Optimal. Leaf size=57 \[ \frac {\sqrt {9-4 x^2}}{18 x^2}+\frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )-\frac {\sqrt {9-4 x^2}}{4 x^4} \]
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Rubi [A] time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 206} \[ \frac {\sqrt {9-4 x^2}}{18 x^2}-\frac {\sqrt {9-4 x^2}}{4 x^4}+\frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 206
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {9-4 x^2}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {9-4 x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{4 x^4}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {9-4 x} x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{4 x^4}+\frac {\sqrt {9-4 x^2}}{18 x^2}-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{\sqrt {9-4 x} x} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{4 x^4}+\frac {\sqrt {9-4 x^2}}{18 x^2}+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{\frac {9}{4}-\frac {x^2}{4}} \, dx,x,\sqrt {9-4 x^2}\right )\\ &=-\frac {\sqrt {9-4 x^2}}{4 x^4}+\frac {\sqrt {9-4 x^2}}{18 x^2}+\frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 32, normalized size = 0.56 \[ -\frac {16 \left (9-4 x^2\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};1-\frac {4 x^2}{9}\right )}{2187} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 45, normalized size = 0.79 \[ -\frac {8 \, x^{4} \log \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{x}\right ) - 3 \, {\left (2 \, x^{2} - 9\right )} \sqrt {-4 \, x^{2} + 9}}{108 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 57, normalized size = 1.00 \[ -\frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} + 9 \, \sqrt {-4 \, x^{2} + 9}}{72 \, x^{4}} + \frac {1}{27} \, \log \left (\sqrt {-4 \, x^{2} + 9} + 3\right ) - \frac {1}{27} \, \log \left (-\sqrt {-4 \, x^{2} + 9} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 55, normalized size = 0.96 \[ \frac {2 \arctanh \left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )}{27}-\frac {\left (-4 x^{2}+9\right )^{\frac {3}{2}}}{162 x^{2}}-\frac {\left (-4 x^{2}+9\right )^{\frac {3}{2}}}{36 x^{4}}-\frac {2 \sqrt {-4 x^{2}+9}}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 65, normalized size = 1.14 \[ -\frac {2}{81} \, \sqrt {-4 \, x^{2} + 9} - \frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{162 \, x^{2}} - \frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{36 \, x^{4}} + \frac {2}{27} \, \log \left (\frac {6 \, \sqrt {-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 49, normalized size = 0.86 \[ \frac {\sqrt {\frac {9}{4}-x^2}}{9\,x^2}-\frac {2\,\ln \left (\sqrt {\frac {9}{4\,x^2}-1}-\frac {3\,\sqrt {\frac {1}{x^2}}}{2}\right )}{27}-\frac {\sqrt {\frac {9}{4}-x^2}}{2\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.27, size = 139, normalized size = 2.44 \[ \begin {cases} \frac {2 \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{27} - \frac {1}{9 x \sqrt {-1 + \frac {9}{4 x^{2}}}} + \frac {3}{4 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {9}{8 x^{5} \sqrt {-1 + \frac {9}{4 x^{2}}}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\- \frac {2 i \operatorname {asin}{\left (\frac {3}{2 x} \right )}}{27} + \frac {i}{9 x \sqrt {1 - \frac {9}{4 x^{2}}}} - \frac {3 i}{4 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {9 i}{8 x^{5} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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