Optimal. Leaf size=57 \[ -\frac {\sqrt {-4 x^2-9}}{18 x^2}+\frac {2}{27} \tan ^{-1}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right )-\frac {\sqrt {-4 x^2-9}}{4 x^4} \]
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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, 204} \[ -\frac {\sqrt {-4 x^2-9}}{18 x^2}-\frac {\sqrt {-4 x^2-9}}{4 x^4}+\frac {2}{27} \tan ^{-1}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 204
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} \tan ^{-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 \[ \frac {16 \left (-4 x^2-9\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {4 x^2}{9}+1\right )}{2187} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.93, size = 72, normalized size = 1.26 \[ \frac {-4 i \, x^{4} \log \left (-\frac {4 \, {\left (i \, \sqrt {-4 \, x^{2} - 9} + 3\right )}}{27 \, x}\right ) + 4 i \, x^{4} \log \left (-\frac {4 \, {\left (-i \, \sqrt {-4 \, x^{2} - 9} + 3\right )}}{27 \, x}\right ) - 3 \, {\left (2 \, x^{2} + 9\right )} \sqrt {-4 \, x^{2} - 9}}{108 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.11, size = 43, normalized size = 0.75 \[ -\frac {{\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} i + 9 \, \sqrt {-4 \, x^{2} - 9}}{72 \, x^{4}} + \frac {2}{27} \, \arctan \left (\frac {1}{3} \, \sqrt {-4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 0.96 \[ -\frac {2 \arctan \left (\frac {3}{\sqrt {-4 x^{2}-9}}\right )}{27}-\frac {\left (-4 x^{2}-9\right )^{\frac {3}{2}}}{162 x^{2}}+\frac {\left (-4 x^{2}-9\right )^{\frac {3}{2}}}{36 x^{4}}-\frac {2 \sqrt {-4 x^{2}-9}}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.99, size = 65, normalized size = 1.14 \[ -\frac {2}{81} \, \sqrt {-4 \, x^{2} - 9} - \frac {{\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{162 \, x^{2}} + \frac {{\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{36 \, x^{4}} - \frac {2}{27} i \, \log \left (\frac {6 \, \sqrt {4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.69, size = 43, normalized size = 0.75 \[ \frac {2\,\mathrm {atan}\left (\frac {\sqrt {-4\,x^2-9}}{3}\right )}{27}-\frac {\frac {\sqrt {-4\,x^2-9}}{8}-\frac {{\left (-4\,x^2-9\right )}^{3/2}}{72}}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.37, size = 68, normalized size = 1.19 \[ \frac {2 i \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{27} - \frac {i}{9 x \sqrt {1 + \frac {9}{4 x^{2}}}} - \frac {3 i}{4 x^{3} \sqrt {1 + \frac {9}{4 x^{2}}}} - \frac {9 i}{8 x^{5} \sqrt {1 + \frac {9}{4 x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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