Optimal. Leaf size=57 \[ -\frac {\sqrt {9-4 x^2}}{54 x^2}-\frac {2}{81} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )-\frac {\sqrt {9-4 x^2}}{36 x^4} \]
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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 = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 206} \[ -\frac {\sqrt {9-4 x^2}}{54 x^2}-\frac {\sqrt {9-4 x^2}}{36 x^4}-\frac {2}{81} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {9-4 x} x^3} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{36 x^4}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {9-4 x} x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{36 x^4}-\frac {\sqrt {9-4 x^2}}{54 x^2}+\frac {1}{27} \operatorname {Subst}\left (\int \frac {1}{\sqrt {9-4 x} x} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{36 x^4}-\frac {\sqrt {9-4 x^2}}{54 x^2}-\frac {1}{54} \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}}{36 x^4}-\frac {\sqrt {9-4 x^2}}{54 x^2}-\frac {2}{81} \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 \[ -\frac {16}{729} \sqrt {9-4 x^2} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-\frac {4 x^2}{9}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 45, normalized size = 0.79 \[ \frac {8 \, x^{4} \log \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{x}\right ) - 3 \, {\left (2 \, x^{2} + 3\right )} \sqrt {-4 \, x^{2} + 9}}{324 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 57, normalized size = 1.00 \[ \frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} - 15 \, \sqrt {-4 \, x^{2} + 9}}{216 \, x^{4}} - \frac {1}{81} \, \log \left (\sqrt {-4 \, x^{2} + 9} + 3\right ) + \frac {1}{81} \, \log \left (-\sqrt {-4 \, x^{2} + 9} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 44, normalized size = 0.77 \[ -\frac {2 \arctanh \left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )}{81}-\frac {\sqrt {-4 x^{2}+9}}{54 x^{2}}-\frac {\sqrt {-4 x^{2}+9}}{36 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 54, normalized size = 0.95 \[ -\frac {\sqrt {-4 \, x^{2} + 9}}{54 \, x^{2}} - \frac {\sqrt {-4 \, x^{2} + 9}}{36 \, x^{4}} - \frac {2}{81} \, \log \left (\frac {6 \, \sqrt {-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.51, size = 49, normalized size = 0.86 \[ \frac {2\,\ln \left (\sqrt {\frac {9}{4\,x^2}-1}-\sqrt {\frac {9}{4\,x^2}}\right )}{81}-\frac {\sqrt {\frac {9}{4}-x^2}\,\left (\frac {2}{27\,x^2}+\frac {1}{9\,x^4}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.96, size = 136, normalized size = 2.39 \[ \begin {cases} - \frac {2 \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{81} + \frac {1}{27 x \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {1}{36 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {1}{8 x^{5} \sqrt {-1 + \frac {9}{4 x^{2}}}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\\frac {2 i \operatorname {asin}{\left (\frac {3}{2 x} \right )}}{81} - \frac {i}{27 x \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {i}{36 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {i}{8 x^{5} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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