Optimal. Leaf size=39 \[ -\frac {\sqrt {4 x^2+9}}{2 x^2}-\frac {2}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right ) \]
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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, 207} \[ -\frac {\sqrt {4 x^2+9}}{2 x^2}-\frac {2}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 207
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}{x \sqrt {9+4 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} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9+4 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.95 \[ -\frac {\sqrt {4 x^2+9}}{2 x^2}-\frac {2}{3} \tanh ^{-1}\left (\sqrt {\frac {4 x^2}{9}+1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 57, normalized size = 1.46 \[ -\frac {4 \, x^{2} \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} + 3\right ) - 4 \, x^{2} \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} - 3\right ) + 3 \, \sqrt {4 \, x^{2} + 9}}{6 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 43, normalized size = 1.10 \[ -\frac {\sqrt {4 \, x^{2} + 9}}{2 \, x^{2}} - \frac {1}{3} \, \log \left (\sqrt {4 \, x^{2} + 9} + 3\right ) + \frac {1}{3} \, \log \left (\sqrt {4 \, x^{2} + 9} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 41, normalized size = 1.05 \[ -\frac {2 \arctanh \left (\frac {3}{\sqrt {4 x^{2}+9}}\right )}{3}-\frac {\left (4 x^{2}+9\right )^{\frac {3}{2}}}{18 x^{2}}+\frac {2 \sqrt {4 x^{2}+9}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.81, size = 35, normalized size = 0.90 \[ \frac {2}{9} \, \sqrt {4 \, x^{2} + 9} - \frac {{\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{18 \, x^{2}} - \frac {2}{3} \, \operatorname {arsinh}\left (\frac {3}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 25, normalized size = 0.64 \[ -\frac {2\,\mathrm {atanh}\left (\frac {2\,\sqrt {x^2+\frac {9}{4}}}{3}\right )}{3}-\frac {\sqrt {x^2+\frac {9}{4}}}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.67, size = 24, normalized size = 0.62 \[ - \frac {2 \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{3} - \frac {\sqrt {1 + \frac {9}{4 x^{2}}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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