3.2353 \(\int \frac{\sqrt{-2-3 x+5 x^2}}{x} \, dx\)

Optimal. Leaf size=88 \[ \sqrt{5 x^2-3 x-2}+\sqrt{2} \tan ^{-1}\left (\frac{3 x+4}{2 \sqrt{2} \sqrt{5 x^2-3 x-2}}\right )+\frac{3 \tanh ^{-1}\left (\frac{3-10 x}{2 \sqrt{5} \sqrt{5 x^2-3 x-2}}\right )}{2 \sqrt{5}} \]

[Out]

Sqrt[-2 - 3*x + 5*x^2] + Sqrt[2]*ArcTan[(4 + 3*x)/(2*Sqrt[2]*Sqrt[-2 - 3*x + 5*x
^2])] + (3*ArcTanh[(3 - 10*x)/(2*Sqrt[5]*Sqrt[-2 - 3*x + 5*x^2])])/(2*Sqrt[5])

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Rubi [A]  time = 0.135453, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \sqrt{5 x^2-3 x-2}+\sqrt{2} \tan ^{-1}\left (\frac{3 x+4}{2 \sqrt{2} \sqrt{5 x^2-3 x-2}}\right )+\frac{3 \tanh ^{-1}\left (\frac{3-10 x}{2 \sqrt{5} \sqrt{5 x^2-3 x-2}}\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-2 - 3*x + 5*x^2]/x,x]

[Out]

Sqrt[-2 - 3*x + 5*x^2] + Sqrt[2]*ArcTan[(4 + 3*x)/(2*Sqrt[2]*Sqrt[-2 - 3*x + 5*x
^2])] + (3*ArcTanh[(3 - 10*x)/(2*Sqrt[5]*Sqrt[-2 - 3*x + 5*x^2])])/(2*Sqrt[5])

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Rubi in Sympy [A]  time = 14.2988, size = 83, normalized size = 0.94 \[ \sqrt{5 x^{2} - 3 x - 2} + \sqrt{2} \operatorname{atan}{\left (- \frac{\sqrt{2} \left (- 3 x - 4\right )}{4 \sqrt{5 x^{2} - 3 x - 2}} \right )} - \frac{3 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (10 x - 3\right )}{10 \sqrt{5 x^{2} - 3 x - 2}} \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(5*x**2 - 3*x - 2) + sqrt(2)*atan(-sqrt(2)*(-3*x - 4)/(4*sqrt(5*x**2 - 3*x -
2))) - 3*sqrt(5)*atanh(sqrt(5)*(10*x - 3)/(10*sqrt(5*x**2 - 3*x - 2)))/10

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Mathematica [A]  time = 0.0688043, size = 81, normalized size = 0.92 \[ \sqrt{5 x^2-3 x-2}-\frac{3 \log \left (-2 \sqrt{5} \sqrt{5 x^2-3 x-2}-10 x+3\right )}{2 \sqrt{5}}+\sqrt{2} \tan ^{-1}\left (\frac{3 x+4}{2 \sqrt{10 x^2-6 x-4}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-2 - 3*x + 5*x^2] + Sqrt[2]*ArcTan[(4 + 3*x)/(2*Sqrt[-4 - 6*x + 10*x^2])] -
(3*Log[3 - 10*x - 2*Sqrt[5]*Sqrt[-2 - 3*x + 5*x^2]])/(2*Sqrt[5])

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Maple [A]  time = 0.01, size = 71, normalized size = 0.8 \[ \sqrt{5\,{x}^{2}-3\,x-2}-{\frac{3\,\sqrt{5}}{10}\ln \left ({\frac{\sqrt{5}}{5} \left ( -{\frac{3}{2}}+5\,x \right ) }+\sqrt{5\,{x}^{2}-3\,x-2} \right ) }-\sqrt{2}\arctan \left ({\frac{ \left ( -3\,x-4 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{5\,{x}^{2}-3\,x-2}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2-3*x-2)^(1/2)/x,x)

[Out]

(5*x^2-3*x-2)^(1/2)-3/10*ln(1/5*(-3/2+5*x)*5^(1/2)+(5*x^2-3*x-2)^(1/2))*5^(1/2)-
2^(1/2)*arctan(1/4*(-3*x-4)*2^(1/2)/(5*x^2-3*x-2)^(1/2))

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Maxima [A]  time = 0.75077, size = 81, normalized size = 0.92 \[ \sqrt{2} \arcsin \left (\frac{3 \, x}{7 \,{\left | x \right |}} + \frac{4}{7 \,{\left | x \right |}}\right ) - \frac{3}{10} \, \sqrt{5} \log \left (2 \, \sqrt{5} \sqrt{5 \, x^{2} - 3 \, x - 2} + 10 \, x - 3\right ) + \sqrt{5 \, x^{2} - 3 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*arcsin(3/7*x/abs(x) + 4/7/abs(x)) - 3/10*sqrt(5)*log(2*sqrt(5)*sqrt(5*x^
2 - 3*x - 2) + 10*x - 3) + sqrt(5*x^2 - 3*x - 2)

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Fricas [A]  time = 0.234141, size = 123, normalized size = 1.4 \[ \frac{1}{20} \, \sqrt{5}{\left (4 \, \sqrt{5} \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (3 \, x + 4\right )}}{4 \, \sqrt{5 \, x^{2} - 3 \, x - 2}}\right ) + 4 \, \sqrt{5} \sqrt{5 \, x^{2} - 3 \, x - 2} + 3 \, \log \left (\sqrt{5}{\left (200 \, x^{2} - 120 \, x - 31\right )} - 20 \, \sqrt{5 \, x^{2} - 3 \, x - 2}{\left (10 \, x - 3\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*sqrt(5)*(4*sqrt(5)*sqrt(2)*arctan(1/4*sqrt(2)*(3*x + 4)/sqrt(5*x^2 - 3*x -
2)) + 4*sqrt(5)*sqrt(5*x^2 - 3*x - 2) + 3*log(sqrt(5)*(200*x^2 - 120*x - 31) - 2
0*sqrt(5*x^2 - 3*x - 2)*(10*x - 3)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (x - 1\right ) \left (5 x + 2\right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x - 1)*(5*x + 2))/x, x)

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GIAC/XCAS [A]  time = 0.234988, size = 104, normalized size = 1.18 \[ -2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{5} x - \sqrt{5 \, x^{2} - 3 \, x - 2}\right )}\right ) + \frac{3}{10} \, \sqrt{5}{\rm ln}\left ({\left | -10 \, \sqrt{5} x + 3 \, \sqrt{5} + 10 \, \sqrt{5 \, x^{2} - 3 \, x - 2} \right |}\right ) + \sqrt{5 \, x^{2} - 3 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(5)*x - sqrt(5*x^2 - 3*x - 2))) + 3/10*sqrt(
5)*ln(abs(-10*sqrt(5)*x + 3*sqrt(5) + 10*sqrt(5*x^2 - 3*x - 2))) + sqrt(5*x^2 -
3*x - 2)