Optimal. Leaf size=104 \[ \frac {13 \sqrt {x^4+5 x^2+3}}{108 x^2}-\frac {\sqrt {x^4+5 x^2+3}}{54 x^4}-\frac {61 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{216 \sqrt {3}}-\frac {\sqrt {x^4+5 x^2+3}}{9 x^6} \]
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Rubi [A] time = 0.09, 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 = {1251, 834, 806, 724, 206} \[ \frac {13 \sqrt {x^4+5 x^2+3}}{108 x^2}-\frac {\sqrt {x^4+5 x^2+3}}{54 x^4}-\frac {\sqrt {x^4+5 x^2+3}}{9 x^6}-\frac {61 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{216 \sqrt {3}} \]
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^7 \sqrt {3+5 x^2+x^4}} \, dx &=\frac {1}{2} \operatorname {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} \operatorname {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} \operatorname {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} \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}}{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} \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}}{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 = 0.02, size = 77, normalized size = 0.74 \[ \frac {6 \sqrt {x^4+5 x^2+3} \left (13 x^4-2 x^2-12\right )-61 \sqrt {3} x^6 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{648 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 90, normalized size = 0.87 \[ \frac {61 \, \sqrt {3} x^{6} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) + 78 \, x^{6} + 6 \, {\left (13 \, x^{4} - 2 \, x^{2} - 12\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{648 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 167, normalized size = 1.61 \[ \frac {61}{648} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) - \frac {61 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{5} - 920 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} - 2052 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 1449 \, x^{2} + 1449 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 108}{108 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 83, normalized size = 0.80 \[ -\frac {61 \sqrt {3}\, \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right )}{648}+\frac {13 \sqrt {x^{4}+5 x^{2}+3}}{108 x^{2}}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{54 x^{4}}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{9 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 85, normalized size = 0.82 \[ -\frac {61}{648} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) + \frac {13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{108 \, x^{2}} - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{54 \, x^{4}} - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{9 \, x^{6}} \]
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 \[ \int \frac {3\,x^2+2}{x^7\,\sqrt {x^4+5\,x^2+3}} \,d x \]
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 \[ \int \frac {3 x^{2} + 2}{x^{7} \sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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