Optimal. Leaf size=90 \[ \frac {-7-8 x^2}{39 x^2 \sqrt {3+5 x^2+x^4}}-\frac {2 \sqrt {3+5 x^2+x^4}}{39 x^2}+\frac {\tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{3 \sqrt {3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1265, 836, 820,
738, 212} \begin {gather*} -\frac {8 x^2+7}{39 x^2 \sqrt {x^4+5 x^2+3}}-\frac {2 \sqrt {x^4+5 x^2+3}}{39 x^2}+\frac {\tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 820
Rule 836
Rule 1265
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x^3 \left (3+5 x^2+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {2+3 x}{x^2 \left (3+5 x+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {7+8 x^2}{39 x^2 \sqrt {3+5 x^2+x^4}}-\frac {1}{39} \text {Subst}\left (\int \frac {-6+8 x}{x^2 \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {7+8 x^2}{39 x^2 \sqrt {3+5 x^2+x^4}}-\frac {2 \sqrt {3+5 x^2+x^4}}{39 x^2}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {7+8 x^2}{39 x^2 \sqrt {3+5 x^2+x^4}}-\frac {2 \sqrt {3+5 x^2+x^4}}{39 x^2}+\frac {2}{3} \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 {7+8 x^2}{39 x^2 \sqrt {3+5 x^2+x^4}}-\frac {2 \sqrt {3+5 x^2+x^4}}{39 x^2}+\frac {\tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 70, normalized size = 0.78 \begin {gather*} \frac {-13-18 x^2-2 x^4}{39 x^2 \sqrt {3+5 x^2+x^4}}-\frac {2 \tanh ^{-1}\left (\frac {x^2-\sqrt {3+5 x^2+x^4}}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 84, normalized size = 0.93
method | result | size |
risch | \(-\frac {2 x^{4}+18 x^{2}+13}{39 x^{2} \sqrt {x^{4}+5 x^{2}+3}}+\frac {\arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{9}\) | \(61\) |
trager | \(-\frac {2 x^{4}+18 x^{2}+13}{39 x^{2} \sqrt {x^{4}+5 x^{2}+3}}-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {-5 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \sqrt {x^{4}+5 x^{2}+3}-6 \RootOf \left (\textit {\_Z}^{2}-3\right )}{x^{2}}\right )}{9}\) | \(79\) |
default | \(-\frac {1}{3 x^{2} \sqrt {x^{4}+5 x^{2}+3}}-\frac {1}{3 \sqrt {x^{4}+5 x^{2}+3}}-\frac {2 x^{2}+5}{39 \sqrt {x^{4}+5 x^{2}+3}}+\frac {\arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{9}\) | \(84\) |
elliptic | \(-\frac {1}{3 x^{2} \sqrt {x^{4}+5 x^{2}+3}}-\frac {1}{3 \sqrt {x^{4}+5 x^{2}+3}}-\frac {2 x^{2}+5}{39 \sqrt {x^{4}+5 x^{2}+3}}+\frac {\arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{9}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 82, normalized size = 0.91 \begin {gather*} -\frac {2 \, x^{2}}{39 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {1}{9} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) - \frac {6}{13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} - \frac {1}{3 \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 124, normalized size = 1.38 \begin {gather*} -\frac {6 \, x^{6} + 30 \, x^{4} - 13 \, \sqrt {3} {\left (x^{6} + 5 \, x^{4} + 3 \, x^{2}\right )} \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 ) + 18 \, x^{2} + 3 \, {\left (2 \, x^{4} + 18 \, x^{2} + 13\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{117 \, {\left (x^{6} + 5 \, x^{4} + 3 \, x^{2}\right )}} \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^{3} \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.47, size = 122, normalized size = 1.36 \begin {gather*} -\frac {1}{9} \, \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 {7 \, x^{2} + 11}{117 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {5 \, x^{2} - 5 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 6}{9 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}} \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^3\,{\left (x^4+5\,x^2+3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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