Optimal. Leaf size=40 \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.0454216, antiderivative size = 40, 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 \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)*Sqrt[x]),x]
[Out]
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Rubi in Sympy [A] time = 8.22622, size = 36, normalized size = 0.9 \[ \frac{2 \sqrt{x}}{a} - \frac{2 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0208891, size = 40, normalized size = 1. \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)*Sqrt[x]),x]
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Maple [A] time = 0.008, size = 32, normalized size = 0.8 \[ 2\,{\frac{\sqrt{x}}{a}}-2\,{\frac{b}{a\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)/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(1/((a + b/x)*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239034, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \, \sqrt{x}}{a}, -\frac{2 \,{\left (\sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) - \sqrt{x}\right )}}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.90047, size = 92, normalized size = 2.3 \[ \begin{cases} \frac{2 \sqrt{x}}{a} + \frac{i \sqrt{b} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{2} \sqrt{\frac{1}{a}}} - \frac{i \sqrt{b} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{2} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.220006, size = 42, normalized size = 1.05 \[ -\frac{2 \, b \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{2 \, \sqrt{x}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)*sqrt(x)),x, algorithm="giac")
[Out]