Optimal. Leaf size=57 \[ \frac {\sqrt {4 x^2+9}}{54 x^2}-\frac {2}{81} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right )-\frac {\sqrt {4 x^2+9}}{36 x^4} \]
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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, 207} \[ \frac {\sqrt {4 x^2+9}}{54 x^2}-\frac {\sqrt {4 x^2+9}}{36 x^4}-\frac {2}{81} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 207
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}{x^3 \sqrt {9+4 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9+4 x^2}}{36 x^4}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {9+4 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}{27} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {9+4 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 \[ -\frac {16}{729} \sqrt {4 x^2+9} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {4 x^2}{9}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 64, normalized size = 1.12 \[ -\frac {8 \, x^{4} \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} + 3\right ) - 8 \, x^{4} \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} - 3\right ) - 3 \, \sqrt {4 \, x^{2} + 9} {\left (2 \, x^{2} - 3\right )}}{324 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 55, normalized size = 0.96 \[ \frac {{\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} - 15 \, \sqrt {4 \, x^{2} + 9}}{216 \, x^{4}} - \frac {1}{81} \, \log \left (\sqrt {4 \, x^{2} + 9} + 3\right ) + \frac {1}{81} \, \log \left (\sqrt {4 \, x^{2} + 9} - 3\right ) \]
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 = 2.95, size = 38, normalized size = 0.67 \[ \frac {\sqrt {4 \, x^{2} + 9}}{54 \, x^{2}} - \frac {\sqrt {4 \, x^{2} + 9}}{36 \, x^{4}} - \frac {2}{81} \, \operatorname {arsinh}\left (\frac {3}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 33, normalized size = 0.58 \[ \frac {\sqrt {x^2+\frac {9}{4}}\,\left (\frac {2}{27\,x^2}-\frac {1}{9\,x^4}\right )}{2}-\frac {2\,\mathrm {atanh}\left (\frac {2\,\sqrt {x^2+\frac {9}{4}}}{3}\right )}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.89, size = 63, normalized size = 1.11 \[ - \frac {2 \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{81} + \frac {1}{27 x \sqrt {1 + \frac {9}{4 x^{2}}}} + \frac {1}{36 x^{3} \sqrt {1 + \frac {9}{4 x^{2}}}} - \frac {1}{8 x^{5} \sqrt {1 + \frac {9}{4 x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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