Optimal. Leaf size=77 \[ \frac{a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{a-b x}-\frac{a \sqrt{x} \sqrt{a-b x}}{4 b} \]
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Rubi [A] time = 0.0536221, antiderivative size = 77, 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 \[ \frac{a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{a-b x}-\frac{a \sqrt{x} \sqrt{a-b x}}{4 b} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*Sqrt[a - b*x],x]
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Rubi in Sympy [A] time = 7.71018, size = 65, normalized size = 0.84 \[ \frac{a^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{4 b^{\frac{3}{2}}} + \frac{a \sqrt{x} \sqrt{a - b x}}{4 b} - \frac{\sqrt{x} \left (a - b x\right )^{\frac{3}{2}}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)*(-b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0477632, size = 65, normalized size = 0.84 \[ \frac{a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )+\sqrt{b} \sqrt{x} \sqrt{a-b x} (2 b x-a)}{4 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*Sqrt[a - b*x],x]
[Out]
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Maple [A] time = 0.007, size = 86, normalized size = 1.1 \[{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{-bx+a}}-{\frac{a}{4\,b}\sqrt{x}\sqrt{-bx+a}}+{\frac{{a}^{2}}{8}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)*(-b*x+a)^(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")
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Fricas [A] time = 0.224612, size = 1, normalized size = 0.01 \[ \left [\frac{a^{2} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) + 2 \,{\left (2 \, b x - a\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{8 \, \sqrt{-b} b}, -\frac{a^{2} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (2 \, b x - a\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{4 \, b^{\frac{3}{2}}}\right ] \]
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 = 11.8246, size = 207, normalized size = 2.69 \[ \begin{cases} \frac{i a^{\frac{3}{2}} \sqrt{x}}{4 b \sqrt{-1 + \frac{b x}{a}}} - \frac{3 i \sqrt{a} x^{\frac{3}{2}}}{4 \sqrt{-1 + \frac{b x}{a}}} - \frac{i a^{2} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} + \frac{i b x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{a^{\frac{3}{2}} \sqrt{x}}{4 b \sqrt{1 - \frac{b x}{a}}} + \frac{3 \sqrt{a} x^{\frac{3}{2}}}{4 \sqrt{1 - \frac{b x}{a}}} + \frac{a^{2} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} - \frac{b x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)*(-b*x+a)**(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]