Optimal. Leaf size=61 \[ -\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{3 x \sqrt{a+\frac{b}{x}}}{a^2}-\frac{2 x}{a \sqrt{a+\frac{b}{x}}} \]
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Rubi [A] time = 0.0809007, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{3 x \sqrt{a+\frac{b}{x}}}{a^2}-\frac{2 x}{a \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x)^(-3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.13374, size = 51, normalized size = 0.84 \[ - \frac{2 x}{a \sqrt{a + \frac{b}{x}}} + \frac{3 x \sqrt{a + \frac{b}{x}}}{a^{2}} - \frac{3 b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(3/2),x)
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Mathematica [A] time = 0.116646, size = 67, normalized size = 1.1 \[ \frac{x \sqrt{a+\frac{b}{x}} (a x+3 b)}{a^2 (a x+b)}-\frac{3 b \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x)^(-3/2),x]
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Maple [B] time = 0.013, size = 203, normalized size = 3.3 \[ -{\frac{x}{2\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( -6\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}+4\,{a}^{7/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}-12\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }xb+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{4}b-6\,{a}^{5/2}\sqrt{x \left ( ax+b \right ) }{b}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{3}{b}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{2}{b}^{3} \right ){a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(3/2),x)
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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 = 0.255671, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b \sqrt{\frac{a x + b}{x}} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (a x + 3 \, b\right )} \sqrt{a}}{2 \, a^{\frac{5}{2}} \sqrt{\frac{a x + b}{x}}}, \frac{3 \, b \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (a x + 3 \, b\right )} \sqrt{-a}}{\sqrt{-a} a^{2} \sqrt{\frac{a x + b}{x}}}\right ] \]
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 = 10.9909, size = 71, normalized size = 1.16 \[ \frac{x^{\frac{3}{2}}}{a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{3 \sqrt{b} \sqrt{x}}{a^{2} \sqrt{\frac{a x}{b} + 1}} - \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.270231, size = 116, normalized size = 1.9 \[ b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \, a - \frac{3 \,{\left (a x + b\right )}}{x}}{{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )} a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(-3/2),x, algorithm="giac")
[Out]