Optimal. Leaf size=34 \[ -\frac{2 a}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^2} \]
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Rubi [A] time = 0.0169184, antiderivative size = 34, 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} \[ -\frac{2 a}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^2} \]
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 )^{3/2} x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{3/2}}+\frac{1}{b \sqrt{a+b x}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0173901, size = 25, normalized size = 0.74 \[ -\frac{2 (2 a x+b)}{b^2 x \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 31, normalized size = 0.9 \begin{align*} -2\,{\frac{ \left ( ax+b \right ) \left ( 2\,ax+b \right ) }{{b}^{2}{x}^{2}} \left ({\frac{ax+b}{x}} \right ) ^{-3/2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986999, size = 41, normalized size = 1.21 \begin{align*} -\frac{2 \, \sqrt{a + \frac{b}{x}}}{b^{2}} - \frac{2 \, a}{\sqrt{a + \frac{b}{x}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47794, size = 68, normalized size = 2. \begin{align*} -\frac{2 \,{\left (2 \, a x + b\right )} \sqrt{\frac{a x + b}{x}}}{a b^{2} x + b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.25887, size = 42, normalized size = 1.24 \begin{align*} \begin{cases} - \frac{4 a}{b^{2} \sqrt{a + \frac{b}{x}}} - \frac{2}{b x \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{3}{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 = 1.24599, size = 47, normalized size = 1.38 \begin{align*} -2 \, b{\left (\frac{a}{b^{3} \sqrt{\frac{a x + b}{x}}} + \frac{\sqrt{\frac{a x + b}{x}}}{b^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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