Optimal. Leaf size=97 \[ -\frac {\sqrt {x^4+5 x^2+3} \left (2-3 x^2\right )}{2 x^2}+\frac {19}{4} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-\frac {7 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.08, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1251, 812, 843, 621, 206, 724} \[ -\frac {\sqrt {x^4+5 x^2+3} \left (2-3 x^2\right )}{2 x^2}+\frac {19}{4} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-\frac {7 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 812
Rule 843
Rule 1251
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(2+3 x) \sqrt {3+5 x+x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (2-3 x^2\right ) \sqrt {3+5 x^2+x^4}}{2 x^2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {-28-19 x}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (2-3 x^2\right ) \sqrt {3+5 x^2+x^4}}{2 x^2}+\frac {19}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )+7 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (2-3 x^2\right ) \sqrt {3+5 x^2+x^4}}{2 x^2}+\frac {19}{2} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {5+2 x^2}{\sqrt {3+5 x^2+x^4}}\right )-14 \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 {\left (2-3 x^2\right ) \sqrt {3+5 x^2+x^4}}{2 x^2}+\frac {19}{4} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )-\frac {7 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 97, normalized size = 1.00 \[ \frac {\sqrt {x^4+5 x^2+3} \left (3 x^2-2\right )}{2 x^2}+\frac {19}{4} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-\frac {7 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 112, normalized size = 1.15 \[ \frac {56 \, \sqrt {3} x^{2} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) - 114 \, x^{2} \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) + 21 \, x^{2} + 12 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} - 2\right )}}{24 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 138, normalized size = 1.42 \[ \frac {7}{3} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) + \frac {3}{2} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {5 \, x^{2} - 5 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 6}{{\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3} - \frac {19}{4} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 104, normalized size = 1.07 \[ -\frac {7 \sqrt {3}\, \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right )}{3}+\frac {19 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}-\frac {\left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{3 x^{2}}+\frac {7 \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {\left (2 x^{2}+5\right ) \sqrt {x^{4}+5 x^{2}+3}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 89, normalized size = 0.92 \[ -\frac {7}{3} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) + \frac {3}{2} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {19}{4} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 84, normalized size = 0.87 \[ \frac {19\,\ln \left (\sqrt {x^4+5\,x^2+3}+x^2+\frac {5}{2}\right )}{4}-\frac {\sqrt {x^4+5\,x^2+3}}{x^2}-\frac {7\,\sqrt {3}\,\ln \left (\frac {3}{x^2}+\frac {\sqrt {3}\,\sqrt {x^4+5\,x^2+3}}{x^2}+\frac {5}{2}\right )}{3}+\frac {3\,\sqrt {x^4+5\,x^2+3}}{2} \]
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 \[ \int \frac {\left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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