Optimal. Leaf size=54 \[ x \left (a+\frac{b}{x}\right )^{3/2}-3 b \sqrt{a+\frac{b}{x}}+3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0253246, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {242, 47, 50, 63, 208} \[ x \left (a+\frac{b}{x}\right )^{3/2}-3 b \sqrt{a+\frac{b}{x}}+3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 242
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^{3/2} \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\left (a+\frac{b}{x}\right )^{3/2} x-\frac{1}{2} (3 b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-3 b \sqrt{a+\frac{b}{x}}+\left (a+\frac{b}{x}\right )^{3/2} x-\frac{1}{2} (3 a b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-3 b \sqrt{a+\frac{b}{x}}+\left (a+\frac{b}{x}\right )^{3/2} x-(3 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )\\ &=-3 b \sqrt{a+\frac{b}{x}}+\left (a+\frac{b}{x}\right )^{3/2} x+3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0191561, size = 46, normalized size = 0.85 \[ \sqrt{a+\frac{b}{x}} (a x-2 b)+3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 100, normalized size = 1.9 \begin{align*} -{\frac{1}{2\,x}\sqrt{{\frac{ax+b}{x}}} \left ( -6\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{x}^{2}-3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}ab+4\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a} \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{\frac{1}{\sqrt{a}}}} \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 = 1.52256, size = 246, normalized size = 4.56 \begin{align*} \left [\frac{3}{2} \, \sqrt{a} b \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) +{\left (a x - 2 \, b\right )} \sqrt{\frac{a x + b}{x}}, -3 \, \sqrt{-a} b \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (a x - 2 \, b\right )} \sqrt{\frac{a x + b}{x}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.37587, size = 92, normalized size = 1.7 \begin{align*} 3 \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} + \frac{a^{2} x^{\frac{3}{2}}}{\sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{a \sqrt{b} \sqrt{x}}{\sqrt{\frac{a x}{b} + 1}} - \frac{2 b^{\frac{3}{2}}}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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