Optimal. Leaf size=18 \[ -\frac{\left (4 x^2+9\right )^{3/2}}{27 x^3} \]
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Rubi [A] time = 0.002985, 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 = 0.0034218, 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 = 0.003, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{27\,{x}^{3}} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.61069, 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 = 1.51408, size = 53, normalized size = 2.94 \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 = 1.00305, size = 34, normalized size = 1.89 \begin{align*} - \frac{8 \sqrt{1 + \frac{9}{4 x^{2}}}}{27} - \frac{2 \sqrt{1 + \frac{9}{4 x^{2}}}}{3 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.45481, 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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