Optimal. Leaf size=82 \[ \frac{1}{3} \sqrt{x} (2-b x)^{5/2}+\frac{5}{6} \sqrt{x} (2-b x)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{2-b x}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.058617, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{3} \sqrt{x} (2-b x)^{5/2}+\frac{5}{6} \sqrt{x} (2-b x)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{2-b x}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[(2 - b*x)^(5/2)/Sqrt[x],x]
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Rubi in Sympy [A] time = 9.90206, size = 73, normalized size = 0.89 \[ \frac{\sqrt{x} \left (- b x + 2\right )^{\frac{5}{2}}}{3} + \frac{5 \sqrt{x} \left (- b x + 2\right )^{\frac{3}{2}}}{6} + \frac{5 \sqrt{x} \sqrt{- b x + 2}}{2} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x+2)**(5/2)/x**(1/2),x)
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Mathematica [A] time = 0.0586113, size = 58, normalized size = 0.71 \[ \frac{1}{6} \sqrt{x} \sqrt{2-b x} \left (2 b^2 x^2-13 b x+33\right )+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 - b*x)^(5/2)/Sqrt[x],x]
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Maple [A] time = 0.009, size = 91, normalized size = 1.1 \[{\frac{1}{3} \left ( -bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{5}{6} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5}{2}\sqrt{x}\sqrt{-bx+2}}+{\frac{5}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x+2)^(5/2)/x^(1/2),x)
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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((-b*x + 2)^(5/2)/sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.227477, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, b^{2} x^{2} - 13 \, b x + 33\right )} \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 15 \, \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{6 \, \sqrt{-b}}, \frac{{\left (2 \, b^{2} x^{2} - 13 \, b x + 33\right )} \sqrt{-b x + 2} \sqrt{b} \sqrt{x} - 30 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{6 \, \sqrt{b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x + 2)^(5/2)/sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 59.5573, size = 209, normalized size = 2.55 \[ \begin{cases} \frac{i b^{3} x^{\frac{7}{2}}}{3 \sqrt{b x - 2}} - \frac{17 i b^{2} x^{\frac{5}{2}}}{6 \sqrt{b x - 2}} + \frac{59 i b x^{\frac{3}{2}}}{6 \sqrt{b x - 2}} - \frac{11 i \sqrt{x}}{\sqrt{b x - 2}} - \frac{5 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{b^{3} x^{\frac{7}{2}}}{3 \sqrt{- b x + 2}} + \frac{17 b^{2} x^{\frac{5}{2}}}{6 \sqrt{- b x + 2}} - \frac{59 b x^{\frac{3}{2}}}{6 \sqrt{- b x + 2}} + \frac{11 \sqrt{x}}{\sqrt{- b x + 2}} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x+2)**(5/2)/x**(1/2),x)
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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((-b*x + 2)^(5/2)/sqrt(x),x, algorithm="giac")
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