Optimal. Leaf size=53 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}-\frac{2 b \sqrt{x}}{a^2}+\frac{2 x^{3/2}}{3 a} \]
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Rubi [A] time = 0.0602211, antiderivative size = 53, 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 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}-\frac{2 b \sqrt{x}}{a^2}+\frac{2 x^{3/2}}{3 a} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/(a + b/x),x]
[Out]
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Rubi in Sympy [A] time = 10.8104, size = 49, normalized size = 0.92 \[ \frac{2 x^{\frac{3}{2}}}{3 a} - \frac{2 b \sqrt{x}}{a^{2}} + \frac{2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(a+b/x),x)
[Out]
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Mathematica [A] time = 0.0368486, size = 49, normalized size = 0.92 \[ \frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{5/2}}+\frac{2 \sqrt{x} (a x-3 b)}{3 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/(a + b/x),x]
[Out]
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Maple [A] time = 0.007, size = 43, normalized size = 0.8 \[{\frac{2}{3\,a}{x}^{{\frac{3}{2}}}}-2\,{\frac{b\sqrt{x}}{{a}^{2}}}+2\,{\frac{{b}^{2}}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(a+b/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(sqrt(x)/(a + b/x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239363, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b \sqrt{-\frac{b}{a}} \log \left (\frac{a x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (a x - 3 \, b\right )} \sqrt{x}}{3 \, a^{2}}, \frac{2 \,{\left (3 \, b \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) +{\left (a x - 3 \, b\right )} \sqrt{x}\right )}}{3 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(a + b/x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.73743, size = 105, normalized size = 1.98 \[ \begin{cases} \frac{2 x^{\frac{3}{2}}}{3 a} - \frac{2 b \sqrt{x}}{a^{2}} - \frac{i b^{\frac{3}{2}} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{3} \sqrt{\frac{1}{a}}} + \frac{i b^{\frac{3}{2}} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{3} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{5}{2}}}{5 b} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(a+b/x),x)
[Out]
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GIAC/XCAS [A] time = 0.223446, size = 61, normalized size = 1.15 \[ \frac{2 \, b^{2} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{2 \,{\left (a^{2} x^{\frac{3}{2}} - 3 \, a b \sqrt{x}\right )}}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(a + b/x),x, algorithm="giac")
[Out]