Optimal. Leaf size=57 \[ -\frac {\sqrt {9+4 x^2}}{36 x^4}+\frac {\sqrt {9+4 x^2}}{54 x^2}-\frac {2}{81} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9+4 x^2}\right ) \]
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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 = {272, 44, 65,
213} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {9+4 x^2}} \, dx &=\frac {1}{2} \text {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} \text {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} \text {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} \text {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 [A]
time = 0.04, size = 46, normalized size = 0.81 \begin {gather*} \frac {\left (-3+2 x^2\right ) \sqrt {9+4 x^2}}{108 x^4}-\frac {2}{81} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9+4 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 44, normalized size = 0.77
method | result | size |
trager | \(\frac {\left (2 x^{2}-3\right ) \sqrt {4 x^{2}+9}}{108 x^{4}}+\frac {2 \ln \left (\frac {\sqrt {4 x^{2}+9}-3}{x}\right )}{81}\) | \(41\) |
risch | \(\frac {8 x^{4}+6 x^{2}-27}{108 x^{4} \sqrt {4 x^{2}+9}}-\frac {2 \arctanh \left (\frac {3}{\sqrt {4 x^{2}+9}}\right )}{81}\) | \(42\) |
default | \(-\frac {\sqrt {4 x^{2}+9}}{36 x^{4}}+\frac {\sqrt {4 x^{2}+9}}{54 x^{2}}-\frac {2 \arctanh \left (\frac {3}{\sqrt {4 x^{2}+9}}\right )}{81}\) | \(44\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \left (-\frac {112}{81} x^{4}-\frac {32}{9} x^{2}+8\right )}{96 x^{4}}-\frac {\sqrt {\pi }\, \left (-\frac {16 x^{2}}{3}+8\right ) \sqrt {1+\frac {4 x^{2}}{9}}}{96 x^{4}}-\frac {2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {4 x^{2}}{9}}}{2}\right )}{81}+\frac {\left (\frac {7}{6}+2 \ln \left (x \right )-2 \ln \left (3\right )\right ) \sqrt {\pi }}{81}-\frac {\sqrt {\pi }}{12 x^{4}}+\frac {\sqrt {\pi }}{27 x^{2}}}{\sqrt {\pi }}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 38, normalized size = 0.67 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.44, size = 64, normalized size = 1.12 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.48, size = 63, normalized size = 1.11 \begin {gather*} - \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}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.23, size = 55, normalized size = 0.96 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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