Optimal. Leaf size=57 \[ -\frac{\sqrt{-4 x^2-9}}{18 x^2}-\frac{\sqrt{-4 x^2-9}}{4 x^4}+\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
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Rubi [A] time = 0.0211147, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 204} \[ -\frac{\sqrt{-4 x^2-9}}{18 x^2}-\frac{\sqrt{-4 x^2-9}}{4 x^4}+\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 63
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{-9-4 x^2}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{-9-4 x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{-9-4 x^2}}{4 x^4}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x^2} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{-9-4 x^2}}{4 x^4}-\frac{\sqrt{-9-4 x^2}}{18 x^2}+\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{-9-4 x^2}}{4 x^4}-\frac{\sqrt{-9-4 x^2}}{18 x^2}-\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{-\frac{9}{4}-\frac{x^2}{4}} \, dx,x,\sqrt{-9-4 x^2}\right )\\ &=-\frac{\sqrt{-9-4 x^2}}{4 x^4}-\frac{\sqrt{-9-4 x^2}}{18 x^2}+\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{-9-4 x^2}\right )\\ \end{align*}
Mathematica [C] time = 0.0049775, size = 32, normalized size = 0.56 \[ \frac{16 \left (-4 x^2-9\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{4 x^2}{9}+1\right )}{2187} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 55, normalized size = 1. \begin{align*}{\frac{1}{36\,{x}^{4}} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{162\,{x}^{2}} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}-{\frac{2}{81}\sqrt{-4\,{x}^{2}-9}}-{\frac{2}{27}\arctan \left ( 3\,{\frac{1}{\sqrt{-4\,{x}^{2}-9}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 4.01728, size = 88, normalized size = 1.54 \begin{align*} -\frac{2}{81} \, \sqrt{-4 \, x^{2} - 9} - \frac{{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{162 \, x^{2}} + \frac{{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{36 \, x^{4}} - \frac{2}{27} i \, \log \left (\frac{6 \, \sqrt{4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.319, size = 192, normalized size = 3.37 \begin{align*} \frac{-4 i \, x^{4} \log \left (-\frac{4 \,{\left (i \, \sqrt{-4 \, x^{2} - 9} + 3\right )}}{27 \, x}\right ) + 4 i \, x^{4} \log \left (-\frac{4 \,{\left (-i \, \sqrt{-4 \, x^{2} - 9} + 3\right )}}{27 \, x}\right ) - 3 \,{\left (2 \, x^{2} + 9\right )} \sqrt{-4 \, x^{2} - 9}}{108 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.53655, size = 68, normalized size = 1.19 \begin{align*} \frac{2 i \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{27} - \frac{i}{9 x \sqrt{1 + \frac{9}{4 x^{2}}}} - \frac{3 i}{4 x^{3} \sqrt{1 + \frac{9}{4 x^{2}}}} - \frac{9 i}{8 x^{5} \sqrt{1 + \frac{9}{4 x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.92472, size = 58, normalized size = 1.02 \begin{align*} -\frac{i \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} + 9 i \, \sqrt{4 \, x^{2} + 9}}{72 \, x^{4}} + \frac{2}{27} \, \arctan \left (\frac{1}{3} i \, \sqrt{4 \, x^{2} + 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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