Optimal. Leaf size=39 \[ -\frac {\sqrt {-4 x^2-9}}{2 x^2}-\frac {2}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 47, 63, 204} \[ -\frac {\sqrt {-4 x^2-9}}{2 x^2}-\frac {2}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 204
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {-9-4 x^2}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {-9-4 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {-9-4 x^2}}{2 x^2}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {-9-4 x} x} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {-9-4 x^2}}{2 x^2}+\frac {1}{2} \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}}{2 x^2}-\frac {2}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9-4 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 55, normalized size = 1.41 \[ \frac {12 x^2+4 \sqrt {4 x^2+9} x^2 \tanh ^{-1}\left (\sqrt {\frac {4 x^2}{9}+1}\right )+27}{6 x^2 \sqrt {-4 x^2-9}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.89, size = 65, normalized size = 1.67 \[ \frac {-2 i \, x^{2} \log \left (-\frac {4 \, {\left (i \, \sqrt {-4 \, x^{2} - 9} - 3\right )}}{3 \, x}\right ) + 2 i \, x^{2} \log \left (-\frac {4 \, {\left (-i \, \sqrt {-4 \, x^{2} - 9} - 3\right )}}{3 \, x}\right ) - 3 \, \sqrt {-4 \, x^{2} - 9}}{6 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 29, normalized size = 0.74 \[ -\frac {\sqrt {-4 \, x^{2} - 9}}{2 \, x^{2}} - \frac {2}{3} \, \arctan \left (\frac {1}{3} \, \sqrt {-4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 1.05 \[ \frac {2 \arctan \left (\frac {3}{\sqrt {-4 x^{2}-9}}\right )}{3}+\frac {\left (-4 x^{2}-9\right )^{\frac {3}{2}}}{18 x^{2}}+\frac {2 \sqrt {-4 x^{2}-9}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.90, size = 51, normalized size = 1.31 \[ \frac {2}{9} \, \sqrt {-4 \, x^{2} - 9} + \frac {{\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{18 \, x^{2}} + \frac {2}{3} i \, \log \left (\frac {6 \, \sqrt {4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.79, size = 29, normalized size = 0.74 \[ -\frac {2\,\mathrm {atan}\left (\frac {\sqrt {-4\,x^2-9}}{3}\right )}{3}-\frac {\sqrt {-4\,x^2-9}}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.70, size = 27, normalized size = 0.69 \[ - \frac {2 i \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{3} - \frac {i \sqrt {1 + \frac {9}{4 x^{2}}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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