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 = 0.0350443, 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, Rules used = {337, 277, 195, 217, 206} \[ 2 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x}}}{\sqrt{x}}-3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 337
Rule 277
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{\sqrt{x}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{x^2} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=2 \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}-(6 b) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{3 b \sqrt{a+\frac{b}{x}}}{\sqrt{x}}+2 \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}-(3 a b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{3 b \sqrt{a+\frac{b}{x}}}{\sqrt{x}}+2 \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}-(3 a b) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )\\ &=-\frac{3 b \sqrt{a+\frac{b}{x}}}{\sqrt{x}}+2 \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}-3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0129486, size = 52, normalized size = 0.75 \[ \frac{2 a \sqrt{x} \sqrt{a+\frac{b}{x}} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b}{a x}\right )}{\sqrt{\frac{b}{a x}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 70, normalized size = 1. \begin{align*} -{\sqrt{{\frac{ax+b}{x}}} \left ({b}^{{\frac{3}{2}}}\sqrt{ax+b}-2\,xa\sqrt{b}\sqrt{ax+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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52228, size = 312, normalized size = 4.52 \begin{align*} \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{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) +{\left (2 \, a x - b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.5422, size = 92, normalized size = 1.33 \begin{align*} \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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25774, size = 68, normalized size = 0.99 \begin{align*}{\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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