Optimal. Leaf size=625 \[ -\frac {2 \sqrt {1+2 x^2+2 x^4}}{x}-\frac {\left (1-8 x^2\right ) \sqrt {1+2 x^2+2 x^4}}{9 x^3}+\frac {\sqrt {2} x \sqrt {1+2 x^2+2 x^4}}{9 \left (1+\sqrt {2} x^2\right )}+\frac {17}{18} \sqrt {\frac {17}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )-\frac {\sqrt [4]{2} \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{9 \sqrt {1+2 x^2+2 x^4}}+\frac {289 \left (3-\sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{126 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}-\frac {17 \left (5+\sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{18 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}+\frac {\sqrt [4]{2} \left (9+5 \sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{9 \sqrt {1+2 x^2+2 x^4}}-\frac {289 \left (11-6 \sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \Pi \left (\frac {1}{24} \left (12+11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{756 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}} \]
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Rubi [A]
time = 0.23, antiderivative size = 625, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1323, 1285,
1295, 1211, 1117, 1209, 1222, 1230, 1720} \begin {gather*} \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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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}}+\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 {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 {\left (1-8 x^2\right ) \sqrt {2 x^4+2 x^2+1}}{9 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 1117
Rule 1209
Rule 1211
Rule 1222
Rule 1230
Rule 1285
Rule 1295
Rule 1323
Rule 1720
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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.17, size = 219, normalized size = 0.35 \begin {gather*} \frac {-6-72 x^2-132 x^4-120 x^6-6 i \sqrt {1-i} x^3 \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 )-(195-201 i) \sqrt {1-i} x^3 \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 )+289 (1-i)^{3/2} x^3 \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 )}{54 x^3 \sqrt {1+2 x^2+2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.15, size = 530, normalized size = 0.85
method | result | size |
risch | \(-\frac {20 x^{6}+22 x^{4}+12 x^{2}+1}{9 x^{3} \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\left (-\frac {1}{9}+\frac {i}{9}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (\EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-\EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {65 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{9 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{27 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\) | \(258\) |
elliptic | \(-\frac {\sqrt {2 x^{4}+2 x^{2}+1}}{9 x^{3}}-\frac {10 \sqrt {2 x^{4}+2 x^{2}+1}}{9 x}-\frac {22 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{3 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{9 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{9 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{9 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{27 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\) | \(379\) |
default | \(-\frac {10 \sqrt {2 x^{4}+2 x^{2}+1}}{9 x}+\frac {44 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{15 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {103 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{45 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {\sqrt {2 x^{4}+2 x^{2}+1}}{9 x^{3}}+\frac {\left (-\frac {12}{5}+\frac {12 i}{5}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (\EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-\EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {118 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticF \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{15 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {103 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{45 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {103 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticE \left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{45 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {289 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{27 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\) | \(530\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\int \frac {{\left (2\,x^4+2\,x^2+1\right )}^{3/2}}{x^4\,\left (2\,x^2-3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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