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