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

Optimal. Leaf size=59 \[ \frac{1}{6} (3 x+2) \sqrt{3 x^2+4 x-2}-\frac{5 \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]

[Out]

((2 + 3*x)*Sqrt[-2 + 4*x + 3*x^2])/6 - (5*ArcTanh[(2 + 3*x)/(Sqrt[3]*Sqrt[-2 + 4
*x + 3*x^2])])/(3*Sqrt[3])

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Rubi [A]  time = 0.0322648, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{1}{6} (3 x+2) \sqrt{3 x^2+4 x-2}-\frac{5 \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-2 + 4*x + 3*x^2],x]

[Out]

((2 + 3*x)*Sqrt[-2 + 4*x + 3*x^2])/6 - (5*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 = 2.00368, size = 54, normalized size = 0.92 \[ \frac{\left (6 x + 4\right ) \sqrt{3 x^{2} + 4 x - 2}}{12} - \frac{5 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 4\right )}{6 \sqrt{3 x^{2} + 4 x - 2}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(6*x + 4)*sqrt(3*x**2 + 4*x - 2)/12 - 5*sqrt(3)*atanh(sqrt(3)*(6*x + 4)/(6*sqrt(
3*x**2 + 4*x - 2)))/9

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Mathematica [A]  time = 0.0381253, size = 53, normalized size = 0.9 \[ \frac{1}{6} (3 x+2) \sqrt{3 x^2+4 x-2}-\frac{5 \log \left (\sqrt{9 x^2+12 x-6}+3 x+2\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

((2 + 3*x)*Sqrt[-2 + 4*x + 3*x^2])/6 - (5*Log[2 + 3*x + Sqrt[-6 + 12*x + 9*x^2]]
)/(3*Sqrt[3])

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Maple [A]  time = 0.005, size = 50, normalized size = 0.9 \[{\frac{6\,x+4}{12}\sqrt{3\,{x}^{2}+4\,x-2}}-{\frac{5\,\sqrt{3}}{9}\ln \left ({\frac{ \left ( 2+3\,x \right ) \sqrt{3}}{3}}+\sqrt{3\,{x}^{2}+4\,x-2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+4*x-2)^(1/2),x)

[Out]

1/12*(6*x+4)*(3*x^2+4*x-2)^(1/2)-5/9*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 = 0.826497, size = 78, normalized size = 1.32 \[ \frac{1}{2} \, \sqrt{3 \, x^{2} + 4 \, x - 2} x - \frac{5}{9} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 4 \, x - 2} + 6 \, x + 4\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(3*x^2 + 4*x - 2)*x - 5/9*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 = 0.226902, size = 86, normalized size = 1.46 \[ \frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \sqrt{3 \, x^{2} + 4 \, x - 2}{\left (3 \, x + 2\right )} + 5 \, \log \left (\sqrt{3}{\left (9 \, x^{2} + 12 \, x - 1\right )} - 3 \, \sqrt{3 \, x^{2} + 4 \, x - 2}{\left (3 \, x + 2\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(sqrt(3)*sqrt(3*x^2 + 4*x - 2)*(3*x + 2) + 5*log(sqrt(3)*(9*x^2 + 1
2*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. \[ \int \sqrt{3 x^{2} + 4 x - 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(3*x**2 + 4*x - 2), x)

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GIAC/XCAS [A]  time = 0.212271, size = 73, normalized size = 1.24 \[ \frac{1}{6} \, \sqrt{3 \, x^{2} + 4 \, x - 2}{\left (3 \, x + 2\right )} + \frac{5}{9} \, \sqrt{3}{\rm ln}\left ({\left | -\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x - 2}\right )} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(3*x^2 + 4*x - 2)*(3*x + 2) + 5/9*sqrt(3)*ln(abs(-sqrt(3)*(sqrt(3)*x - s
qrt(3*x^2 + 4*x - 2)) - 2))