Optimal. Leaf size=257 \[ \frac {3 x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}+\frac {\sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {3}{2} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}} \]
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Rubi [A] time = 0.08, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1189, 1099, 1135} \[ \frac {3 x \left (2 x^2+\sqrt {13}+5\right )}{2 \sqrt {x^4+5 x^2+3}}+\frac {\sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {3}{2} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1189
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{\sqrt {3+5 x^2+x^4}} \, dx &=2 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx+3 \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=\frac {3 x \left (5+\sqrt {13}+2 x^2\right )}{2 \sqrt {3+5 x^2+x^4}}-\frac {\sqrt {\frac {3}{2} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{2 \sqrt {3+5 x^2+x^4}}+\frac {\sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {3+5 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 159, normalized size = 0.62 \[ \frac {i \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} \left (\left (11-3 \sqrt {13}\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+3 \left (\sqrt {13}-5\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )\right )}{2 \sqrt {2} \sqrt {x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 194, normalized size = 0.75 \[ \frac {12 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {108 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )+\EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (\sqrt {13}+5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {3\,x^2+2}{\sqrt {x^4+5\,x^2+3}} \,d x \]
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 \[ \int \frac {3 x^{2} + 2}{\sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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