Optimal. Leaf size=63 \[ \frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{1-a x}-\frac{\sqrt{x} \sqrt{1-a x}}{4 a} \]
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Rubi [A] time = 0.0490371, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{1-a x}-\frac{\sqrt{x} \sqrt{1-a x}}{4 a} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*Sqrt[1 - a*x],x]
[Out]
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Rubi in Sympy [A] time = 8.20873, size = 51, normalized size = 0.81 \[ - \frac{\sqrt{x} \left (- a x + 1\right )^{\frac{3}{2}}}{2 a} + \frac{\sqrt{x} \sqrt{- a x + 1}}{4 a} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{4 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)*(-a*x+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0406145, size = 49, normalized size = 0.78 \[ \frac{\sqrt{a} \sqrt{x} \sqrt{1-a x} (2 a x-1)+\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*Sqrt[1 - a*x],x]
[Out]
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Maple [A] time = 0.007, size = 79, normalized size = 1.3 \[{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{-ax+1}}-{\frac{1}{4\,a}\sqrt{x}\sqrt{-ax+1}}+{\frac{1}{8}\sqrt{ \left ( -ax+1 \right ) x}\arctan \left ({1\sqrt{a} \left ( x-{\frac{1}{2\,a}} \right ){\frac{1}{\sqrt{-a{x}^{2}+x}}}} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-ax+1}}}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)*(-a*x+1)^(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")
[Out]
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Fricas [A] time = 0.287385, size = 1, normalized size = 0.02 \[ \left [\frac{2 \,{\left (2 \, a x - 1\right )} \sqrt{-a x + 1} \sqrt{-a} \sqrt{x} + \log \left (-2 \, \sqrt{-a x + 1} a \sqrt{x} -{\left (2 \, a x - 1\right )} \sqrt{-a}\right )}{8 \, \sqrt{-a} a}, \frac{{\left (2 \, a x - 1\right )} \sqrt{-a x + 1} \sqrt{a} \sqrt{x} - \arctan \left (\frac{\sqrt{-a x + 1}}{\sqrt{a} \sqrt{x}}\right )}{4 \, a^{\frac{3}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-a*x + 1)*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.2354, size = 148, normalized size = 2.35 \[ \begin{cases} \frac{i a x^{\frac{5}{2}}}{2 \sqrt{a x - 1}} - \frac{3 i x^{\frac{3}{2}}}{4 \sqrt{a x - 1}} + \frac{i \sqrt{x}}{4 a \sqrt{a x - 1}} - \frac{i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{4 a^{\frac{3}{2}}} & \text{for}\: \left |{a x}\right | > 1 \\- \frac{a x^{\frac{5}{2}}}{2 \sqrt{- a x + 1}} + \frac{3 x^{\frac{3}{2}}}{4 \sqrt{- a x + 1}} - \frac{\sqrt{x}}{4 a \sqrt{- a x + 1}} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)*(-a*x+1)**(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]