Optimal. Leaf size=55 \[ 2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 a \sqrt{a+\frac{b}{x}}-\frac{2}{3} \left (a+\frac{b}{x}\right )^{3/2} \]
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Rubi [A] time = 0.0865256, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ 2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 a \sqrt{a+\frac{b}{x}}-\frac{2}{3} \left (a+\frac{b}{x}\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x)^(3/2)/x,x]
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Rubi in Sympy [A] time = 8.54465, size = 44, normalized size = 0.8 \[ 2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - 2 a \sqrt{a + \frac{b}{x}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**(3/2)/x,x)
[Out]
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Mathematica [A] time = 0.0855327, size = 57, normalized size = 1.04 \[ a^{3/2} \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )-\frac{2 \sqrt{a+\frac{b}{x}} (4 a x+b)}{3 x} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x)^(3/2)/x,x]
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Maple [B] time = 0.016, size = 115, normalized size = 2.1 \[{\frac{1}{3\,b{x}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 3\,{a}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}b+6\,{a}^{2}\sqrt{a{x}^{2}+bx}{x}^{3}-6\,a \left ( a{x}^{2}+bx \right ) ^{3/2}x-2\, \left ( a{x}^{2}+bx \right ) ^{3/2}b \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^(3/2)/x,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,x, algorithm="maxima")
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Fricas [A] time = 0.237224, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a^{\frac{3}{2}} x \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (4 \, a x + b\right )} \sqrt{\frac{a x + b}{x}}}{3 \, x}, \frac{2 \,{\left (3 \, \sqrt{-a} a x \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) -{\left (4 \, a x + b\right )} \sqrt{\frac{a x + b}{x}}\right )}}{3 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(3/2)/x,x, algorithm="fricas")
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Sympy [A] time = 7.53769, size = 71, normalized size = 1.29 \[ - \frac{8 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}}{3} - a^{\frac{3}{2}} \log{\left (\frac{b}{a x} \right )} + 2 a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )} - \frac{2 \sqrt{a} b \sqrt{1 + \frac{b}{a x}}}{3 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**(3/2)/x,x)
[Out]
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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,x, algorithm="giac")
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