3.336 \(\int \frac{1}{x \sqrt{-25+2 x}} \, dx\)

Optimal. Leaf size=18 \[ \frac{2}{5} \tan ^{-1}\left (\frac{1}{5} \sqrt{2 x-25}\right ) \]

[Out]

(2*ArcTan[Sqrt[-25 + 2*x]/5])/5

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Rubi [A]  time = 0.0120941, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2}{5} \tan ^{-1}\left (\frac{1}{5} \sqrt{2 x-25}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[-25 + 2*x]),x]

[Out]

(2*ArcTan[Sqrt[-25 + 2*x]/5])/5

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Rubi in Sympy [A]  time = 1.12354, size = 14, normalized size = 0.78 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{2 x - 25}}{5} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-25+2*x)**(1/2),x)

[Out]

2*atan(sqrt(2*x - 25)/5)/5

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Mathematica [A]  time = 0.00637822, size = 18, normalized size = 1. \[ \frac{2}{5} \tan ^{-1}\left (\frac{1}{5} \sqrt{2 x-25}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[-25 + 2*x]),x]

[Out]

(2*ArcTan[Sqrt[-25 + 2*x]/5])/5

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Maple [A]  time = 0.008, size = 13, normalized size = 0.7 \[{\frac{2}{5}\arctan \left ({\frac{1}{5}\sqrt{-25+2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-25+2*x)^(1/2),x)

[Out]

2/5*arctan(1/5*(-25+2*x)^(1/2))

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Maxima [A]  time = 1.51821, size = 16, normalized size = 0.89 \[ \frac{2}{5} \, \arctan \left (\frac{1}{5} \, \sqrt{2 \, x - 25}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*x - 25)*x),x, algorithm="maxima")

[Out]

2/5*arctan(1/5*sqrt(2*x - 25))

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Fricas [A]  time = 0.210336, size = 16, normalized size = 0.89 \[ \frac{2}{5} \, \arctan \left (\frac{1}{5} \, \sqrt{2 \, x - 25}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*x - 25)*x),x, algorithm="fricas")

[Out]

2/5*arctan(1/5*sqrt(2*x - 25))

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Sympy [A]  time = 1.67547, size = 46, normalized size = 2.56 \[ \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{5 \sqrt{2}}{2 \sqrt{x}} \right )}}{5} & \text{for}\: \frac{25 \left |{\frac{1}{x}}\right |}{2} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{5 \sqrt{2}}{2 \sqrt{x}} \right )}}{5} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-25+2*x)**(1/2),x)

[Out]

Piecewise((2*I*acosh(5*sqrt(2)/(2*sqrt(x)))/5, 25*Abs(1/x)/2 > 1), (-2*asin(5*sq
rt(2)/(2*sqrt(x)))/5, True))

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GIAC/XCAS [A]  time = 0.216565, size = 16, normalized size = 0.89 \[ \frac{2}{5} \, \arctan \left (\frac{1}{5} \, \sqrt{2 \, x - 25}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*x - 25)*x),x, algorithm="giac")

[Out]

2/5*arctan(1/5*sqrt(2*x - 25))