Optimal. Leaf size=36 \[ \frac{2}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 a}{3 b^2 \left (a+\frac{b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.0172852, antiderivative size = 36, 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} \[ \frac{2}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 a}{3 b^2 \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{(a+b x)^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{5/2}}+\frac{1}{b (a+b x)^{3/2}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a}{3 b^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2}{b^2 \sqrt{a+\frac{b}{x}}}\\ \end{align*}
Mathematica [A] time = 0.0217715, size = 33, normalized size = 0.92 \[ \frac{4 a x+6 b}{3 b^2 \sqrt{a+\frac{b}{x}} (a x+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 33, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax+3\,b \right ) }{3\,{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 = 1.28351, size = 41, normalized size = 1.14 \begin{align*} \frac{2}{\sqrt{a + \frac{b}{x}} b^{2}} - \frac{2 \, a}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.592, size = 99, normalized size = 2.75 \begin{align*} \frac{2 \,{\left (2 \, a x^{2} + 3 \, b x\right )} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.98322, size = 82, normalized size = 2.28 \begin{align*} \begin{cases} \frac{4 a x}{3 a b^{2} x \sqrt{a + \frac{b}{x}} + 3 b^{3} \sqrt{a + \frac{b}{x}}} + \frac{6 b}{3 a b^{2} x \sqrt{a + \frac{b}{x}} + 3 b^{3} \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{5}{2}} x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28229, size = 49, normalized size = 1.36 \begin{align*} -\frac{2 \,{\left (a - \frac{3 \,{\left (a x + b\right )}}{x}\right )} x}{3 \,{\left (a x + b\right )} b^{2} \sqrt{\frac{a x + b}{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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