Optimal. Leaf size=54 \[ x \left (a+\frac{b}{x}\right )^{3/2}-3 b \sqrt{a+\frac{b}{x}}+3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0767684, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ x \left (a+\frac{b}{x}\right )^{3/2}-3 b \sqrt{a+\frac{b}{x}}+3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b/x)^(3/2),x]
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Rubi in Sympy [A] time = 7.71145, size = 44, normalized size = 0.81 \[ 3 \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - 3 b \sqrt{a + \frac{b}{x}} + x \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**(3/2),x)
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Mathematica [A] time = 0.0458004, size = 56, normalized size = 1.04 \[ \sqrt{a+\frac{b}{x}} (a x-2 b)+\frac{3}{2} \sqrt{a} b \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x)^(3/2),x]
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Maple [B] time = 0.006, size = 94, normalized size = 1.7 \[{\frac{1}{2\,x}\sqrt{{\frac{ax+b}{x}}} \left ( 3\,\sqrt{a}b\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}+6\,a\sqrt{a{x}^{2}+bx}{x}^{2}-4\, \left ( a{x}^{2}+bx \right ) ^{3/2} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^(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((a + b/x)^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.263802, size = 1, normalized size = 0.02 \[ \left [\frac{3}{2} \, \sqrt{a} b \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) +{\left (a x - 2 \, b\right )} \sqrt{\frac{a x + b}{x}}, 3 \, \sqrt{-a} b \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) +{\left (a x - 2 \, b\right )} \sqrt{\frac{a x + b}{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(3/2),x, algorithm="fricas")
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Sympy [A] time = 9.18015, size = 92, normalized size = 1.7 \[ 3 \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} + \frac{a^{2} x^{\frac{3}{2}}}{\sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{a \sqrt{b} \sqrt{x}}{\sqrt{\frac{a x}{b} + 1}} - \frac{2 b^{\frac{3}{2}}}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**(3/2),x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(3/2),x, algorithm="giac")
[Out]