Optimal. Leaf size=104 \[ -\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {\sqrt {3+5 x^2+x^4}}{54 x^4}+\frac {13 \sqrt {3+5 x^2+x^4}}{108 x^2}-\frac {61 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{216 \sqrt {3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1265, 848, 820,
738, 212} \begin {gather*} \frac {13 \sqrt {x^4+5 x^2+3}}{108 x^2}-\frac {\sqrt {x^4+5 x^2+3}}{54 x^4}-\frac {61 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{216 \sqrt {3}}-\frac {\sqrt {x^4+5 x^2+3}}{9 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 820
Rule 848
Rule 1265
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x^7 \sqrt {3+5 x^2+x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {2+3 x}{x^4 \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {1}{18} \text {Subst}\left (\int \frac {-2+4 x}{x^3 \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {\sqrt {3+5 x^2+x^4}}{54 x^4}+\frac {1}{108} \text {Subst}\left (\int \frac {-39-2 x}{x^2 \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {\sqrt {3+5 x^2+x^4}}{54 x^4}+\frac {13 \sqrt {3+5 x^2+x^4}}{108 x^2}+\frac {61}{216} \text {Subst}\left (\int \frac {1}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {\sqrt {3+5 x^2+x^4}}{54 x^4}+\frac {13 \sqrt {3+5 x^2+x^4}}{108 x^2}-\frac {61}{108} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {6+5 x^2}{\sqrt {3+5 x^2+x^4}}\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {\sqrt {3+5 x^2+x^4}}{54 x^4}+\frac {13 \sqrt {3+5 x^2+x^4}}{108 x^2}-\frac {61 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{216 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 75, normalized size = 0.72 \begin {gather*} \frac {\sqrt {3+5 x^2+x^4} \left (-12-2 x^2+13 x^4\right )}{108 x^6}+\frac {61 \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}-\frac {\sqrt {3+5 x^2+x^4}}{\sqrt {3}}\right )}{108 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 83, normalized size = 0.80
method | result | size |
risch | \(\frac {13 x^{8}+63 x^{6}+17 x^{4}-66 x^{2}-36}{108 x^{6} \sqrt {x^{4}+5 x^{2}+3}}-\frac {61 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{648}\) | \(71\) |
trager | \(\frac {\left (13 x^{4}-2 x^{2}-12\right ) \sqrt {x^{4}+5 x^{2}+3}}{108 x^{6}}+\frac {61 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {-5 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \sqrt {x^{4}+5 x^{2}+3}-6 \RootOf \left (\textit {\_Z}^{2}-3\right )}{x^{2}}\right )}{648}\) | \(79\) |
default | \(-\frac {61 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{648}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{9 x^{6}}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{54 x^{4}}+\frac {13 \sqrt {x^{4}+5 x^{2}+3}}{108 x^{2}}\) | \(83\) |
elliptic | \(-\frac {61 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{648}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{9 x^{6}}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{54 x^{4}}+\frac {13 \sqrt {x^{4}+5 x^{2}+3}}{108 x^{2}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 85, normalized size = 0.82 \begin {gather*} -\frac {61}{648} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) + \frac {13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{108 \, x^{2}} - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{54 \, x^{4}} - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{9 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 90, normalized size = 0.87 \begin {gather*} \frac {61 \, \sqrt {3} x^{6} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) + 78 \, x^{6} + 6 \, {\left (13 \, x^{4} - 2 \, x^{2} - 12\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{648 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x^{2} + 2}{x^{7} \sqrt {x^{4} + 5 x^{2} + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs.
\(2 (82) = 164\).
time = 3.63, size = 167, normalized size = 1.61 \begin {gather*} \frac {61}{648} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) - \frac {61 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{5} - 920 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} - 2052 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 1449 \, x^{2} + 1449 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 108}{108 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,x^2+2}{x^7\,\sqrt {x^4+5\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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