Optimal. Leaf size=57 \[ -\frac {\sqrt {4 x^2+9}}{18 x^2}+\frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right )-\frac {\sqrt {4 x^2+9}}{4 x^4} \]
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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, 207} \[ -\frac {\sqrt {4 x^2+9}}{18 x^2}-\frac {\sqrt {4 x^2+9}}{4 x^4}+\frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 207
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}{x^2 \sqrt {9+4 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}{9} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {9+4 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 \[ -\frac {16 \left (4 x^2+9\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {4 x^2}{9}+1\right )}{2187} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 64, normalized size = 1.12 \[ \frac {8 \, x^{4} \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} + 3\right ) - 8 \, x^{4} \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} - 3\right ) - 3 \, \sqrt {4 \, x^{2} + 9} {\left (2 \, x^{2} + 9\right )}}{108 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 55, normalized size = 0.96 \[ -\frac {{\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} + 9 \, \sqrt {4 \, x^{2} + 9}}{72 \, x^{4}} + \frac {1}{27} \, \log \left (\sqrt {4 \, x^{2} + 9} + 3\right ) - \frac {1}{27} \, \log \left (\sqrt {4 \, x^{2} + 9} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, 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 = 3.01, size = 49, normalized size = 0.86 \[ -\frac {2}{81} \, \sqrt {4 \, x^{2} + 9} + \frac {{\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{162 \, x^{2}} - \frac {{\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{36 \, x^{4}} + \frac {2}{27} \, \operatorname {arsinh}\left (\frac {3}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 45, normalized size = 0.79 \[ \frac {2\,\mathrm {atanh}\left (\frac {2\,\sqrt {x^2+\frac {9}{4}}}{3}\right )}{27}+\frac {\sqrt {x^2+\frac {9}{4}}\,\left (\frac {2}{3\,x^2}-\frac {1}{x^4}\right )}{2}-\frac {4\,\sqrt {x^2+\frac {9}{4}}}{9\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.20, size = 63, normalized size = 1.11 \[ \frac {2 \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{27} - \frac {1}{9 x \sqrt {1 + \frac {9}{4 x^{2}}}} - \frac {3}{4 x^{3} \sqrt {1 + \frac {9}{4 x^{2}}}} - \frac {9}{8 x^{5} \sqrt {1 + \frac {9}{4 x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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