Optimal. Leaf size=42 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{a+\frac{b}{x}}} \]
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Rubi [A] time = 0.0714656, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^(3/2)*x),x]
[Out]
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Rubi in Sympy [A] time = 6.9679, size = 32, normalized size = 0.76 \[ - \frac{2}{a \sqrt{a + \frac{b}{x}}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(3/2)/x,x)
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Mathematica [A] time = 0.0763326, size = 57, normalized size = 1.36 \[ \frac{\log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{3/2}}-\frac{2 x \sqrt{a+\frac{b}{x}}}{a (a x+b)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^(3/2)*x),x]
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Maple [B] time = 0.011, size = 198, normalized size = 4.7 \[{\frac{x}{b \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( \ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{a}^{2}b-2\,{a}^{5/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}+2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) xa{b}^{2}+2\,{a}^{3/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}-4\,{a}^{3/2}\sqrt{x \left ( ax+b \right ) }xb+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){b}^{3}-2\,\sqrt{a}\sqrt{x \left ( ax+b \right ) }{b}^{2} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(3/2)/x,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(1/((a + b/x)^(3/2)*x),x, algorithm="maxima")
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Fricas [A] time = 0.248218, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \, \sqrt{a}}{a^{\frac{3}{2}} \sqrt{\frac{a x + b}{x}}}, -\frac{2 \,{\left (\sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) + \sqrt{-a}\right )}}{\sqrt{-a} a \sqrt{\frac{a x + b}{x}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*x),x, algorithm="fricas")
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Sympy [A] time = 6.63779, size = 148, normalized size = 3.52 \[ - \frac{2 a^{3} x \sqrt{1 + \frac{b}{a x}}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} - \frac{a^{3} x \log{\left (\frac{b}{a x} \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} + \frac{2 a^{3} x \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x} \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(3/2)/x,x)
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GIAC/XCAS [A] time = 0.263957, size = 70, normalized size = 1.67 \[ -2 \, b{\left (\frac{\arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b} + \frac{1}{a b \sqrt{\frac{a x + b}{x}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*x),x, algorithm="giac")
[Out]