Optimal. Leaf size=48 \[ \frac{2 x^{3/2} \sqrt{a+\frac{b}{x}}}{3 a}-\frac{4 b \sqrt{x} \sqrt{a+\frac{b}{x}}}{3 a^2} \]
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Rubi [A] time = 0.0125094, antiderivative size = 48, 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, Rules used = {271, 264} \[ \frac{2 x^{3/2} \sqrt{a+\frac{b}{x}}}{3 a}-\frac{4 b \sqrt{x} \sqrt{a+\frac{b}{x}}}{3 a^2} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{\sqrt{a+\frac{b}{x}}} \, dx &=\frac{2 \sqrt{a+\frac{b}{x}} x^{3/2}}{3 a}-\frac{(2 b) \int \frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}} \, dx}{3 a}\\ &=-\frac{4 b \sqrt{a+\frac{b}{x}} \sqrt{x}}{3 a^2}+\frac{2 \sqrt{a+\frac{b}{x}} x^{3/2}}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0168336, size = 30, normalized size = 0.62 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x-2 b)}{3 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( ax-2\,b \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948021, size = 46, normalized size = 0.96 \begin{align*} \frac{2 \,{\left ({\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}} - 3 \, \sqrt{a + \frac{b}{x}} b \sqrt{x}\right )}}{3 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42539, size = 63, normalized size = 1.31 \begin{align*} \frac{2 \,{\left (a x - 2 \, b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.47776, size = 42, normalized size = 0.88 \begin{align*} \frac{2 \sqrt{b} x \sqrt{\frac{a x}{b} + 1}}{3 a} - \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1704, size = 43, normalized size = 0.9 \begin{align*} \frac{4 \, b^{\frac{3}{2}}}{3 \, a^{2}} + \frac{2 \,{\left ({\left (a x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x + b} b\right )}}{3 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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