Optimal. Leaf size=18 \[ \frac {\left (4 x^2-9\right )^{3/2}}{27 x^3} \]
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Rubi [A] time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \[ \frac {\left (4 x^2-9\right )^{3/2}}{27 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 20, normalized size = 1.11 \[ \frac {8 \, x^{3} + {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{27 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.10, size = 42, normalized size = 2.33 \[ \frac {16 \, {\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 9}\right )}^{4} + 27\right )}}{{\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 9}\right )}^{2} + 9\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 1.39 \[ \frac {\left (2 x -3\right ) \left (2 x +3\right ) \sqrt {4 x^{2}-9}}{27 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 14, normalized size = 0.78 \[ \frac {{\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{27 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.10, size = 31, normalized size = 1.72 \[ \frac {4\,x^2\,\sqrt {4\,x^2-9}-9\,\sqrt {4\,x^2-9}}{27\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.98, size = 76, normalized size = 4.22 \[ \begin {cases} \frac {8 i \sqrt {-1 + \frac {9}{4 x^{2}}}}{27} - \frac {2 i \sqrt {-1 + \frac {9}{4 x^{2}}}}{3 x^{2}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\\frac {8 \sqrt {1 - \frac {9}{4 x^{2}}}}{27} - \frac {2 \sqrt {1 - \frac {9}{4 x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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