Optimal. Leaf size=18 \[ \frac{\left (4 x^2-9\right )^{3/2}}{27 x^3} \]
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Rubi [A] time = 0.0030039, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ \frac{\left (4 x^2-9\right )^{3/2}}{27 x^3} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt{-9+4 x^2}}{x^4} \, dx &=\frac{\left (-9+4 x^2\right )^{3/2}}{27 x^3}\\ \end{align*}
Mathematica [A] time = 0.0024081, size = 18, normalized size = 1. \[ \frac{\left (4 x^2-9\right )^{3/2}}{27 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 1.4 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) }{27\,{x}^{3}}\sqrt{4\,{x}^{2}-9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.60186, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{27 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45217, size = 51, normalized size = 2.83 \begin{align*} \frac{8 \, x^{3} +{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{27 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.05356, size = 76, normalized size = 4.22 \begin{align*} \begin{cases} \frac{8 i \sqrt{-1 + \frac{9}{4 x^{2}}}}{27} - \frac{2 i \sqrt{-1 + \frac{9}{4 x^{2}}}}{3 x^{2}} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\\frac{8 \sqrt{1 - \frac{9}{4 x^{2}}}}{27} - \frac{2 \sqrt{1 - \frac{9}{4 x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.18852, size = 57, normalized size = 3.17 \begin{align*} \frac{16 \,{\left ({\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{4} + 27\right )}}{{\left ({\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2} + 9\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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