3.2422 \(\int \frac{1}{x \sqrt{2+5 x-3 x^2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0372239, 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 \[ -\frac{\tanh ^{-1}\left (\frac{5 x+4}{2 \sqrt{2} \sqrt{-3 x^2+5 x+2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.14897, size = 34, normalized size = 0.94 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (5 x + 4\right )}{4 \sqrt{- 3 x^{2} + 5 x + 2}} \right )}}{2} \]

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

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Mathematica [A]  time = 0.0356781, size = 33, normalized size = 0.92 \[ \frac{\log (x)-\log \left (2 \sqrt{-6 x^2+10 x+4}+5 x+4\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 29, normalized size = 0.8 \[ -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 4+5\,x \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{-3\,{x}^{2}+5\,x+2}}}} \right ) } \]

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

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Maxima [A]  time = 0.750307, size = 47, normalized size = 1.31 \[ -\frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{-3 \, x^{2} + 5 \, x + 2}}{{\left | x \right |}} + \frac{4}{{\left | x \right |}} + 5\right ) \]

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)*log(2*sqrt(2)*sqrt(-3*x^2 + 5*x + 2)/abs(x) + 4/abs(x) + 5)

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Fricas [A]  time = 0.227158, size = 58, normalized size = 1.61 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 5 \, x + 2}{\left (5 \, x + 4\right )} - x^{2} - 80 \, x - 32}{x^{2}}\right ) \]

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]

1/4*sqrt(2)*log(-(4*sqrt(2)*sqrt(-3*x^2 + 5*x + 2)*(5*x + 4) - x^2 - 80*x - 32)/
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 = 0.247735, size = 113, normalized size = 3.14 \[ -\frac{1}{6} \, \sqrt{6} \sqrt{3}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{6} + \frac{10 \,{\left (2 \, \sqrt{3} \sqrt{-3 \, x^{2} + 5 \, x + 2} - 7\right )}}{6 \, x - 5} - 14 \right |}}{{\left | 4 \, \sqrt{6} + \frac{10 \,{\left (2 \, \sqrt{3} \sqrt{-3 \, x^{2} + 5 \, x + 2} - 7\right )}}{6 \, x - 5} - 14 \right |}}\right ) \]

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]

-1/6*sqrt(6)*sqrt(3)*ln(abs(-4*sqrt(6) + 10*(2*sqrt(3)*sqrt(-3*x^2 + 5*x + 2) -
7)/(6*x - 5) - 14)/abs(4*sqrt(6) + 10*(2*sqrt(3)*sqrt(-3*x^2 + 5*x + 2) - 7)/(6*
x - 5) - 14))