Optimal. Leaf size=46 \[ \frac{\left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^2} \]
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Rubi [A] time = 0.0373817, antiderivative size = 46, 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} \[ \frac{\left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}}}{x^3} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int (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) \sqrt{c+d x}}{d}+\frac{b (c+d x)^{3/2}}{d}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{(b c-a d) \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^2}\\ \end{align*}
Mathematica [A] time = 0.0162534, size = 47, normalized size = 1.02 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (5 a d x^2-2 b c x^2+3 b d\right )}{15 d^2 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 48, normalized size = 1. \begin{align*} -{\frac{ \left ( 5\,ad{x}^{2}-2\,bc{x}^{2}+3\,bd \right ) \left ( c{x}^{2}+d \right ) }{15\,{d}^{2}{x}^{4}}\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 = 0.940565, size = 66, normalized size = 1.43 \begin{align*} -\frac{1}{15} \, b{\left (\frac{3 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{d^{2}} - \frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c}{d^{2}}\right )} - \frac{a{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32251, size = 132, normalized size = 2.87 \begin{align*} \frac{{\left ({\left (2 \, b c^{2} - 5 \, a c d\right )} x^{4} - 3 \, b d^{2} -{\left (b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \, d^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.30036, size = 58, normalized size = 1.26 \begin{align*} - \frac{a \left (\begin{cases} \frac{\sqrt{c}}{x^{2}} & \text{for}\: d = 0 \\\frac{2 \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right )}{2} - \frac{b \left (- \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.19649, size = 338, normalized size = 7.35 \begin{align*} \frac{2 \,{\left (15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) - 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{3}{2}} d \mathrm{sgn}\left (x\right ) + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{5}{2}} d \mathrm{sgn}\left (x\right ) + 20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{3}{2}} d^{2} \mathrm{sgn}\left (x\right ) + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{5}{2}} d^{2} \mathrm{sgn}\left (x\right ) - 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{3}{2}} d^{3} \mathrm{sgn}\left (x\right ) - 2 \, b c^{\frac{5}{2}} d^{3} \mathrm{sgn}\left (x\right ) + 5 \, a c^{\frac{3}{2}} d^{4} \mathrm{sgn}\left (x\right )\right )}}{15 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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