Optimal. Leaf size=65 \[ -\frac{6 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}+\frac{3 \sqrt{x} \sqrt{2-b x}}{b^2}+\frac{2 x^{3/2}}{b \sqrt{2-b x}} \]
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Rubi [A] time = 0.0520545, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{6 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}}+\frac{3 \sqrt{x} \sqrt{2-b x}}{b^2}+\frac{2 x^{3/2}}{b \sqrt{2-b x}} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/(2 - b*x)^(3/2),x]
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Rubi in Sympy [A] time = 8.70608, size = 60, normalized size = 0.92 \[ \frac{2 x^{\frac{3}{2}}}{b \sqrt{- b x + 2}} + \frac{3 \sqrt{x} \sqrt{- b x + 2}}{b^{2}} - \frac{6 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(-b*x+2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0757573, size = 50, normalized size = 0.77 \[ \frac{\sqrt{x} (6-b x)}{b^2 \sqrt{2-b x}}-\frac{6 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/(2 - b*x)^(3/2),x]
[Out]
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Maple [B] time = 0.036, size = 133, normalized size = 2.1 \[ -{\frac{bx-2}{{b}^{2}}\sqrt{x}\sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{-bx+2}}}}-{1 \left ( 3\,{\frac{1}{{b}^{5/2}}\arctan \left ({\frac{\sqrt{b}}{\sqrt{-b{x}^{2}+2\,x}} \left ( x-{b}^{-1} \right ) } \right ) }+4\,{\frac{1}{{b}^{3}}\sqrt{-b \left ( x-2\,{b}^{-1} \right ) ^{2}-2\,x+4\,{b}^{-1}} \left ( x-2\,{b}^{-1} \right ) ^{-1}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(-b*x+2)^(3/2),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(x^(3/2)/(-b*x + 2)^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.22152, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{-b x + 2} \sqrt{x} \log \left (\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right ) -{\left (b x^{2} - 6 \, x\right )} \sqrt{-b}}{\sqrt{-b x + 2} \sqrt{-b} b^{2} \sqrt{x}}, \frac{6 \, \sqrt{-b x + 2} \sqrt{x} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) -{\left (b x^{2} - 6 \, x\right )} \sqrt{b}}{\sqrt{-b x + 2} b^{\frac{5}{2}} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(-b*x + 2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 14.7449, size = 128, normalized size = 1.97 \[ \begin{cases} \frac{i x^{\frac{3}{2}}}{b \sqrt{b x - 2}} - \frac{6 i \sqrt{x}}{b^{2} \sqrt{b x - 2}} + \frac{6 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{x^{\frac{3}{2}}}{b \sqrt{- b x + 2}} + \frac{6 \sqrt{x}}{b^{2} \sqrt{- b x + 2}} - \frac{6 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)/(-b*x+2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.218386, size = 162, normalized size = 2.49 \[ \frac{{\left (\frac{3 \, \sqrt{-b}{\rm ln}\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{b} + \frac{\sqrt{{\left (b x - 2\right )} b + 2 \, b} \sqrt{-b x + 2}}{b} - \frac{16 \, \sqrt{-b}}{{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b}\right )}{\left | b \right |}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(-b*x + 2)^(3/2),x, algorithm="giac")
[Out]