Optimal. Leaf size=422 \[ \frac {3 \sqrt {2 x^4+2 x^2+1} x}{5 \sqrt {2} \left (\sqrt {2} x^2+1\right )}-\frac {\left (3 x^2+1\right ) x}{5 \sqrt {2 x^4+2 x^2+1}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )}{5 \sqrt {15}}+\frac {\left (2+\sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{2\ 2^{3/4} \left (3 \sqrt {2}-2\right ) \sqrt {2 x^4+2 x^2+1}}-\frac {3 \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{5\ 2^{3/4} \sqrt {2 x^4+2 x^2+1}}+\frac {\left (3+\sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \Pi \left (\frac {1}{24} \left (12-11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{15\ 2^{3/4} \left (2-3 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}} \]
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Rubi [A] time = 0.23, antiderivative size = 501, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1221, 1178, 1197, 1103, 1195, 1216, 1706} \[ \frac {3 \sqrt {2 x^4+2 x^2+1} x}{5 \sqrt {2} \left (\sqrt {2} x^2+1\right )}-\frac {\left (3 x^2+1\right ) x}{5 \sqrt {2 x^4+2 x^2+1}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )}{5 \sqrt {15}}+\frac {\left (3+2 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{10\ 2^{3/4} \sqrt {2 x^4+2 x^2+1}}+\frac {\left (3+\sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{35 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {3 \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{5\ 2^{3/4} \sqrt {2 x^4+2 x^2+1}}-\frac {\left (3+\sqrt {2}\right )^2 \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \Pi \left (\frac {1}{24} \left (12-11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{210 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1178
Rule 1195
Rule 1197
Rule 1216
Rule 1221
Rule 1706
Rubi steps
\begin {align*} \int \frac {1}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx &=\frac {1}{10} \int \frac {2-4 x^2}{\left (1+2 x^2+2 x^4\right )^{3/2}} \, dx+\frac {2}{5} \int \frac {1}{\left (3+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {x \left (1+3 x^2\right )}{5 \sqrt {1+2 x^2+2 x^4}}+\frac {1}{40} \int \frac {16+24 x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx+\frac {1}{35} \left (2 \left (3+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{35} \left (2 \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 {x \left (1+3 x^2\right )}{5 \sqrt {1+2 x^2+2 x^4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )}{5 \sqrt {15}}+\frac {\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 )}{35 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}-\frac {\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 )}{210 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}-\frac {3 \int \frac {1-\sqrt {2} x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx}{5 \sqrt {2}}+\frac {1}{10} \left (4+3 \sqrt {2}\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {x \left (1+3 x^2\right )}{5 \sqrt {1+2 x^2+2 x^4}}+\frac {3 x \sqrt {1+2 x^2+2 x^4}}{5 \sqrt {2} \left (1+\sqrt {2} x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )}{5 \sqrt {15}}-\frac {3 \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 )}{5\ 2^{3/4} \sqrt {1+2 x^2+2 x^4}}+\frac {\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 )}{35 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}+\frac {\left (3+2 \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 )}{10\ 2^{3/4} \sqrt {1+2 x^2+2 x^4}}-\frac {\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 )}{210 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 199, normalized size = 0.47 \[ \frac {-18 x^3+(6+3 i) \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} F\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )-9 i \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} E\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )+2 (1-i)^{3/2} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \Pi \left (\frac {1}{3}+\frac {i}{3};\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )-6 x}{30 \sqrt {2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2 \, x^{4} + 2 \, x^{2} + 1}}{8 \, x^{10} + 28 \, x^{8} + 40 \, x^{6} + 32 \, x^{4} + 14 \, x^{2} + 3}, x\right ) \]
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 \[ \int \frac {1}{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 3\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 366, normalized size = 0.87 \[ \frac {3 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {3 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {3 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{10 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {2 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (\sqrt {-1+i}\, x , \frac {1}{3}+\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{15 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {4 \left (\frac {3}{20} x^{3}+\frac {1}{20} x \right )}{\sqrt {2 x^{4}+2 x^{2}+1}} \]
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 \[ \int \frac {1}{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 3\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (2\,x^2+3\right )\,{\left (2\,x^4+2\,x^2+1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (2 x^{2} + 3\right ) \left (2 x^{4} + 2 x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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