Optimal. Leaf size=34 \[ -\frac{a}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^2} \]
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Rubi [A] time = 0.0217856, 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} \[ -\frac{a}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^5} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \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^2}\right )\right )\\ &=-\frac{a}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0086135, size = 28, normalized size = 0.82 \[ \frac{-2 a x^2-b}{b^2 x^2 \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 37, normalized size = 1.1 \begin{align*} -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 2\,a{x}^{2}+b \right ) }{{b}^{2}{x}^{4}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04155, size = 41, normalized size = 1.21 \begin{align*} -\frac{\sqrt{a + \frac{b}{x^{2}}}}{b^{2}} - \frac{a}{\sqrt{a + \frac{b}{x^{2}}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4559, size = 76, normalized size = 2.24 \begin{align*} -\frac{{\left (2 \, a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a b^{2} x^{2} + b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.90804, size = 48, normalized size = 1.41 \begin{align*} \begin{cases} - \frac{2 a}{b^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{1}{b x^{2} \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{4 a^{\frac{3}{2}} x^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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