3.2426 \(\int \frac{1}{x \sqrt{-2+5 x-3 x^2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac{\tan ^{-1}\left (\frac{4-5 x}{2 \sqrt{2} \sqrt{-3 x^2+5 x-2}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0382044, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\tan ^{-1}\left (\frac{4-5 x}{2 \sqrt{2} \sqrt{-3 x^2+5 x-2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[-2 + 5*x - 3*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.29906, size = 32, normalized size = 0.89 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (5 x - 4\right )}{4 \sqrt{- 3 x^{2} + 5 x - 2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atan(sqrt(2)*(5*x - 4)/(4*sqrt(-3*x**2 + 5*x - 2)))/2

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Mathematica [A]  time = 0.0375922, size = 31, normalized size = 0.86 \[ -\frac{\tan ^{-1}\left (\frac{4-5 x}{2 \sqrt{-6 x^2+10 x-4}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[-2 + 5*x - 3*x^2]),x]

[Out]

-(ArcTan[(4 - 5*x)/(2*Sqrt[-4 + 10*x - 6*x^2])]/Sqrt[2])

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Maple [A]  time = 0.008, size = 29, normalized size = 0.8 \[{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 5\,x-4 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{-3\,{x}^{2}+5\,x-2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-3*x^2+5*x-2)^(1/2),x)

[Out]

1/2*2^(1/2)*arctan(1/4*(5*x-4)*2^(1/2)/(-3*x^2+5*x-2)^(1/2))

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Maxima [A]  time = 0.752635, size = 27, normalized size = 0.75 \[ \frac{1}{2} \, \sqrt{2} \arcsin \left (\frac{5 \, x}{{\left | x \right |}} - \frac{4}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-3*x^2 + 5*x - 2)*x),x, algorithm="maxima")

[Out]

1/2*sqrt(2)*arcsin(5*x/abs(x) - 4/abs(x))

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Fricas [A]  time = 0.223459, size = 38, normalized size = 1.06 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (5 \, x - 4\right )}}{4 \, \sqrt{-3 \, x^{2} + 5 \, x - 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-3*x^2 + 5*x - 2)*x),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*arctan(1/4*sqrt(2)*(5*x - 4)/sqrt(-3*x^2 + 5*x - 2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213894, size = 59, normalized size = 1.64 \[ -\frac{1}{3} \, \sqrt{6} \sqrt{3} \arctan \left (\frac{1}{12} \, \sqrt{6}{\left (\frac{5 \,{\left (2 \, \sqrt{3} \sqrt{-3 \, x^{2} + 5 \, x - 2} - 1\right )}}{6 \, x - 5} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-3*x^2 + 5*x - 2)*x),x, algorithm="giac")

[Out]

-1/3*sqrt(6)*sqrt(3)*arctan(1/12*sqrt(6)*(5*(2*sqrt(3)*sqrt(-3*x^2 + 5*x - 2) -
1)/(6*x - 5) - 1))