Optimal. Leaf size=331 \[ -\frac{353 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right ),\frac{1}{6} \left (5 \sqrt{13}-13\right )\right )}{33 \sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}+\frac{1}{99} \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2} x^3-\frac{\left (890 x^2+911\right ) \sqrt{x^4+5 x^2+3} x^3}{1155}+\frac{353}{99} \sqrt{x^4+5 x^2+3} x-\frac{49949 \left (2 x^2+\sqrt{13}+5\right ) x}{3465 \sqrt{x^4+5 x^2+3}}+\frac{49949 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{3465 \sqrt{x^4+5 x^2+3}} \]
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Rubi [A] time = 0.213838, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1273, 1279, 1189, 1099, 1135} \[ \frac{1}{99} \left (27 x^2+67\right ) \left (x^4+5 x^2+3\right )^{3/2} x^3-\frac{\left (890 x^2+911\right ) \sqrt{x^4+5 x^2+3} x^3}{1155}+\frac{353}{99} \sqrt{x^4+5 x^2+3} x-\frac{49949 \left (2 x^2+\sqrt{13}+5\right ) x}{3465 \sqrt{x^4+5 x^2+3}}-\frac{353 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{33 \sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}+\frac{49949 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{3465 \sqrt{x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1273
Rule 1279
Rule 1189
Rule 1099
Rule 1135
Rubi steps
\begin{align*} \int x^2 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx &=\frac{1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{1}{33} \int x^2 \left (-3-178 x^2\right ) \sqrt{3+5 x^2+x^4} \, dx\\ &=-\frac{x^3 \left (911+890 x^2\right ) \sqrt{3+5 x^2+x^4}}{1155}+\frac{1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{\int \frac{x^2 \left (7884+12355 x^2\right )}{\sqrt{3+5 x^2+x^4}} \, dx}{1155}\\ &=\frac{353}{99} x \sqrt{3+5 x^2+x^4}-\frac{x^3 \left (911+890 x^2\right ) \sqrt{3+5 x^2+x^4}}{1155}+\frac{1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac{\int \frac{37065+99898 x^2}{\sqrt{3+5 x^2+x^4}} \, dx}{3465}\\ &=\frac{353}{99} x \sqrt{3+5 x^2+x^4}-\frac{x^3 \left (911+890 x^2\right ) \sqrt{3+5 x^2+x^4}}{1155}+\frac{1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac{353}{33} \int \frac{1}{\sqrt{3+5 x^2+x^4}} \, dx-\frac{99898 \int \frac{x^2}{\sqrt{3+5 x^2+x^4}} \, dx}{3465}\\ &=-\frac{49949 x \left (5+\sqrt{13}+2 x^2\right )}{3465 \sqrt{3+5 x^2+x^4}}+\frac{353}{99} x \sqrt{3+5 x^2+x^4}-\frac{x^3 \left (911+890 x^2\right ) \sqrt{3+5 x^2+x^4}}{1155}+\frac{1}{99} x^3 \left (67+27 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{49949 \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 )}{3465 \sqrt{3+5 x^2+x^4}}-\frac{353 \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 )}{33 \sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{3+5 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.308416, size = 244, normalized size = 0.74 \[ \frac{i \sqrt{2} \left (49949 \sqrt{13}-212680\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right ),\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+2 x \left (945 x^{12}+11795 x^{10}+50075 x^8+84962 x^6+69535 x^4+74681 x^2+37065\right )-49949 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )}{6930 \sqrt{x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.015, size = 277, normalized size = 0.8 \begin{align*}{\frac{3\,{x}^{9}}{11}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{202\,{x}^{7}}{99}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{2378\,{x}^{5}}{693}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{478\,{x}^{3}}{385}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{353\,x}{99}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{706}{11\,\sqrt{-30+6\,\sqrt{13}}}\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}+{\frac{399592}{385\,\sqrt{-30+6\,\sqrt{13}} \left ( \sqrt{13}+5 \right ) }\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) \right ){\frac{1}{\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{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )} x^{2}\,{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 ({\left (3 \, x^{8} + 17 \, x^{6} + 19 \, x^{4} + 6 \, x^{2}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}, 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 x^{2} \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}\, 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{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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