Optimal. Leaf size=270 \[ \sqrt {x^4+5 x^2+3} x-\frac {4 \left (2 x^2+\sqrt {13}+5\right ) x}{\sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}+\frac {2 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}} \]
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Rubi [A] time = 0.12, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1279, 1189, 1099, 1135} \[ \sqrt {x^4+5 x^2+3} x-\frac {4 \left (2 x^2+\sqrt {13}+5\right ) x}{\sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}+\frac {2 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1189
Rule 1279
Rubi steps
\begin {align*} \int \frac {x^2 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx &=x \sqrt {3+5 x^2+x^4}-\frac {1}{3} \int \frac {9+24 x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=x \sqrt {3+5 x^2+x^4}-3 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx-8 \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {4 x \left (5+\sqrt {13}+2 x^2\right )}{\sqrt {3+5 x^2+x^4}}+x \sqrt {3+5 x^2+x^4}+\frac {2 \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 ) 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 )}{\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 ) 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 = 0.24, size = 222, normalized size = 0.82 \[ \frac {i \sqrt {2} \left (4 \sqrt {13}-17\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-4 i \sqrt {2} \left (\sqrt {13}-5\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+2 x \left (x^4+5 x^2+3\right )}{2 \sqrt {x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {3 \, x^{4} + 2 \, x^{2}}{\sqrt {x^{4} + 5 \, x^{2} + 3}}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x^{2} + 2\right )} x^{2}}{\sqrt {x^{4} + 5 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 208, normalized size = 0.77 \[ \sqrt {x^{4}+5 x^{2}+3}\, x -\frac {18 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {288 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )+\EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (\sqrt {13}+5\right )} \]
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 \[ \int \frac {{\left (3 \, x^{2} + 2\right )} x^{2}}{\sqrt {x^{4} + 5 \, x^{2} + 3}}\,{d x} \]
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 \[ \int \frac {x^2\,\left (3\,x^2+2\right )}{\sqrt {x^4+5\,x^2+3}} \,d x \]
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 \[ \int \frac {x^{2} \left (3 x^{2} + 2\right )}{\sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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