Optimal. Leaf size=424 \[ \frac{\left (263 \sqrt{2}-70\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right ),\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{60\ 2^{3/4} \left (3 \sqrt{2}-2\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{1}{60} \left (13-6 x^2\right ) \sqrt{2 x^4+2 x^2+1} x+\frac{109 \sqrt{2 x^4+2 x^2+1} x}{60 \sqrt{2} \left (\sqrt{2} x^2+1\right )}+\frac{3}{16} \sqrt{15} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{109 \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{60\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{15 \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{16\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}} \]
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Rubi [A] time = 0.561012, antiderivative size = 619, normalized size of antiderivative = 1.46, number of steps used = 17, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {1335, 1091, 1197, 1103, 1195, 1116, 1208, 1216, 1706} \[ \frac{1}{30} \left (3 x^2+1\right ) \sqrt{2 x^4+2 x^2+1} x+\frac{109 \sqrt{2 x^4+2 x^2+1} x}{60 \sqrt{2} \left (\sqrt{2} x^2+1\right )}-\frac{1}{4} \sqrt{2 x^4+2 x^2+1} x+\frac{3}{16} \sqrt{15} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )+\frac{45 \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{112 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\left (1+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{4\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{139 \left (1-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{240 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{109 \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{60\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{15 \left (3+\sqrt{2}\right )^2 \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{224 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1335
Rule 1091
Rule 1197
Rule 1103
Rule 1195
Rule 1116
Rule 1208
Rule 1216
Rule 1706
Rubi steps
\begin{align*} \int \frac{x^4 \sqrt{1+2 x^2+2 x^4}}{3+2 x^2} \, dx &=\int \left (-\frac{3}{4} \sqrt{1+2 x^2+2 x^4}+\frac{1}{2} x^2 \sqrt{1+2 x^2+2 x^4}+\frac{9 \sqrt{1+2 x^2+2 x^4}}{4 \left (3+2 x^2\right )}\right ) \, dx\\ &=\frac{1}{2} \int x^2 \sqrt{1+2 x^2+2 x^4} \, dx-\frac{3}{4} \int \sqrt{1+2 x^2+2 x^4} \, dx+\frac{9}{4} \int \frac{\sqrt{1+2 x^2+2 x^4}}{3+2 x^2} \, dx\\ &=-\frac{1}{4} x \sqrt{1+2 x^2+2 x^4}+\frac{1}{30} x \left (1+3 x^2\right ) \sqrt{1+2 x^2+2 x^4}-\frac{1}{60} \int \frac{2-4 x^2}{\sqrt{1+2 x^2+2 x^4}} \, dx-\frac{1}{4} \int \frac{2+2 x^2}{\sqrt{1+2 x^2+2 x^4}} \, dx-\frac{9}{16} \int \frac{2-4 x^2}{\sqrt{1+2 x^2+2 x^4}} \, dx+\frac{45}{8} \int \frac{1}{\left (3+2 x^2\right ) \sqrt{1+2 x^2+2 x^4}} \, dx\\ &=-\frac{1}{4} x \sqrt{1+2 x^2+2 x^4}+\frac{1}{30} x \left (1+3 x^2\right ) \sqrt{1+2 x^2+2 x^4}-\frac{\int \frac{1-\sqrt{2} x^2}{\sqrt{1+2 x^2+2 x^4}} \, dx}{15 \sqrt{2}}+\frac{\int \frac{1-\sqrt{2} x^2}{\sqrt{1+2 x^2+2 x^4}} \, dx}{2 \sqrt{2}}-\frac{9 \int \frac{1-\sqrt{2} x^2}{\sqrt{1+2 x^2+2 x^4}} \, dx}{4 \sqrt{2}}-\frac{1}{30} \left (1-\sqrt{2}\right ) \int \frac{1}{\sqrt{1+2 x^2+2 x^4}} \, dx-\frac{1}{8} \left (9 \left (1-\sqrt{2}\right )\right ) \int \frac{1}{\sqrt{1+2 x^2+2 x^4}} \, dx-\frac{1}{4} \left (2+\sqrt{2}\right ) \int \frac{1}{\sqrt{1+2 x^2+2 x^4}} \, dx+\frac{1}{56} \left (45 \left (3+\sqrt{2}\right )\right ) \int \frac{1}{\sqrt{1+2 x^2+2 x^4}} \, dx-\frac{1}{56} \left (45 \left (2+3 \sqrt{2}\right )\right ) \int \frac{1+\sqrt{2} x^2}{\left (3+2 x^2\right ) \sqrt{1+2 x^2+2 x^4}} \, dx\\ &=-\frac{1}{4} x \sqrt{1+2 x^2+2 x^4}+\frac{1}{30} x \left (1+3 x^2\right ) \sqrt{1+2 x^2+2 x^4}+\frac{109 x \sqrt{1+2 x^2+2 x^4}}{60 \sqrt{2} \left (1+\sqrt{2} x^2\right )}+\frac{3}{16} \sqrt{15} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{1+2 x^2+2 x^4}}\right )-\frac{109 \left (1+\sqrt{2} x^2\right ) \sqrt{\frac{1+2 x^2+2 x^4}{\left (1+\sqrt{2} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{60\ 2^{3/4} \sqrt{1+2 x^2+2 x^4}}-\frac{139 \left (1-\sqrt{2}\right ) \left (1+\sqrt{2} x^2\right ) \sqrt{\frac{1+2 x^2+2 x^4}{\left (1+\sqrt{2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{240 \sqrt [4]{2} \sqrt{1+2 x^2+2 x^4}}-\frac{\left (1+\sqrt{2}\right ) \left (1+\sqrt{2} x^2\right ) \sqrt{\frac{1+2 x^2+2 x^4}{\left (1+\sqrt{2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{4\ 2^{3/4} \sqrt{1+2 x^2+2 x^4}}+\frac{45 \left (3+\sqrt{2}\right ) \left (1+\sqrt{2} x^2\right ) \sqrt{\frac{1+2 x^2+2 x^4}{\left (1+\sqrt{2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{112 \sqrt [4]{2} \sqrt{1+2 x^2+2 x^4}}-\frac{15 \left (3+\sqrt{2}\right )^2 \left (1+\sqrt{2} x^2\right ) \sqrt{\frac{1+2 x^2+2 x^4}{\left (1+\sqrt{2} x^2\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{224 \sqrt [4]{2} \sqrt{1+2 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.2605, size = 209, normalized size = 0.49 \[ \frac{-(199-417 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{1-i} x\right ),i\right )+48 x^7-56 x^5-80 x^3-218 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+225 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left (\frac{1}{3}+\frac{i}{3};\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )-52 x}{240 \sqrt{2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.069, size = 528, normalized size = 1.3 \begin{align*}{\frac{{x}^{3}}{10}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}-{\frac{13\,x}{60}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}-{\frac{8\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{15\,\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{ \left ({\frac{13}{60}}-{\frac{13\,i}{60}} \right ) \left ({\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) -{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) \right ) }{\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{9\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{4\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{{\frac{9\,i}{8}}{\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{9\,{\it EllipticE} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{8\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{{\frac{9\,i}{8}}{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{15}{8\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\it EllipticPi} \left ( x\sqrt{-1+i},{\frac{1}{3}}+{\frac{i}{3}},{\frac{\sqrt{-1-i}}{\sqrt{-1+i}}} \right ){\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}} \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{2 \, x^{4} + 2 \, x^{2} + 1} x^{4}}{2 \, x^{2} + 3}\,{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{2 \, x^{4} + 2 \, x^{2} + 1} x^{4}}{2 \, 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 \frac{x^{4} \sqrt{2 x^{4} + 2 x^{2} + 1}}{2 x^{2} + 3}\, 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{2 \, x^{4} + 2 \, x^{2} + 1} x^{4}}{2 \, x^{2} + 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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