Optimal. Leaf size=302 \[ \frac {7 x \left (2 x^2+\sqrt {13}+5\right )}{54 \sqrt {x^4+5 x^2+3}}-\frac {7 \sqrt {x^4+5 x^2+3}}{27 x}-\frac {\sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{9 \sqrt {x^4+5 x^2+3}}-\frac {7 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{54 \sqrt {x^4+5 x^2+3}}-\frac {2 \sqrt {x^4+5 x^2+3}}{9 x^3} \]
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Rubi [A] time = 0.16, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1281, 1189, 1099, 1135} \[ \frac {7 x \left (2 x^2+\sqrt {13}+5\right )}{54 \sqrt {x^4+5 x^2+3}}-\frac {7 \sqrt {x^4+5 x^2+3}}{27 x}-\frac {2 \sqrt {x^4+5 x^2+3}}{9 x^3}-\frac {\sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{9 \sqrt {x^4+5 x^2+3}}-\frac {7 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{54 \sqrt {x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1189
Rule 1281
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x^4 \sqrt {3+5 x^2+x^4}} \, dx &=-\frac {2 \sqrt {3+5 x^2+x^4}}{9 x^3}-\frac {1}{9} \int \frac {-7+2 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {2 \sqrt {3+5 x^2+x^4}}{9 x^3}-\frac {7 \sqrt {3+5 x^2+x^4}}{27 x}+\frac {1}{27} \int \frac {-6+7 x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {2 \sqrt {3+5 x^2+x^4}}{9 x^3}-\frac {7 \sqrt {3+5 x^2+x^4}}{27 x}-\frac {2}{9} \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx+\frac {7}{27} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=\frac {7 x \left (5+\sqrt {13}+2 x^2\right )}{54 \sqrt {3+5 x^2+x^4}}-\frac {2 \sqrt {3+5 x^2+x^4}}{9 x^3}-\frac {7 \sqrt {3+5 x^2+x^4}}{27 x}-\frac {7 \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 )}{54 \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 )}{9 \sqrt {3+5 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 237, normalized size = 0.78 \[ \frac {-i \sqrt {2} \left (7 \sqrt {13}-47\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} x^3 F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+7 i \sqrt {2} \left (\sqrt {13}-5\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} x^3 E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-4 \left (7 x^6+41 x^4+51 x^2+18\right )}{108 x^3 \sqrt {x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )}}{x^{8} + 5 \, x^{6} + 3 \, x^{4}}, x\right ) \]
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 \[ \int \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 228, normalized size = 0.75 \[ -\frac {4 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{3 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {7 \sqrt {x^{4}+5 x^{2}+3}}{27 x}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{9 x^{3}}-\frac {28 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )+\EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{3 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (\sqrt {13}+5\right )} \]
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 \[ \int \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3} x^{4}}\,{d x} \]
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 \[ \int \frac {3\,x^2+2}{x^4\,\sqrt {x^4+5\,x^2+3}} \,d x \]
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 \[ \int \frac {3 x^{2} + 2}{x^{4} \sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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