Optimal. Leaf size=42 \[ -\frac{b c-a d}{d^2 \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d^2} \]
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Rubi [A] time = 0.0354627, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {444, 43} \[ -\frac{b c-a d}{d^2 \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{a+\frac{b}{x^2}}{\left (c+\frac{d}{x^2}\right )^{3/2} x^3} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{a+b x}{(c+d x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{-b c+a d}{d (c+d x)^{3/2}}+\frac{b}{d \sqrt{c+d x}}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{b c-a d}{d^2 \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d^2}\\ \end{align*}
Mathematica [A] time = 0.0184008, size = 36, normalized size = 0.86 \[ \frac{a d x^2-b \left (2 c x^2+d\right )}{d^2 x^2 \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 46, normalized size = 1.1 \begin{align*}{\frac{ \left ( ad{x}^{2}-2\,bc{x}^{2}-bd \right ) \left ( c{x}^{2}+d \right ) }{{d}^{2}{x}^{4}} \left ({\frac{c{x}^{2}+d}{{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 = 0.927368, size = 62, normalized size = 1.48 \begin{align*} -b{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{d^{2}} + \frac{c}{\sqrt{c + \frac{d}{x^{2}}} d^{2}}\right )} + \frac{a}{\sqrt{c + \frac{d}{x^{2}}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47852, size = 92, normalized size = 2.19 \begin{align*} -\frac{{\left ({\left (2 \, b c - a d\right )} x^{2} + b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c d^{2} x^{2} + d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.39004, size = 68, normalized size = 1.62 \begin{align*} \begin{cases} \frac{a}{d \sqrt{c + \frac{d}{x^{2}}}} - \frac{2 b c}{d^{2} \sqrt{c + \frac{d}{x^{2}}}} - \frac{b}{d x^{2} \sqrt{c + \frac{d}{x^{2}}}} & \text{for}\: d \neq 0 \\\frac{- \frac{a}{2 x^{2}} - \frac{b}{4 x^{4}}}{c^{\frac{3}{2}}} & \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{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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