Optimal. Leaf size=103 \[ \frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x}}}{8 b \sqrt{x}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{3 x^{3/2}} \]
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Rubi [A] time = 0.053141, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {337, 279, 321, 217, 206} \[ \frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x}}}{8 b \sqrt{x}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 337
Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x^{5/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int x^2 \left (a+b x^2\right )^{3/2} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{3 x^{3/2}}-a \operatorname{Subst}\left (\int x^2 \sqrt{a+b x^2} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{a \sqrt{a+\frac{b}{x}}}{4 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{a \sqrt{a+\frac{b}{x}}}{4 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x}}}{8 b \sqrt{x}}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{8 b}\\ &=-\frac{a \sqrt{a+\frac{b}{x}}}{4 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x}}}{8 b \sqrt{x}}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{8 b}\\ &=-\frac{a \sqrt{a+\frac{b}{x}}}{4 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x}}}{8 b \sqrt{x}}+\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{8 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.122791, size = 90, normalized size = 0.87 \[ \frac{\sqrt{a+\frac{b}{x}} \left (\frac{3 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}}\right )}{\sqrt{\frac{b}{a x}+1}}-\frac{\sqrt{b} \left (3 a^2 x^2+14 a b x+8 b^2\right )}{x^{5/2}}\right )}{24 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 92, normalized size = 0.9 \begin{align*} -{\frac{1}{24}\sqrt{{\frac{ax+b}{x}}} \left ( -3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{3}{x}^{3}+8\,{b}^{5/2}\sqrt{ax+b}+14\,xa{b}^{3/2}\sqrt{ax+b}+3\,{x}^{2}{a}^{2}\sqrt{b}\sqrt{ax+b} \right ){x}^{-{\frac{5}{2}}}{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 = 1.52383, size = 416, normalized size = 4.04 \begin{align*} \left [\frac{3 \, a^{3} \sqrt{b} x^{3} \log \left (\frac{a x + 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) - 2 \,{\left (3 \, a^{2} b x^{2} + 14 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{48 \, b^{2} x^{3}}, -\frac{3 \, a^{3} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) +{\left (3 \, a^{2} b x^{2} + 14 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{24 \, b^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 27.4767, size = 124, normalized size = 1.2 \begin{align*} - \frac{a^{\frac{5}{2}}}{8 b \sqrt{x} \sqrt{1 + \frac{b}{a x}}} - \frac{17 a^{\frac{3}{2}}}{24 x^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}} - \frac{11 \sqrt{a} b}{12 x^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}} + \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{8 b^{\frac{3}{2}}} - \frac{b^{2}}{3 \sqrt{a} x^{\frac{7}{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 = 1.28967, size = 97, normalized size = 0.94 \begin{align*} -\frac{1}{24} \, a^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{3 \,{\left (a x + b\right )}^{\frac{5}{2}} + 8 \,{\left (a x + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{a x + b} b^{2}}{a^{3} b x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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