Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.0174295, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{2 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{b}{x}}}{x^3} \, dx &=-\operatorname{Subst}\left (\int x \sqrt{a+b x} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0146367, size = 34, normalized size = 0.89 \[ \frac{2 \sqrt{a+\frac{b}{x}} (a x+b) (2 a x-3 b)}{15 b^2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 33, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax-3\,b \right ) }{15\,{b}^{2}{x}^{2}}\sqrt{{\frac{ax+b}{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.9994, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}}{5 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73122, size = 84, normalized size = 2.21 \begin{align*} \frac{2 \,{\left (2 \, a^{2} x^{2} - a b x - 3 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{15 \, b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.54957, size = 304, normalized size = 8. \begin{align*} \frac{4 a^{\frac{11}{2}} b^{\frac{3}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + \frac{2 a^{\frac{9}{2}} b^{\frac{5}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{8 a^{\frac{7}{2}} b^{\frac{7}{2}} x \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{6 a^{\frac{5}{2}} b^{\frac{9}{2}} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{6} b x^{\frac{7}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{5} b^{2} x^{\frac{5}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15617, size = 155, normalized size = 4.08 \begin{align*} \frac{2 \,{\left (15 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) + 25 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b \mathrm{sgn}\left (x\right ) + 15 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{2} \mathrm{sgn}\left (x\right ) + 3 \, b^{3} \mathrm{sgn}\left (x\right )\right )}}{15 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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