Optimal. Leaf size=90 \[ \frac {-8 x^2-7}{39 x^2 \sqrt {x^4+5 x^2+3}}-\frac {2 \sqrt {x^4+5 x^2+3}}{39 x^2}+\frac {\tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.07, 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 = {1251, 822, 806, 724, 206} \[ -\frac {8 x^2+7}{39 x^2 \sqrt {x^4+5 x^2+3}}-\frac {2 \sqrt {x^4+5 x^2+3}}{39 x^2}+\frac {\tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 822
Rule 1251
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} \operatorname {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} \operatorname {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} \operatorname {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} \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 {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 = 0.02, size = 88, normalized size = 0.98 \[ \frac {-6 x^4-54 x^2+13 \sqrt {3} \sqrt {x^4+5 x^2+3} x^2 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )-39}{117 x^2 \sqrt {x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 124, normalized size = 1.38 \[ -\frac {6 \, x^{6} + 30 \, x^{4} - 13 \, \sqrt {3} {\left (x^{6} + 5 \, x^{4} + 3 \, x^{2}\right )} \log \left (\frac {25 \, x^{2} + 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} + 6\right )} + 30}{x^{2}}\right ) + 18 \, x^{2} + 3 \, {\left (2 \, x^{4} + 18 \, x^{2} + 13\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{117 \, {\left (x^{6} + 5 \, x^{4} + 3 \, x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 122, normalized size = 1.36 \[ -\frac {1}{9} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) + \frac {7 \, x^{2} + 11}{117 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {5 \, x^{2} - 5 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 6}{9 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 0.93 \[ \frac {\sqrt {3}\, \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right )}{9}-\frac {1}{3 \sqrt {x^{4}+5 x^{2}+3}\, x^{2}}-\frac {1}{3 \sqrt {x^{4}+5 x^{2}+3}}-\frac {2 x^{2}+5}{39 \sqrt {x^{4}+5 x^{2}+3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.09, size = 82, normalized size = 0.91 \[ -\frac {2 \, x^{2}}{39 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {1}{9} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) - \frac {6}{13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} - \frac {1}{3 \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2}} \]
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 \[ \int \frac {3\,x^2+2}{x^3\,{\left (x^4+5\,x^2+3\right )}^{3/2}} \,d x \]
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 \[ \int \frac {3 x^{2} + 2}{x^{3} \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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