Optimal. Leaf size=55 \[ 2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 a \sqrt{a+\frac{b}{x}}-\frac{2}{3} \left (a+\frac{b}{x}\right )^{3/2} \]
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Rubi [A] time = 0.0269337, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ 2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 a \sqrt{a+\frac{b}{x}}-\frac{2}{3} \left (a+\frac{b}{x}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2}{3} \left (a+\frac{b}{x}\right )^{3/2}-a \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-2 a \sqrt{a+\frac{b}{x}}-\frac{2}{3} \left (a+\frac{b}{x}\right )^{3/2}-a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-2 a \sqrt{a+\frac{b}{x}}-\frac{2}{3} \left (a+\frac{b}{x}\right )^{3/2}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{b}\\ &=-2 a \sqrt{a+\frac{b}{x}}-\frac{2}{3} \left (a+\frac{b}{x}\right )^{3/2}+2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0357304, size = 50, normalized size = 0.91 \[ 2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{2 \sqrt{a+\frac{b}{x}} (4 a x+b)}{3 x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 123, normalized size = 2.2 \begin{align*} -{\frac{1}{3\,b{x}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( -6\,\sqrt{a{x}^{2}+bx}{a}^{5/2}{x}^{3}-3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{2}b+6\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}x+2\, \left ( a{x}^{2}+bx \right ) ^{3/2}b\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 = 1.81818, size = 270, normalized size = 4.91 \begin{align*} \left [\frac{3 \, a^{\frac{3}{2}} x \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (4 \, a x + b\right )} \sqrt{\frac{a x + b}{x}}}{3 \, x}, -\frac{2 \,{\left (3 \, \sqrt{-a} a x \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (4 \, a x + b\right )} \sqrt{\frac{a x + b}{x}}\right )}}{3 \, x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.96372, size = 71, normalized size = 1.29 \begin{align*} - \frac{8 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}}{3} - a^{\frac{3}{2}} \log{\left (\frac{b}{a x} \right )} + 2 a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )} - \frac{2 \sqrt{a} b \sqrt{1 + \frac{b}{a x}}}{3 x} \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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