3.326 \(\int \frac{\sqrt{-a+b x}}{x^2} \, dx\)

Optimal. Leaf size=42 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b x-a}}{x} \]

[Out]

-(Sqrt[-a + b*x]/x) + (b*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/Sqrt[a]

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Rubi [A]  time = 0.0371955, antiderivative size = 42, 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 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b x-a}}{x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-a + b*x]/x^2,x]

[Out]

-(Sqrt[-a + b*x]/x) + (b*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/Sqrt[a]

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Rubi in Sympy [A]  time = 5.27163, size = 31, normalized size = 0.74 \[ - \frac{\sqrt{- a + b x}}{x} + \frac{b \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0258018, size = 42, normalized size = 1. \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b x-a}}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-a + b*x]/x^2,x]

[Out]

-(Sqrt[-a + b*x]/x) + (b*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/Sqrt[a]

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Maple [A]  time = 0.012, size = 35, normalized size = 0.8 \[{b\arctan \left ({1\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}}-{\frac{1}{x}\sqrt{bx-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x-a)^(1/2)/x^2,x)

[Out]

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

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227303, size = 1, normalized size = 0.02 \[ \left [\frac{b x \log \left (\frac{{\left (b x - 2 \, a\right )} \sqrt{-a} + 2 \, \sqrt{b x - a} a}{x}\right ) - 2 \, \sqrt{b x - a} \sqrt{-a}}{2 \, \sqrt{-a} x}, -\frac{b x \arctan \left (\frac{\sqrt{a}}{\sqrt{b x - a}}\right ) + \sqrt{b x - a} \sqrt{a}}{\sqrt{a} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(b*x*log(((b*x - 2*a)*sqrt(-a) + 2*sqrt(b*x - a)*a)/x) - 2*sqrt(b*x - a)*sq
rt(-a))/(sqrt(-a)*x), -(b*x*arctan(sqrt(a)/sqrt(b*x - a)) + sqrt(b*x - a)*sqrt(a
))/(sqrt(a)*x)]

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Sympy [A]  time = 6.71264, size = 121, normalized size = 2.88 \[ \begin{cases} - \frac{i a}{\sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{i \sqrt{b}}{\sqrt{x} \sqrt{\frac{a}{b x} - 1}} + \frac{i b \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\- \frac{\sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{\sqrt{x}} - \frac{b \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*a/(sqrt(b)*x**(3/2)*sqrt(a/(b*x) - 1)) + I*sqrt(b)/(sqrt(x)*sqrt(a
/(b*x) - 1)) + I*b*acosh(sqrt(a)/(sqrt(b)*sqrt(x)))/sqrt(a), Abs(a/(b*x)) > 1),
(-sqrt(b)*sqrt(-a/(b*x) + 1)/sqrt(x) - b*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/sqrt(a)
, True))

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GIAC/XCAS [A]  time = 0.211147, size = 55, normalized size = 1.31 \[ \frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}} - \frac{\sqrt{b x - a} b}{x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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