Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^2} \]
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Rubi [A] time = 0.0171105, 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} \[ \frac{2 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^3} \, dx &=-\operatorname{Subst}\left (\int x (a+b x)^{5/2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{5/2}}{b}+\frac{(a+b x)^{7/2}}{b}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.0166454, size = 38, normalized size = 1. \[ \frac{2 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 33, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax-7\,b \right ) }{63\,{b}^{2}{x}^{2}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.36685, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}}}{9 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a}{7 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.69571, size = 130, normalized size = 3.42 \begin{align*} \frac{2 \,{\left (2 \, a^{4} x^{4} - a^{3} b x^{3} - 15 \, a^{2} b^{2} x^{2} - 19 \, a b^{3} x - 7 \, b^{4}\right )} \sqrt{\frac{a x + b}{x}}}{63 \, b^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.49604, size = 416, normalized size = 10.95 \begin{align*} \frac{4 a^{\frac{19}{2}} b^{\frac{3}{2}} x^{5} \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} + \frac{2 a^{\frac{17}{2}} b^{\frac{5}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{32 a^{\frac{15}{2}} b^{\frac{7}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{68 a^{\frac{13}{2}} b^{\frac{9}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{52 a^{\frac{11}{2}} b^{\frac{11}{2}} x \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{14 a^{\frac{9}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x}{b} + 1}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{4 a^{10} b x^{\frac{11}{2}}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} - \frac{4 a^{9} b^{2} x^{\frac{9}{2}}}{63 a^{\frac{11}{2}} b^{3} x^{\frac{11}{2}} + 63 a^{\frac{9}{2}} b^{4} x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18399, size = 323, normalized size = 8.5 \begin{align*} \frac{2 \,{\left (63 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 273 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b \mathrm{sgn}\left (x\right ) + 567 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{2} \mathrm{sgn}\left (x\right ) + 693 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{3} \mathrm{sgn}\left (x\right ) + 525 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{4} \mathrm{sgn}\left (x\right ) + 243 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{5} \mathrm{sgn}\left (x\right ) + 63 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{6} \mathrm{sgn}\left (x\right ) + 7 \, b^{7} \mathrm{sgn}\left (x\right )\right )}}{63 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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