Optimal. Leaf size=43 \[ \frac{\sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^2} \]
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Rubi [A] time = 0.0356831, antiderivative size = 43, 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} \[ \frac{\sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}} x^3} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{a+b x}{\sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{-b c+a d}{d \sqrt{c+d x}}+\frac{b \sqrt{c+d x}}{d}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{(b c-a d) \sqrt{c+\frac{d}{x^2}}}{d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^2}\\ \end{align*}
Mathematica [A] time = 0.0251518, size = 39, normalized size = 0.91 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (3 a d x^2+b \left (d-2 c x^2\right )\right )}{3 d^2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 47, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 3\,ad{x}^{2}-2\,bc{x}^{2}+bd \right ) \left ( c{x}^{2}+d \right ) }{3\,{d}^{2}{x}^{4}}{\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.929946, size = 65, normalized size = 1.51 \begin{align*} -\frac{1}{3} \, b{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}{d^{2}} - \frac{3 \, \sqrt{c + \frac{d}{x^{2}}} c}{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 = 1.32556, size = 88, normalized size = 2.05 \begin{align*} \frac{{\left ({\left (2 \, b c - 3 \, a d\right )} x^{2} - b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \, d^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.26124, size = 139, normalized size = 3.23 \begin{align*} - \frac{\begin{cases} \frac{\frac{a}{x^{2}} + \frac{b}{2 x^{4}}}{\sqrt{c}} & \text{for}\: d = 0 \\- \frac{\frac{2 a c}{\sqrt{c + \frac{d}{x^{2}}}} + 2 a \left (- \frac{c}{\sqrt{c + \frac{d}{x^{2}}}} - \sqrt{c + \frac{d}{x^{2}}}\right ) + \frac{2 b c \left (- \frac{c}{\sqrt{c + \frac{d}{x^{2}}}} - \sqrt{c + \frac{d}{x^{2}}}\right )}{d} + \frac{2 b \left (\frac{c^{2}}{\sqrt{c + \frac{d}{x^{2}}}} + 2 c \sqrt{c + \frac{d}{x^{2}}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3}\right )}{d}}{d} & \text{otherwise} \end{cases}}{2} \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}}}{\sqrt{c + \frac{d}{x^{2}}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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