Optimal. Leaf size=553 \[ \frac {262 \sqrt {2} \sqrt {2 x^4+2 x^2+1} x}{135 \left (\sqrt {2} x^2+1\right )}-\frac {262 \sqrt {2 x^4+2 x^2+1}}{135 x}+\frac {17}{27} \sqrt {\frac {17}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )+\frac {2^{3/4} \left (37+23 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{135 \sqrt {2 x^4+2 x^2+1}}+\frac {85\ 2^{3/4} \left (3-\sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{189 \sqrt {2 x^4+2 x^2+1}}-\frac {262 \sqrt [4]{2} \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{135 \sqrt {2 x^4+2 x^2+1}}-\frac {289 \left (11-6 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \Pi \left (\frac {1}{24} \left (12+11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{1134 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {\left (40 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}{45 x^5}+\frac {74 \sqrt {2 x^4+2 x^2+1}}{135 x^3} \]
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Rubi [A] time = 0.49, antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1309, 1271, 1281, 1197, 1103, 1195, 1311, 1216, 1706} \[ \frac {262 \sqrt {2} \sqrt {2 x^4+2 x^2+1} x}{135 \left (\sqrt {2} x^2+1\right )}-\frac {262 \sqrt {2 x^4+2 x^2+1}}{135 x}+\frac {74 \sqrt {2 x^4+2 x^2+1}}{135 x^3}-\frac {\left (40 x^2+3\right ) \sqrt {2 x^4+2 x^2+1}}{45 x^5}+\frac {17}{27} \sqrt {\frac {17}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )+\frac {2^{3/4} \left (37+23 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{135 \sqrt {2 x^4+2 x^2+1}}+\frac {85\ 2^{3/4} \left (3-\sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{189 \sqrt {2 x^4+2 x^2+1}}-\frac {262 \sqrt [4]{2} \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{135 \sqrt {2 x^4+2 x^2+1}}-\frac {289 \left (11-6 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \Pi \left (\frac {1}{24} \left (12+11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{1134 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1195
Rule 1197
Rule 1216
Rule 1271
Rule 1281
Rule 1309
Rule 1311
Rule 1706
Rubi steps
\begin {align*} \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{x^6 \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^6} \, dx+\frac {34}{9} \int \frac {\sqrt {1+2 x^2+2 x^4}}{x^2 \left (3-2 x^2\right )} \, dx\\ &=-\frac {\left (3+40 x^2\right ) \sqrt {1+2 x^2+2 x^4}}{45 x^5}+\frac {1}{45} \int \frac {-74-68 x^2}{x^4 \sqrt {1+2 x^2+2 x^4}} \, dx-\frac {17}{27} \int \frac {-2+6 x^2}{x^2 \sqrt {1+2 x^2+2 x^4}} \, dx+\frac {578}{27} \int \frac {1}{\left (3-2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx\\ &=\frac {74 \sqrt {1+2 x^2+2 x^4}}{135 x^3}-\frac {34 \sqrt {1+2 x^2+2 x^4}}{27 x}-\frac {\left (3+40 x^2\right ) \sqrt {1+2 x^2+2 x^4}}{45 x^5}-\frac {1}{135} \int \frac {-92-148 x^2}{x^2 \sqrt {1+2 x^2+2 x^4}} \, dx+\frac {17}{27} \int \frac {-6+4 x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{189} \left (578 \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}{189} \left (578 \left (3-\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx\\ &=\frac {74 \sqrt {1+2 x^2+2 x^4}}{135 x^3}-\frac {262 \sqrt {1+2 x^2+2 x^4}}{135 x}-\frac {\left (3+40 x^2\right ) \sqrt {1+2 x^2+2 x^4}}{45 x^5}+\frac {17}{27} \sqrt {\frac {17}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )+\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 )}{189 \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 )}{1134 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}+\frac {1}{135} \int \frac {148+184 x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{27} \left (34 \sqrt {2}\right ) \int \frac {1-\sqrt {2} x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{27} \left (34 \left (3-\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx\\ &=\frac {74 \sqrt {1+2 x^2+2 x^4}}{135 x^3}-\frac {262 \sqrt {1+2 x^2+2 x^4}}{135 x}-\frac {\left (3+40 x^2\right ) \sqrt {1+2 x^2+2 x^4}}{45 x^5}+\frac {34 \sqrt {2} x \sqrt {1+2 x^2+2 x^4}}{27 \left (1+\sqrt {2} x^2\right )}+\frac {17}{27} \sqrt {\frac {17}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )-\frac {34 \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 )}{27 \sqrt {1+2 x^2+2 x^4}}+\frac {85\ 2^{3/4} \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 )}{189 \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 )}{1134 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}-\frac {1}{135} \left (92 \sqrt {2}\right ) \int \frac {1-\sqrt {2} x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx+\frac {1}{135} \left (4 \left (37+23 \sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx\\ &=\frac {74 \sqrt {1+2 x^2+2 x^4}}{135 x^3}-\frac {262 \sqrt {1+2 x^2+2 x^4}}{135 x}-\frac {\left (3+40 x^2\right ) \sqrt {1+2 x^2+2 x^4}}{45 x^5}+\frac {262 \sqrt {2} x \sqrt {1+2 x^2+2 x^4}}{135 \left (1+\sqrt {2} x^2\right )}+\frac {17}{27} \sqrt {\frac {17}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )-\frac {262 \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 )}{135 \sqrt {1+2 x^2+2 x^4}}+\frac {85\ 2^{3/4} \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 )}{189 \sqrt {1+2 x^2+2 x^4}}+\frac {2^{3/4} \left (37+23 \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 )}{135 \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 )}{1134 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 224, normalized size = 0.41 \[ -\frac {1572 x^8+1848 x^6+1116 x^4+192 x^2+(543-1329 i) \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} x^5 F\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )+786 i \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} x^5 E\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )-1445 (1-i)^{3/2} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} x^5 \Pi \left (-\frac {1}{3}-\frac {i}{3};\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )+27}{405 x^5 \sqrt {2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, x^{8} - 3 \, x^{6}}, x\right ) \]
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 \[ \int -\frac {{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (2 \, x^{2} - 3\right )} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 549, normalized size = 0.99 \[ \frac {206 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{135 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {206 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{135 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {236 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{45 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {206 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{135 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {184 \sqrt {\left (1-i\right ) x^{2}+1}\, \sqrt {\left (1+i\right ) x^{2}+1}\, \EllipticF \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{45 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {578 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (\sqrt {-1+i}\, x , -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{81 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {262 \sqrt {2 x^{4}+2 x^{2}+1}}{135 x}-\frac {46 \sqrt {2 x^{4}+2 x^{2}+1}}{135 x^{3}}-\frac {\sqrt {2 x^{4}+2 x^{2}+1}}{15 x^{5}}+\frac {\left (-\frac {52}{15}+\frac {52 i}{15}\right ) \sqrt {\left (1-i\right ) x^{2}+1}\, \sqrt {\left (1+i\right ) x^{2}+1}\, \left (-\EllipticE \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}} \]
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 \[ -\int \frac {{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (2 \, x^{2} - 3\right )} x^{6}}\,{d x} \]
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 \[ -\int \frac {{\left (2\,x^4+2\,x^2+1\right )}^{3/2}}{x^6\,\left (2\,x^2-3\right )} \,d x \]
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 \[ - \int \frac {\sqrt {2 x^{4} + 2 x^{2} + 1}}{2 x^{8} - 3 x^{6}}\, dx - \int \frac {2 x^{2} \sqrt {2 x^{4} + 2 x^{2} + 1}}{2 x^{8} - 3 x^{6}}\, dx - \int \frac {2 x^{4} \sqrt {2 x^{4} + 2 x^{2} + 1}}{2 x^{8} - 3 x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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