Optimal. Leaf size=35 \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.032716, antiderivative size = 35, 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 \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - a*x]/Sqrt[x],x]
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Rubi in Sympy [A] time = 5.92644, size = 29, normalized size = 0.83 \[ \sqrt{x} \sqrt{- a x + 1} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-a*x+1)**(1/2)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0265896, size = 35, normalized size = 1. \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - a*x]/Sqrt[x],x]
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Maple [B] time = 0.026, size = 62, normalized size = 1.8 \[ \sqrt{x}\sqrt{-ax+1}+{\frac{1}{2}\sqrt{ \left ( -ax+1 \right ) x}\arctan \left ({1\sqrt{a} \left ( x-{\frac{1}{2\,a}} \right ){\frac{1}{\sqrt{-a{x}^{2}+x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-ax+1}}}{\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-a*x+1)^(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(-a*x + 1)/sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.287702, size = 1, normalized size = 0.03 \[ \left [\frac{2 \, \sqrt{-a x + 1} \sqrt{-a} \sqrt{x} + \log \left (-2 \, \sqrt{-a x + 1} a \sqrt{x} -{\left (2 \, a x - 1\right )} \sqrt{-a}\right )}{2 \, \sqrt{-a}}, \frac{\sqrt{-a x + 1} \sqrt{a} \sqrt{x} - \arctan \left (\frac{\sqrt{-a x + 1}}{\sqrt{a} \sqrt{x}}\right )}{\sqrt{a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-a*x + 1)/sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 5.87043, size = 83, normalized size = 2.37 \[ \begin{cases} \frac{i a x^{\frac{3}{2}}}{\sqrt{a x - 1}} - \frac{i \sqrt{x}}{\sqrt{a x - 1}} - \frac{i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} & \text{for}\: \left |{a x}\right | > 1 \\\sqrt{x} \sqrt{- a x + 1} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-a*x+1)**(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(-a*x + 1)/sqrt(x),x, algorithm="giac")
[Out]