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 = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 264
Rule 271
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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.76, size = 28, normalized size = 0.76 \[ \frac {16 \, x^{3} + {\left (8 \, x^{2} - 9\right )} \sqrt {4 \, x^{2} + 9}}{243 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 42, normalized size = 1.14 \[ \frac {32 \, {\left ({\left (2 \, x - \sqrt {4 \, x^{2} + 9}\right )}^{2} - 3\right )}}{{\left ({\left (2 \, x - \sqrt {4 \, x^{2} + 9}\right )}^{2} - 9\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.59 \[ \frac {\sqrt {4 x^{2}+9}\, \left (8 x^{2}-9\right )}{243 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.86, size = 29, normalized size = 0.78 \[ \frac {8 \, \sqrt {4 \, x^{2} + 9}}{243 \, x} - \frac {\sqrt {4 \, x^{2} + 9}}{27 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 19, normalized size = 0.51 \[ \sqrt {x^2+\frac {9}{4}}\,\left (\frac {16}{243\,x}-\frac {2}{27\,x^3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.29, size = 32, normalized size = 0.86 \[ \frac {16 \sqrt {1 + \frac {9}{4 x^{2}}}}{243} - \frac {2 \sqrt {1 + \frac {9}{4 x^{2}}}}{27 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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