3.2406 \(\int \frac{x}{\sqrt{-2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=57 \[ \frac{1}{3} \sqrt{3 x^2+5 x-2}-\frac{5 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x-2}}\right )}{6 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0443317, antiderivative size = 57, 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 \[ \frac{1}{3} \sqrt{3 x^2+5 x-2}-\frac{5 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x-2}}\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[-2 + 5*x + 3*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.59416, size = 49, normalized size = 0.86 \[ \frac{\sqrt{3 x^{2} + 5 x - 2}}{3} - \frac{5 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x - 2}} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(3*x**2+5*x-2)**(1/2),x)

[Out]

sqrt(3*x**2 + 5*x - 2)/3 - 5*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*
x - 2)))/18

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Mathematica [A]  time = 0.0395272, size = 50, normalized size = 0.88 \[ \frac{1}{18} \left (6 \sqrt{3 x^2+5 x-2}-5 \sqrt{3} \log \left (2 \sqrt{9 x^2+15 x-6}+6 x+5\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[-2 + 5*x + 3*x^2],x]

[Out]

(6*Sqrt[-2 + 5*x + 3*x^2] - 5*Sqrt[3]*Log[5 + 6*x + 2*Sqrt[-6 + 15*x + 9*x^2]])/
18

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Maple [A]  time = 0.007, size = 45, normalized size = 0.8 \[{\frac{1}{3}\sqrt{3\,{x}^{2}+5\,x-2}}-{\frac{5\,\sqrt{3}}{18}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x-2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(3*x^2+5*x-2)^(1/2),x)

[Out]

1/3*(3*x^2+5*x-2)^(1/2)-5/18*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x-2)^(1/2))*3^(1/
2)

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Maxima [A]  time = 0.760555, size = 58, normalized size = 1.02 \[ -\frac{5}{18} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x - 2} + 6 \, x + 5\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(3*x^2 + 5*x - 2),x, algorithm="maxima")

[Out]

-5/18*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x - 2) + 6*x + 5) + 1/3*sqrt(3*x^2 +
5*x - 2)

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Fricas [A]  time = 0.22448, size = 81, normalized size = 1.42 \[ \frac{1}{36} \, \sqrt{3}{\left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x - 2} + 5 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 1\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x - 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(3*x^2 + 5*x - 2),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(4*sqrt(3)*sqrt(3*x^2 + 5*x - 2) + 5*log(sqrt(3)*(72*x^2 + 120*x +
1) - 12*sqrt(3*x^2 + 5*x - 2)*(6*x + 5)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{\left (x + 2\right ) \left (3 x - 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(3*x**2+5*x-2)**(1/2),x)

[Out]

Integral(x/sqrt((x + 2)*(3*x - 1)), x)

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GIAC/XCAS [A]  time = 0.214079, size = 66, normalized size = 1.16 \[ \frac{5}{18} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x - 2}\right )} - 5 \right |}\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 5 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(3*x^2 + 5*x - 2),x, algorithm="giac")

[Out]

5/18*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x - 2)) - 5)) + 1/3*s
qrt(3*x^2 + 5*x - 2)