Optimal. Leaf size=305 \[ -\frac {4}{3} \sqrt {x^4+5 x^2+3} x+\frac {1247 \left (2 x^2+\sqrt {13}+5\right ) x}{210 \sqrt {x^4+5 x^2+3}}+\frac {2 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {1247 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{210 \sqrt {x^4+5 x^2+3}}+\frac {1}{35} \left (15 x^2+29\right ) \sqrt {x^4+5 x^2+3} x^3 \]
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Rubi [A] time = 0.20, antiderivative size = 305, 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 = {1273, 1279, 1189, 1099, 1135} \[ \frac {1}{35} \left (15 x^2+29\right ) \sqrt {x^4+5 x^2+3} x^3-\frac {4}{3} \sqrt {x^4+5 x^2+3} x+\frac {1247 \left (2 x^2+\sqrt {13}+5\right ) x}{210 \sqrt {x^4+5 x^2+3}}+\frac {2 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {1247 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{210 \sqrt {x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1189
Rule 1273
Rule 1279
Rubi steps
\begin {align*} \int x^2 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx &=\frac {1}{35} x^3 \left (29+15 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{35} \int \frac {x^2 \left (-51-140 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {4}{3} x \sqrt {3+5 x^2+x^4}+\frac {1}{35} x^3 \left (29+15 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1}{105} \int \frac {-420-1247 x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {4}{3} x \sqrt {3+5 x^2+x^4}+\frac {1}{35} x^3 \left (29+15 x^2\right ) \sqrt {3+5 x^2+x^4}+4 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx+\frac {1247}{105} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=\frac {1247 x \left (5+\sqrt {13}+2 x^2\right )}{210 \sqrt {3+5 x^2+x^4}}-\frac {4}{3} x \sqrt {3+5 x^2+x^4}+\frac {1}{35} x^3 \left (29+15 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1247 \sqrt {\frac {1}{6} \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 )}{210 \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 ) 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 = 0.26, size = 234, normalized size = 0.77 \[ \frac {-i \sqrt {2} \left (1247 \sqrt {13}-5395\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+1247 i \sqrt {2} \left (\sqrt {13}-5\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+4 x \left (45 x^8+312 x^6+430 x^4-439 x^2-420\right )}{420 \sqrt {x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (3 \, x^{4} + 2 \, x^{2}\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}, x\right ) \]
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 \[ \int \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 243, normalized size = 0.80 \[ \frac {3 \sqrt {x^{4}+5 x^{2}+3}\, x^{5}}{7}+\frac {29 \sqrt {x^{4}+5 x^{2}+3}\, x^{3}}{35}-\frac {4 \sqrt {x^{4}+5 x^{2}+3}\, x}{3}+\frac {24 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {14964 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )+\EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (\sqrt {13}+5\right )} \]
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 \[ \int \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )} x^{2}\,{d x} \]
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 \[ \int x^2\,\left (3\,x^2+2\right )\,\sqrt {x^4+5\,x^2+3} \,d x \]
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 \[ \int x^{2} \left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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