Optimal. Leaf size=69 \[ 2 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x}}}{\sqrt{x}}-3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right ) \]
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Rubi [A] time = 0.0947284, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ 2 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x}}}{\sqrt{x}}-3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b/x)^(3/2)/Sqrt[x],x]
[Out]
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Rubi in Sympy [A] time = 8.35803, size = 60, normalized size = 0.87 \[ - 3 a \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )} - \frac{3 b \sqrt{a + \frac{b}{x}}}{\sqrt{x}} + 2 \sqrt{x} \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**(3/2)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.180888, size = 71, normalized size = 1.03 \[ \frac{\sqrt{a+\frac{b}{x}} (2 a x-b)}{\sqrt{x}}-3 a \sqrt{b} \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+\frac{3}{2} a \sqrt{b} \log (x) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x)^(3/2)/Sqrt[x],x]
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Maple [A] time = 0.022, size = 70, normalized size = 1. \[ -{1\sqrt{{\frac{ax+b}{x}}} \left ({b}^{{\frac{3}{2}}}\sqrt{ax+b}-2\,xa\sqrt{ax+b}\sqrt{b}+3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) xab \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{ax+b}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^(3/2)/x^(1/2),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((a + b/x)^(3/2)/sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.244022, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a \sqrt{b} x \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) + 2 \,{\left (2 \, a x - b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{2 \, x}, -\frac{3 \, a \sqrt{-b} x \arctan \left (\frac{\sqrt{x} \sqrt{\frac{a x + b}{x}}}{\sqrt{-b}}\right ) -{\left (2 \, a x - b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(3/2)/sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 48.0271, size = 92, normalized size = 1.33 \[ \frac{2 a^{\frac{3}{2}} \sqrt{x}}{\sqrt{1 + \frac{b}{a x}}} + \frac{\sqrt{a} b}{\sqrt{x} \sqrt{1 + \frac{b}{a x}}} - 3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )} - \frac{b^{2}}{\sqrt{a} x^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**(3/2)/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.26841, size = 68, normalized size = 0.99 \[{\left (\frac{3 \, b \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \, \sqrt{a x + b} - \frac{\sqrt{a x + b} b}{a x}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(3/2)/sqrt(x),x, algorithm="giac")
[Out]