Optimal. Leaf size=37 \[ \frac{8 \sqrt{4 x^2+9}}{243 x}-\frac{\sqrt{4 x^2+9}}{27 x^3} \]
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Rubi [A] time = 0.0071159, antiderivative size = 37, 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 = {271, 264} \[ \frac{8 \sqrt{4 x^2+9}}{243 x}-\frac{\sqrt{4 x^2+9}}{27 x^3} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt{9+4 x^2}} \, dx &=-\frac{\sqrt{9+4 x^2}}{27 x^3}-\frac{8}{27} \int \frac{1}{x^2 \sqrt{9+4 x^2}} \, dx\\ &=-\frac{\sqrt{9+4 x^2}}{27 x^3}+\frac{8 \sqrt{9+4 x^2}}{243 x}\\ \end{align*}
Mathematica [A] time = 0.0041132, size = 27, normalized size = 0.73 \[ -\frac{\left (9-8 x^2\right ) \sqrt{\frac{4 x^2}{9}+1}}{81 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 0.6 \begin{align*}{\frac{8\,{x}^{2}-9}{243\,{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 = 3.56123, size = 39, normalized size = 1.05 \begin{align*} \frac{8 \, \sqrt{4 \, x^{2} + 9}}{243 \, x} - \frac{\sqrt{4 \, x^{2} + 9}}{27 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25628, size = 68, normalized size = 1.84 \begin{align*} \frac{16 \, x^{3} +{\left (8 \, x^{2} - 9\right )} \sqrt{4 \, x^{2} + 9}}{243 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.45535, size = 32, normalized size = 0.86 \begin{align*} \frac{16 \sqrt{1 + \frac{9}{4 x^{2}}}}{243} - \frac{2 \sqrt{1 + \frac{9}{4 x^{2}}}}{27 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.93105, size = 57, normalized size = 1.54 \begin{align*} \frac{32 \,{\left ({\left (2 \, x - \sqrt{4 \, x^{2} + 9}\right )}^{2} - 3\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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