Optimal. Leaf size=73 \[ -2 a^2 \sqrt{a+\frac{b}{x}}+2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{2}{3} a \left (a+\frac{b}{x}\right )^{3/2}-\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2} \]
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Rubi [A] time = 0.033514, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ -2 a^2 \sqrt{a+\frac{b}{x}}+2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{2}{3} a \left (a+\frac{b}{x}\right )^{3/2}-\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2} \]
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 )^{5/2}}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2}-a \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2}{3} a \left (a+\frac{b}{x}\right )^{3/2}-\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2}-a^2 \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-2 a^2 \sqrt{a+\frac{b}{x}}-\frac{2}{3} a \left (a+\frac{b}{x}\right )^{3/2}-\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2}-a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-2 a^2 \sqrt{a+\frac{b}{x}}-\frac{2}{3} a \left (a+\frac{b}{x}\right )^{3/2}-\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2}-\frac{\left (2 a^3\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^2 \sqrt{a+\frac{b}{x}}-\frac{2}{3} a \left (a+\frac{b}{x}\right )^{3/2}-\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2}+2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0469411, size = 63, normalized size = 0.86 \[ 2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{2 \sqrt{a+\frac{b}{x}} \left (23 a^2 x^2+11 a b x+3 b^2\right )}{15 x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 145, normalized size = 2. \begin{align*} -{\frac{1}{15\,b{x}^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( -30\,\sqrt{a{x}^{2}+bx}{a}^{7/2}{x}^{4}-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{4}{a}^{3}b+30\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}{x}^{2}+16\,{a}^{3/2} \left ( a{x}^{2}+bx \right ) ^{3/2}bx+6\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}{b}^{2} \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 = 1.71498, size = 343, normalized size = 4.7 \begin{align*} \left [\frac{15 \, a^{\frac{5}{2}} x^{2} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (23 \, a^{2} x^{2} + 11 \, a b x + 3 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{15 \, x^{2}}, -\frac{2 \,{\left (15 \, \sqrt{-a} a^{2} x^{2} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (23 \, a^{2} x^{2} + 11 \, a b x + 3 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}\right )}}{15 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.89614, size = 97, normalized size = 1.33 \begin{align*} - \frac{46 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}}{15} - a^{\frac{5}{2}} \log{\left (\frac{b}{a x} \right )} + 2 a^{\frac{5}{2}} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )} - \frac{22 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x}}}{15 x} - \frac{2 \sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x}}}{5 x^{2}} \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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