Optimal. Leaf size=100 \[ -\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 \sqrt{x}}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 \sqrt{b}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{12 \sqrt{x}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{3 \sqrt{x}} \]
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Rubi [A] time = 0.045123, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {337, 195, 217, 206} \[ -\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 \sqrt{x}}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 \sqrt{b}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{12 \sqrt{x}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 337
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^{3/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \left (a+b x^2\right )^{5/2} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{3 \sqrt{x}}-\frac{1}{3} (5 a) \operatorname{Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{12 \sqrt{x}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{3 \sqrt{x}}-\frac{1}{4} \left (5 a^2\right ) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 \sqrt{x}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{12 \sqrt{x}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{3 \sqrt{x}}-\frac{1}{8} \left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 \sqrt{x}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{12 \sqrt{x}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{3 \sqrt{x}}-\frac{1}{8} \left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 \sqrt{x}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{12 \sqrt{x}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{3 \sqrt{x}}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{8 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.159139, size = 85, normalized size = 0.85 \[ \frac{1}{24} \sqrt{a+\frac{b}{x}} \left (-\frac{33 a^2 x^2+26 a b x+8 b^2}{x^{5/2}}-\frac{15 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}}\right )}{\sqrt{b} \sqrt{\frac{b}{a x}+1}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 92, normalized size = 0.9 \begin{align*} -{\frac{1}{24}\sqrt{{\frac{ax+b}{x}}} \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{3}{x}^{3}+33\,{x}^{2}{a}^{2}\sqrt{b}\sqrt{ax+b}+26\,xa{b}^{3/2}\sqrt{ax+b}+8\,{b}^{5/2}\sqrt{ax+b} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ax+b}}}{\frac{1}{\sqrt{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 = 1.55714, size = 414, normalized size = 4.14 \begin{align*} \left [\frac{15 \, 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 (33 \, a^{2} b x^{2} + 26 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{48 \, b x^{3}}, \frac{15 \, a^{3} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) -{\left (33 \, a^{2} b x^{2} + 26 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{24 \, b x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 70.646, size = 104, normalized size = 1.04 \begin{align*} - \frac{11 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}}{8 \sqrt{x}} - \frac{13 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x}}}{12 x^{\frac{3}{2}}} - \frac{\sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x}}}{3 x^{\frac{5}{2}}} - \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{8 \sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31902, size = 90, normalized size = 0.9 \begin{align*} \frac{1}{24} \, a^{3}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{33 \,{\left (a x + b\right )}^{\frac{5}{2}} - 40 \,{\left (a x + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{a x + b} b^{2}}{a^{3} x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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