt(3*x**2 + 4*x - 2), x)

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GIAC/XCAS [A]  time = 0.212709, size = 66, normalized size = 1.22 \[ \frac{2}{9} \, \sqrt{3}{\rm ln}\left ({\left | -\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x - 2}\right )} - 2 \right |}\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(3*x^2 + 4*x - 2),x, algorithm="giac")

[Out]

2/9*sqrt(3)*ln(abs(-sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 4*x - 2)) - 2)) + 1/3*sqrt
(3*x^2 + 4*x - 2)