Optimal. Leaf size=52 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{b \sqrt{x}} \]
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Rubi [A] time = 0.0260261, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {337, 321, 217, 206} \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{b \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 337
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x}} x^{5/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{b \sqrt{x}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{b}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{b \sqrt{x}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{b}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{b \sqrt{x}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0585977, size = 77, normalized size = 1.48 \[ \frac{a^{3/2} x^{3/2} \sqrt{\frac{b}{a x}+1} \sinh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}}\right )-\sqrt{b} (a x+b)}{b^{3/2} x^{3/2} \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 55, normalized size = 1.1 \begin{align*} -{\sqrt{{\frac{ax+b}{x}}} \left ( -{\it Artanh} \left ({\sqrt{ax+b}{\frac{1}{\sqrt{b}}}} \right ) ax+\sqrt{ax+b}\sqrt{b} \right ){\frac{1}{\sqrt{x}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax+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 = 1.54058, size = 297, normalized size = 5.71 \begin{align*} \left [\frac{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 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}}}{2 \, b^{2} x}, -\frac{a \sqrt{-b} x \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) + b \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b^{2} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.6402, size = 44, normalized size = 0.85 \begin{align*} - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x}}}{b \sqrt{x}} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{b^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23219, size = 59, normalized size = 1.13 \begin{align*} -a{\left (\frac{\arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{\sqrt{a x + b}}{a b x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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