Optimal. Leaf size=44 \[ \frac{4 b}{a^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{2 \sqrt{x}}{a \sqrt{a+\frac{b}{x}}} \]
[Out]
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Rubi [A] time = 0.0531805, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{4 b}{a^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{2 \sqrt{x}}{a \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^(3/2)*Sqrt[x]),x]
[Out]
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Rubi in Sympy [A] time = 4.29511, size = 36, normalized size = 0.82 \[ \frac{2 \sqrt{x}}{a \sqrt{a + \frac{b}{x}}} + \frac{4 b}{a^{2} \sqrt{x} \sqrt{a + \frac{b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(3/2)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0438773, size = 35, normalized size = 0.8 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+2 b)}{a^2 (a x+b)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^(3/2)*Sqrt[x]),x]
[Out]
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Maple [A] time = 0.004, size = 32, normalized size = 0.7 \[ 2\,{\frac{ \left ( ax+b \right ) \left ( ax+2\,b \right ) }{{a}^{2}{x}^{3/2}} \left ({\frac{ax+b}{x}} \right ) ^{-3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(3/2)/x^(1/2),x)
[Out]
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Maxima [A] time = 1.44857, size = 49, normalized size = 1.11 \[ \frac{2 \, \sqrt{a + \frac{b}{x}} \sqrt{x}}{a^{2}} + \frac{2 \, b}{\sqrt{a + \frac{b}{x}} a^{2} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234281, size = 35, normalized size = 0.8 \[ \frac{2 \,{\left (a x + 2 \, b\right )}}{a^{2} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 26.7802, size = 39, normalized size = 0.89 \[ \frac{2 x}{a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{4 \sqrt{b}}{a^{2} \sqrt{\frac{a x}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(3/2)/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.229112, size = 42, normalized size = 0.95 \[ \frac{2 \,{\left (\sqrt{a x + b} + \frac{b}{\sqrt{a x + b}}\right )}}{a^{2}} - \frac{4 \, \sqrt{b}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*sqrt(x)),x, algorithm="giac")
[Out]