3.2425 \(\int \frac{1}{x \sqrt{-2+5 x+3 x^2}} \, dx\)

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 = 0.0366422, 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 = 5.1378, 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 = 0.0294029, size = 30, normalized size = 0.83 \[ \frac{\tan ^{-1}\left (\frac{5 x-4}{2 \sqrt{6 x^2+10 x-4}}\right )}{\sqrt{2}} \]

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 = 0.007, 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 = 0.745549, size = 27, normalized size = 0.75 \[ \frac{1}{2} \, \sqrt{2} \arcsin \left (\frac{5 \, x}{7 \,{\left | x \right |}} - \frac{4}{7 \,{\left | x \right |}}\right ) \]

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(5/7*x/abs(x) - 4/7/abs(x))

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Fricas [A]  time = 0.224767, 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. \[ \int \frac{1}{x \sqrt{\left (x + 2\right ) \left (3 x - 1\right )}}\, dx \]

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 + 2)*(3*x - 1))), x)

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GIAC/XCAS [A]  time = 0.211433, size = 41, normalized size = 1.14 \[ \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x - 2}\right )}\right ) \]

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]

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