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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]