3.2400 \(\int \frac{x}{\sqrt{2+4 x+3 x^2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{1}{3} \sqrt{3 x^2+4 x+2}-\frac{2 \sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.048943, antiderivative size = 40, 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 \[ \frac{1}{3} \sqrt{3 x^2+4 x+2}-\frac{2 \sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{3 \sqrt{3}} \]

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*ArcSinh[(2 + 3*x)/Sqrt[2]])/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 4.6974, size = 49, normalized size = 1.22 \[ \frac{\sqrt{3 x^{2} + 4 x + 2}}{3} - \frac{2 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 4\right )}{6 \sqrt{3 x^{2} + 4 x + 2}} \right )}}{9} \]

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 = 0.0237098, size = 40, normalized size = 1. \[ \frac{1}{3} \sqrt{3 x^2+4 x+2}-\frac{2 \sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 30, normalized size = 0.8 \[{\frac{1}{3}\sqrt{3\,{x}^{2}+4\,x+2}}-{\frac{2\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{3\,\sqrt{2}}{2} \left ( x+{\frac{2}{3}} \right ) } \right ) } \]

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*3^(1/2)*arcsinh(3/2*2^(1/2)*(x+2/3))

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Maxima [A]  time = 0.75471, size = 42, normalized size = 1.05 \[ -\frac{2}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231749, size = 78, normalized size = 1.95 \[ \frac{1}{9} \, \sqrt{3}{\left (\sqrt{3} \sqrt{3 \, x^{2} + 4 \, x + 2} + \log \left (-\sqrt{3}{\left (9 \, x^{2} + 12 \, x + 5\right )} + 3 \, \sqrt{3 \, x^{2} + 4 \, x + 2}{\left (3 \, x + 2\right )}\right )\right )} \]

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 + 5) + 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 = 0.211363, size = 65, normalized size = 1.62 \[ \frac{2}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x + 2}\right )} - 2\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 4 \, x + 2} \]

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