Optimal. Leaf size=89 \[ -\frac{15 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}+\frac{15 \sqrt{x} \sqrt{2-b x}}{2 b^3}+\frac{5 x^{3/2} \sqrt{2-b x}}{2 b^2}+\frac{2 x^{5/2}}{b \sqrt{2-b x}} \]
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Rubi [A] time = 0.070138, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{15 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}+\frac{15 \sqrt{x} \sqrt{2-b x}}{2 b^3}+\frac{5 x^{3/2} \sqrt{2-b x}}{2 b^2}+\frac{2 x^{5/2}}{b \sqrt{2-b x}} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)/(2 - b*x)^(3/2),x]
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Rubi in Sympy [A] time = 10.5232, size = 82, normalized size = 0.92 \[ \frac{2 x^{\frac{5}{2}}}{b \sqrt{- b x + 2}} + \frac{5 x^{\frac{3}{2}} \sqrt{- b x + 2}}{2 b^{2}} + \frac{15 \sqrt{x} \sqrt{- b x + 2}}{2 b^{3}} - \frac{15 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)/(-b*x+2)**(3/2),x)
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Mathematica [A] time = 0.10218, size = 60, normalized size = 0.67 \[ -\frac{15 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{\sqrt{x} \left (b^2 x^2+5 b x-30\right )}{2 b^3 \sqrt{2-b x}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)/(2 - b*x)^(3/2),x]
[Out]
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Maple [B] time = 0.042, size = 138, normalized size = 1.6 \[ -{\frac{ \left ( bx+7 \right ) \left ( bx-2 \right ) }{2\,{b}^{3}}\sqrt{x}\sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{-bx+2}}}}-{1 \left ({\frac{15}{2}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){b}^{-{\frac{7}{2}}}}+8\,{\frac{1}{{b}^{4}}\sqrt{-b \left ( x-2\,{b}^{-1} \right ) ^{2}-2\,x+4\,{b}^{-1}} \left ( x-2\,{b}^{-1} \right ) ^{-1}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)/(-b*x+2)^(3/2),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)/(-b*x + 2)^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.226447, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{-b x + 2} \sqrt{x} \log \left (\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right ) -{\left (b^{2} x^{3} + 5 \, b x^{2} - 30 \, x\right )} \sqrt{-b}}{2 \, \sqrt{-b x + 2} \sqrt{-b} b^{3} \sqrt{x}}, \frac{30 \, \sqrt{-b x + 2} \sqrt{x} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) -{\left (b^{2} x^{3} + 5 \, b x^{2} - 30 \, x\right )} \sqrt{b}}{2 \, \sqrt{-b x + 2} b^{\frac{7}{2}} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(-b*x + 2)^(3/2),x, algorithm="fricas")
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Sympy [A] time = 82.162, size = 173, normalized size = 1.94 \[ \begin{cases} \frac{i x^{\frac{5}{2}}}{2 b \sqrt{b x - 2}} + \frac{5 i x^{\frac{3}{2}}}{2 b^{2} \sqrt{b x - 2}} - \frac{15 i \sqrt{x}}{b^{3} \sqrt{b x - 2}} + \frac{15 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{x^{\frac{5}{2}}}{2 b \sqrt{- b x + 2}} - \frac{5 x^{\frac{3}{2}}}{2 b^{2} \sqrt{- b x + 2}} + \frac{15 \sqrt{x}}{b^{3} \sqrt{- b x + 2}} - \frac{15 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)/(-b*x+2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221612, size = 184, normalized size = 2.07 \[ \frac{{\left (\sqrt{{\left (b x - 2\right )} b + 2 \, b} \sqrt{-b x + 2}{\left (\frac{b x - 2}{b^{3}} + \frac{9}{b^{3}}\right )} - \frac{15 \,{\rm ln}\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt{-b} b^{2}} + \frac{64}{{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} \sqrt{-b} b}\right )}{\left | b \right |}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(-b*x + 2)^(3/2),x, algorithm="giac")
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