Optimal. Leaf size=83 \[ -\frac {\sqrt {x^4+5 x^2+3}}{12 x^2}-\frac {\sqrt {x^4+5 x^2+3}}{6 x^4}+\frac {1}{8} \sqrt {3} \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 83, 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 = {1251, 834, 806, 724, 206} \[ -\frac {\sqrt {x^4+5 x^2+3}}{12 x^2}-\frac {\sqrt {x^4+5 x^2+3}}{6 x^4}+\frac {1}{8} \sqrt {3} \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 834
Rule 1251
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x^5 \sqrt {3+5 x^2+x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {2+3 x}{x^3 \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{6 x^4}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {-3+2 x}{x^2 \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{6 x^4}-\frac {\sqrt {3+5 x^2+x^4}}{12 x^2}-\frac {3}{8} \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}}{6 x^4}-\frac {\sqrt {3+5 x^2+x^4}}{12 x^2}+\frac {3}{4} \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}}{6 x^4}-\frac {\sqrt {3+5 x^2+x^4}}{12 x^2}+\frac {1}{8} \sqrt {3} \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 67, normalized size = 0.81 \[ \frac {1}{8} \sqrt {3} \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )-\frac {\left (x^2+2\right ) \sqrt {x^4+5 x^2+3}}{12 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 83, normalized size = 1.00 \[ \frac {3 \, \sqrt {3} x^{4} \log \left (\frac {25 \, x^{2} + 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} + 6\right )} + 30}{x^{2}}\right ) - 2 \, x^{4} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (x^{2} + 2\right )}}{24 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 145, normalized size = 1.75 \[ -\frac {1}{8} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) + \frac {9 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} + 36 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} + 47 \, x^{2} - 47 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 12}{12 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 0.80 \[ \frac {\sqrt {3}\, \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right )}{8}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{12 x^{2}}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{6 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 68, normalized size = 0.82 \[ \frac {1}{8} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{12 \, x^{2}} - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{6 \, x^{4}} \]
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 \[ \int \frac {3\,x^2+2}{x^5\,\sqrt {x^4+5\,x^2+3}} \,d x \]
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 \[ \int \frac {3 x^{2} + 2}{x^{5} \sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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