Optimal. Leaf size=25 \[ -\frac{\sqrt{9-4 x^2}}{x}-2 \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
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Rubi [A] time = 0.0050271, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {277, 216} \[ -\frac{\sqrt{9-4 x^2}}{x}-2 \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 277
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{9-4 x^2}}{x^2} \, dx &=-\frac{\sqrt{9-4 x^2}}{x}-4 \int \frac{1}{\sqrt{9-4 x^2}} \, dx\\ &=-\frac{\sqrt{9-4 x^2}}{x}-2 \sin ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0064594, size = 25, normalized size = 1. \[ -\frac{\sqrt{9-4 x^2}}{x}-2 \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 34, normalized size = 1.4 \begin{align*} -{\frac{1}{9\,x} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{4\,x}{9}\sqrt{-4\,{x}^{2}+9}}-2\,\arcsin \left ( 2/3\,x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.75001, size = 28, normalized size = 1.12 \begin{align*} -\frac{\sqrt{-4 \, x^{2} + 9}}{x} - 2 \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5143, size = 88, normalized size = 3.52 \begin{align*} \frac{4 \, x \arctan \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{2 \, x}\right ) - \sqrt{-4 \, x^{2} + 9}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.231266, size = 20, normalized size = 0.8 \begin{align*} - 2 \operatorname{asin}{\left (\frac{2 x}{3} \right )} - \frac{\sqrt{9 - 4 x^{2}}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.33949, size = 53, normalized size = 2.12 \begin{align*} \frac{2 \, x}{\sqrt{-4 \, x^{2} + 9} - 3} - \frac{\sqrt{-4 \, x^{2} + 9} - 3}{2 \, x} - 2 \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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