3.325 \(\int \frac{\sqrt{-a+b x}}{x} \, dx\)

Optimal. Leaf size=39 \[ 2 \sqrt{b x-a}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]

[Out]

2*Sqrt[-a + b*x] - 2*Sqrt[a]*ArcTan[Sqrt[-a + b*x]/Sqrt[a]]

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Rubi [A]  time = 0.0361587, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 2 \sqrt{b x-a}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-a + b*x]/x,x]

[Out]

2*Sqrt[-a + b*x] - 2*Sqrt[a]*ArcTan[Sqrt[-a + b*x]/Sqrt[a]]

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Rubi in Sympy [A]  time = 5.13146, size = 31, normalized size = 0.79 \[ - 2 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )} + 2 \sqrt{- a + b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x-a)**(1/2)/x,x)

[Out]

-2*sqrt(a)*atan(sqrt(-a + b*x)/sqrt(a)) + 2*sqrt(-a + b*x)

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Mathematica [A]  time = 0.0203416, size = 39, normalized size = 1. \[ 2 \sqrt{b x-a}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-a + b*x]/x,x]

[Out]

2*Sqrt[-a + b*x] - 2*Sqrt[a]*ArcTan[Sqrt[-a + b*x]/Sqrt[a]]

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Maple [A]  time = 0.01, size = 32, normalized size = 0.8 \[ -2\,\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) \sqrt{a}+2\,\sqrt{bx-a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x-a)^(1/2)/x,x)

[Out]

-2*arctan((b*x-a)^(1/2)/a^(1/2))*a^(1/2)+2*(b*x-a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x - a)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230187, size = 1, normalized size = 0.03 \[ \left [\sqrt{-a} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + 2 \, \sqrt{b x - a}, -2 \, \sqrt{a} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + 2 \, \sqrt{b x - a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x - a)/x,x, algorithm="fricas")

[Out]

[sqrt(-a)*log((b*x - 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) + 2*sqrt(b*x - a), -2*sq
rt(a)*arctan(sqrt(b*x - a)/sqrt(a)) + 2*sqrt(b*x - a)]

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Sympy [A]  time = 5.43088, size = 148, normalized size = 3.79 \[ \begin{cases} - 2 i \sqrt{a} \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 i a}{\sqrt{b} \sqrt{x} \sqrt{\frac{a}{b x} - 1}} - \frac{2 i \sqrt{b} \sqrt{x}}{\sqrt{\frac{a}{b x} - 1}} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\2 \sqrt{a} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} - \frac{2 a}{\sqrt{b} \sqrt{x} \sqrt{- \frac{a}{b x} + 1}} + \frac{2 \sqrt{b} \sqrt{x}}{\sqrt{- \frac{a}{b x} + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x-a)**(1/2)/x,x)

[Out]

Piecewise((-2*I*sqrt(a)*acosh(sqrt(a)/(sqrt(b)*sqrt(x))) + 2*I*a/(sqrt(b)*sqrt(x
)*sqrt(a/(b*x) - 1)) - 2*I*sqrt(b)*sqrt(x)/sqrt(a/(b*x) - 1), Abs(a/(b*x)) > 1),
(2*sqrt(a)*asin(sqrt(a)/(sqrt(b)*sqrt(x))) - 2*a/(sqrt(b)*sqrt(x)*sqrt(-a/(b*x)
+ 1)) + 2*sqrt(b)*sqrt(x)/sqrt(-a/(b*x) + 1), True))

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GIAC/XCAS [A]  time = 0.207293, size = 42, normalized size = 1.08 \[ -2 \, \sqrt{a} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + 2 \, \sqrt{b x - a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x - a)/x,x, algorithm="giac")

[Out]

-2*sqrt(a)*arctan(sqrt(b*x - a)/sqrt(a)) + 2*sqrt(b*x - a)