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

Optimal. Leaf size=46 \[ \sqrt{x} \sqrt{a-b x}+\frac{a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0350356, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \sqrt{x} \sqrt{a-b x}+\frac{a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a - b*x]/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.24705, size = 39, normalized size = 0.85 \[ - \frac{a \operatorname{atan}{\left (\frac{\sqrt{a - b x}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{b}} + \sqrt{x} \sqrt{a - b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0259138, size = 46, normalized size = 1. \[ \sqrt{x} \sqrt{a-b x}+\frac{a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a - b*x]/Sqrt[x],x]

[Out]

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

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Maple [A]  time = 0.007, size = 66, normalized size = 1.4 \[ \sqrt{x}\sqrt{-bx+a}+{\frac{a}{2}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+a)^(1/2)/x^(1/2),x)

[Out]

x^(1/2)*(-b*x+a)^(1/2)+1/2*a*(x*(-b*x+a))^(1/2)/(-b*x+a)^(1/2)/x^(1/2)/b^(1/2)*a
rctan(b^(1/2)*(x-1/2*a/b)/(-b*x^2+a*x)^(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)/sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231128, size = 1, normalized size = 0.02 \[ \left [\frac{a \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) + 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{2 \, \sqrt{-b}}, -\frac{a \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) - \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{\sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.6744, size = 119, normalized size = 2.59 \[ \begin{cases} - \frac{i \sqrt{a} \sqrt{x}}{\sqrt{-1 + \frac{b x}{a}}} - \frac{i a \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{b}} + \frac{i b x^{\frac{3}{2}}}{\sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\\sqrt{a} \sqrt{x} \sqrt{1 - \frac{b x}{a}} + \frac{a \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError