Optimal. Leaf size=91 \[ \frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{5 \sqrt{x} \sqrt{2-b x}}{2 b^3}-\frac{5 x^{3/2} \sqrt{2-b x}}{6 b^2}-\frac{x^{5/2} \sqrt{2-b x}}{3 b} \]
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Rubi [A] time = 0.0702088, antiderivative size = 91, 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 \[ \frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{5 \sqrt{x} \sqrt{2-b x}}{2 b^3}-\frac{5 x^{3/2} \sqrt{2-b x}}{6 b^2}-\frac{x^{5/2} \sqrt{2-b x}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)/Sqrt[2 - b*x],x]
[Out]
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Rubi in Sympy [A] time = 9.61548, size = 82, normalized size = 0.9 \[ - \frac{x^{\frac{5}{2}} \sqrt{- b x + 2}}{3 b} - \frac{5 x^{\frac{3}{2}} \sqrt{- b x + 2}}{6 b^{2}} - \frac{5 \sqrt{x} \sqrt{- b x + 2}}{2 b^{3}} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)/(-b*x+2)**(1/2),x)
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Mathematica [A] time = 0.071209, size = 61, normalized size = 0.67 \[ \frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{\sqrt{x} \sqrt{2-b x} \left (2 b^2 x^2+5 b x+15\right )}{6 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)/Sqrt[2 - b*x],x]
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Maple [A] time = 0.007, size = 100, normalized size = 1.1 \[ -{\frac{1}{3\,b}{x}^{{\frac{5}{2}}}\sqrt{-bx+2}}-{\frac{5}{6\,{b}^{2}}{x}^{{\frac{3}{2}}}\sqrt{-bx+2}}-{\frac{5}{2\,{b}^{3}}\sqrt{x}\sqrt{-bx+2}}+{\frac{5}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)/(-b*x+2)^(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(x^(5/2)/sqrt(-b*x + 2),x, algorithm="maxima")
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Fricas [A] time = 0.244485, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, b^{2} x^{2} + 5 \, b x + 15\right )} \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} - 15 \, \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{6 \, \sqrt{-b} b^{3}}, -\frac{{\left (2 \, b^{2} x^{2} + 5 \, b x + 15\right )} \sqrt{-b x + 2} \sqrt{b} \sqrt{x} + 30 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{6 \, b^{\frac{7}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/sqrt(-b*x + 2),x, algorithm="fricas")
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Sympy [A] time = 65.4969, size = 206, normalized size = 2.26 \[ \begin{cases} - \frac{i x^{\frac{7}{2}}}{3 \sqrt{b x - 2}} - \frac{i x^{\frac{5}{2}}}{6 b \sqrt{b x - 2}} - \frac{5 i x^{\frac{3}{2}}}{6 b^{2} \sqrt{b x - 2}} + \frac{5 i \sqrt{x}}{b^{3} \sqrt{b x - 2}} - \frac{5 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\\frac{x^{\frac{7}{2}}}{3 \sqrt{- b x + 2}} + \frac{x^{\frac{5}{2}}}{6 b \sqrt{- b x + 2}} + \frac{5 x^{\frac{3}{2}}}{6 b^{2} \sqrt{- b x + 2}} - \frac{5 \sqrt{x}}{b^{3} \sqrt{- b x + 2}} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)/(-b*x+2)**(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(x^(5/2)/sqrt(-b*x + 2),x, algorithm="giac")
[Out]