Optimal. Leaf size=625 \[ \frac{\sqrt [4]{2} \left (9+5 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right ),\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt{2 x^4+2 x^2+1}}-\frac{17 \left (5+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right ),\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{18 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}+\frac{289 \left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right ),\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{126 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}+\frac{\sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{9 \left (\sqrt{2} x^2+1\right )}-\frac{2 \sqrt{2 x^4+2 x^2+1}}{x}-\frac{\left (1-8 x^2\right ) \sqrt{2 x^4+2 x^2+1}}{9 x^3}+\frac{17}{18} \sqrt{\frac{17}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{\sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left (11-6 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12+11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{756 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}} \]
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Rubi [A] time = 0.366185, antiderivative size = 625, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {1309, 1271, 1281, 1197, 1103, 1195, 1208, 1216, 1706} \[ \frac{\sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{9 \left (\sqrt{2} x^2+1\right )}-\frac{2 \sqrt{2 x^4+2 x^2+1}}{x}-\frac{\left (1-8 x^2\right ) \sqrt{2 x^4+2 x^2+1}}{9 x^3}+\frac{17}{18} \sqrt{\frac{17}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )+\frac{\sqrt [4]{2} \left (9+5 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt{2 x^4+2 x^2+1}}-\frac{17 \left (5+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{18 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}+\frac{289 \left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{126 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left (11-6 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12+11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{756 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1309
Rule 1271
Rule 1281
Rule 1197
Rule 1103
Rule 1195
Rule 1208
Rule 1216
Rule 1706
Rubi steps
\begin{align*} \int \frac{\left (1+2 x^2+2 x^4\right )^{3/2}}{x^4 \left (3-2 x^2\right )} \, dx &=\frac{1}{9} \int \frac{\left (3+8 x^2\right ) \sqrt{1+2 x^2+2 x^4}}{x^4} \, dx+\frac{34}{9} \int \frac{\sqrt{1+2 x^2+2 x^4}}{3-2 x^2} \, dx\\ &=-\frac{\left (1-8 x^2\right ) \sqrt{1+2 x^2+2 x^4}}{9 x^3}-\frac{1}{27} \int \frac{-54-60 x^2}{x^2 \sqrt{1+2 x^2+2 x^4}} \, dx-\frac{17}{18} \int \frac{10+4 x^2}{\sqrt{1+2 x^2+2 x^4}} \, dx+\frac{289}{9} \int \frac{1}{\left (3-2 x^2\right ) \sqrt{1+2 x^2+2 x^4}} \, dx\\ &=-\frac{2 \sqrt{1+2 x^2+2 x^4}}{x}-\frac{\left (1-8 x^2\right ) \sqrt{1+2 x^2+2 x^4}}{9 x^3}+\frac{1}{27} \int \frac{60+108 x^2}{\sqrt{1+2 x^2+2 x^4}} \, dx+\frac{1}{9} \left (17 \sqrt{2}\right ) \int \frac{1-\sqrt{2} x^2}{\sqrt{1+2 x^2+2 x^4}} \, dx-\frac{1}{63} \left (289 \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}{63} \left (289 \left (3-\sqrt{2}\right )\right ) \int \frac{1}{\sqrt{1+2 x^2+2 x^4}} \, dx-\frac{1}{9} \left (17 \left (5+\sqrt{2}\right )\right ) \int \frac{1}{\sqrt{1+2 x^2+2 x^4}} \, dx\\ &=-\frac{2 \sqrt{1+2 x^2+2 x^4}}{x}-\frac{\left (1-8 x^2\right ) \sqrt{1+2 x^2+2 x^4}}{9 x^3}-\frac{17 \sqrt{2} x \sqrt{1+2 x^2+2 x^4}}{9 \left (1+\sqrt{2} x^2\right )}+\frac{17}{18} \sqrt{\frac{17}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{17}{3}} x}{\sqrt{1+2 x^2+2 x^4}}\right )+\frac{17 \sqrt [4]{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}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt{1+2 x^2+2 x^4}}+\frac{289 \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 )}{126 \sqrt [4]{2} \sqrt{1+2 x^2+2 x^4}}-\frac{17 \left (5+\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 )}{18 \sqrt [4]{2} \sqrt{1+2 x^2+2 x^4}}-\frac{289 \left (11-6 \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}} \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 )}{756 \sqrt [4]{2} \sqrt{1+2 x^2+2 x^4}}-\left (2 \sqrt{2}\right ) \int \frac{1-\sqrt{2} x^2}{\sqrt{1+2 x^2+2 x^4}} \, dx+\frac{1}{9} \left (2 \left (10+9 \sqrt{2}\right )\right ) \int \frac{1}{\sqrt{1+2 x^2+2 x^4}} \, dx\\ &=-\frac{2 \sqrt{1+2 x^2+2 x^4}}{x}-\frac{\left (1-8 x^2\right ) \sqrt{1+2 x^2+2 x^4}}{9 x^3}+\frac{\sqrt{2} x \sqrt{1+2 x^2+2 x^4}}{9 \left (1+\sqrt{2} x^2\right )}+\frac{17}{18} \sqrt{\frac{17}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{17}{3}} x}{\sqrt{1+2 x^2+2 x^4}}\right )-\frac{\sqrt [4]{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}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt{1+2 x^2+2 x^4}}+\frac{289 \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 )}{126 \sqrt [4]{2} \sqrt{1+2 x^2+2 x^4}}-\frac{17 \left (5+\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 )}{18 \sqrt [4]{2} \sqrt{1+2 x^2+2 x^4}}+\frac{\sqrt [4]{2} \left (9+5 \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 )}{9 \sqrt{1+2 x^2+2 x^4}}-\frac{289 \left (11-6 \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}} \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 )}{756 \sqrt [4]{2} \sqrt{1+2 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.234868, size = 219, normalized size = 0.35 \[ \frac{-(195-201 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{1-i} x\right ),i\right )-120 x^6-132 x^4-72 x^2-6 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 E\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+289 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 \Pi \left (-\frac{1}{3}-\frac{i}{3};\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )-6}{54 x^3 \sqrt{2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 530, normalized size = 0.9 \begin{align*} -{\frac{118\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{15\,\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{ \left ({\frac{12}{5}}-{\frac{12\,i}{5}} \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{103\,{\it EllipticE} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{45\,\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{103\,i}{45}}{\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{289}{27\,\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}}}}-{\frac{10}{9\,x}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}+{\frac{44\,{\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{{\frac{103\,i}{45}}{\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{1}{9\,{x}^{3}}\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{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (2 \, x^{2} - 3\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{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac{3}{2}}}{2 \, x^{6} - 3 \, 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{\sqrt{2 x^{4} + 2 x^{2} + 1}}{2 x^{6} - 3 x^{4}}\, dx - \int \frac{2 x^{2} \sqrt{2 x^{4} + 2 x^{2} + 1}}{2 x^{6} - 3 x^{4}}\, dx - \int \frac{2 x^{4} \sqrt{2 x^{4} + 2 x^{2} + 1}}{2 x^{6} - 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{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (2 \, x^{2} - 3\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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