Optimal. Leaf size=82 \[ -\frac{2 (2-b x)^{5/2}}{\sqrt{x}}-\frac{5}{2} b \sqrt{x} (2-b x)^{3/2}-\frac{15}{2} b \sqrt{x} \sqrt{2-b x}-15 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0595271, antiderivative size = 82, 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 \[ -\frac{2 (2-b x)^{5/2}}{\sqrt{x}}-\frac{5}{2} b \sqrt{x} (2-b x)^{3/2}-\frac{15}{2} b \sqrt{x} \sqrt{2-b x}-15 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(2 - b*x)^(5/2)/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 9.78192, size = 78, normalized size = 0.95 \[ - 15 \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{5 b \sqrt{x} \left (- b x + 2\right )^{\frac{3}{2}}}{2} - \frac{15 b \sqrt{x} \sqrt{- b x + 2}}{2} - \frac{2 \left (- b x + 2\right )^{\frac{5}{2}}}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x+2)**(5/2)/x**(3/2),x)
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Mathematica [A] time = 0.0501641, size = 57, normalized size = 0.7 \[ \frac{\sqrt{2-b x} \left (b^2 x^2-9 b x-16\right )}{2 \sqrt{x}}-15 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(2 - b*x)^(5/2)/x^(3/2),x]
[Out]
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Maple [A] time = 0.027, size = 106, normalized size = 1.3 \[ -{\frac{{b}^{3}{x}^{3}-11\,{b}^{2}{x}^{2}+2\,bx+32}{2}\sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{-x \left ( bx-2 \right ) }}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}}-{\frac{15}{2}\sqrt{b}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x+2)^(5/2)/x^(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((-b*x + 2)^(5/2)/x^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.227175, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{-b} x \log \left (-b x + \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right ) +{\left (b^{2} x^{2} - 9 \, b x - 16\right )} \sqrt{-b x + 2} \sqrt{x}}{2 \, x}, \frac{30 \, \sqrt{b} x \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) +{\left (b^{2} x^{2} - 9 \, b x - 16\right )} \sqrt{-b x + 2} \sqrt{x}}{2 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x + 2)^(5/2)/x^(3/2),x, algorithm="fricas")
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Sympy [A] time = 76.389, size = 202, normalized size = 2.46 \[ \begin{cases} 15 i \sqrt{b} \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} + \frac{i b^{3} x^{\frac{5}{2}}}{2 \sqrt{b x - 2}} - \frac{11 i b^{2} x^{\frac{3}{2}}}{2 \sqrt{b x - 2}} + \frac{i b \sqrt{x}}{\sqrt{b x - 2}} + \frac{16 i}{\sqrt{x} \sqrt{b x - 2}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- 15 \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{b^{3} x^{\frac{5}{2}}}{2 \sqrt{- b x + 2}} + \frac{11 b^{2} x^{\frac{3}{2}}}{2 \sqrt{- b x + 2}} - \frac{b \sqrt{x}}{\sqrt{- b x + 2}} - \frac{16}{\sqrt{x} \sqrt{- b x + 2}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x+2)**(5/2)/x**(3/2),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((-b*x + 2)^(5/2)/x^(3/2),x, algorithm="giac")
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