Optimal. Leaf size=305 \[ \frac{49 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right ),\frac{1}{6} \left (5 \sqrt{13}-13\right )\right )}{3 \sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}-\frac{\sqrt{x^4+5 x^2+3} \left (2-9 x^2\right )}{3 x^3}-\frac{64 \sqrt{x^4+5 x^2+3}}{9 x}+\frac{32 x \left (2 x^2+\sqrt{13}+5\right )}{9 \sqrt{x^4+5 x^2+3}}-\frac{16 \sqrt{\frac{2}{3} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{9 \sqrt{x^4+5 x^2+3}} \]
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Rubi [A] time = 0.158304, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1271, 1281, 1189, 1099, 1135} \[ -\frac{\sqrt{x^4+5 x^2+3} \left (2-9 x^2\right )}{3 x^3}-\frac{64 \sqrt{x^4+5 x^2+3}}{9 x}+\frac{32 x \left (2 x^2+\sqrt{13}+5\right )}{9 \sqrt{x^4+5 x^2+3}}+\frac{49 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{3 \sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}-\frac{16 \sqrt{\frac{2}{3} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{9 \sqrt{x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1271
Rule 1281
Rule 1189
Rule 1099
Rule 1135
Rubi steps
\begin{align*} \int \frac{\left (2+3 x^2\right ) \sqrt{3+5 x^2+x^4}}{x^4} \, dx &=-\frac{\left (2-9 x^2\right ) \sqrt{3+5 x^2+x^4}}{3 x^3}-\frac{1}{3} \int \frac{-64-49 x^2}{x^2 \sqrt{3+5 x^2+x^4}} \, dx\\ &=-\frac{64 \sqrt{3+5 x^2+x^4}}{9 x}-\frac{\left (2-9 x^2\right ) \sqrt{3+5 x^2+x^4}}{3 x^3}+\frac{1}{9} \int \frac{147+64 x^2}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=-\frac{64 \sqrt{3+5 x^2+x^4}}{9 x}-\frac{\left (2-9 x^2\right ) \sqrt{3+5 x^2+x^4}}{3 x^3}+\frac{64}{9} \int \frac{x^2}{\sqrt{3+5 x^2+x^4}} \, dx+\frac{49}{3} \int \frac{1}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=\frac{32 x \left (5+\sqrt{13}+2 x^2\right )}{9 \sqrt{3+5 x^2+x^4}}-\frac{64 \sqrt{3+5 x^2+x^4}}{9 x}-\frac{\left (2-9 x^2\right ) \sqrt{3+5 x^2+x^4}}{3 x^3}-\frac{16 \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 )}{9 \sqrt{3+5 x^2+x^4}}+\frac{49 \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 )}{3 \sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{3+5 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.313315, size = 237, normalized size = 0.78 \[ \frac{-i \sqrt{2} \left (32 \sqrt{13}-13\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right ),\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )-2 \left (37 x^6+191 x^4+141 x^2+18\right )+32 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )}{18 x^3 \sqrt{x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.017, size = 228, normalized size = 0.8 \begin{align*} -{\frac{37}{9\,x}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+98\,{\frac{\sqrt{1- \left ( -5/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -5/6-1/6\,\sqrt{13} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) }{\sqrt{-30+6\,\sqrt{13}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}}-256\,{\frac{\sqrt{1- \left ( -5/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -5/6-1/6\,\sqrt{13} \right ){x}^{2}} \left ({\it EllipticF} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) -{\it EllipticE} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) \right ) }{\sqrt{-30+6\,\sqrt{13}}\sqrt{{x}^{4}+5\,{x}^{2}+3} \left ( \sqrt{13}+5 \right ) }}-{\frac{2}{3\,{x}^{3}}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )}}{x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x^{2} + 2\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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