Optimal. Leaf size=62 \[ -\frac {\sqrt {x^4+5 x^2+3}}{3 x^2}-\frac {2 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1251, 806, 724, 206} \[ -\frac {\sqrt {x^4+5 x^2+3}}{3 x^2}-\frac {2 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 1251
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x^3 \sqrt {3+5 x^2+x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {2+3 x}{x^2 \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{3 x^2}+\frac {2}{3} \operatorname {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}}{3 x^2}-\frac {4}{3} \operatorname {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}}{3 x^2}-\frac {2 \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 = 0.01, size = 62, normalized size = 1.00 \[ -\frac {\sqrt {x^4+5 x^2+3}}{3 x^2}-\frac {2 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 78, normalized size = 1.26 \[ \frac {2 \, \sqrt {3} x^{2} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) - 3 \, x^{2} - 3 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{9 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 101, normalized size = 1.63 \[ \frac {2}{9} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) + \frac {5 \, x^{2} - 5 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 6}{3 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 49, normalized size = 0.79 \[ -\frac {2 \sqrt {3}\, \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right )}{9}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{3 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.02, size = 51, normalized size = 0.82 \[ -\frac {2}{9} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{3 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 83, normalized size = 1.34 \[ \frac {5\,\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\left (5\,x^2+6\right )}{6\,\sqrt {x^4+5\,x^2+3}}\right )}{18}-\frac {\sqrt {x^4+5\,x^2+3}}{3\,x^2}-\frac {\sqrt {3}\,\left (\ln \left (\frac {1}{x^2}\right )+\ln \left (2\,\sqrt {3}\,\sqrt {x^4+5\,x^2+3}+5\,x^2+6\right )\right )}{2} \]
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 \[ \int \frac {3 x^{2} + 2}{x^{3} \sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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