Optimal. Leaf size=154 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{3 x^4+2}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{3 x^4+2}}\right )+\frac {4 \sqrt [4]{3 x^4+2} \left (6 x^4-1\right )}{5 x^5}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{3 x^4+2}}{\sqrt {3 x^4+2}-x^2}\right )}{2 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{3 x^4+2}}{\sqrt {3 x^4+2}+x^2}\right )}{2 \sqrt {2}} \]
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Rubi [C] time = 0.97, antiderivative size = 157, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {6725, 264, 277, 331, 298, 203, 206, 510} \begin {gather*} \frac {10}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-x^4,-\frac {3 x^4}{2}\right )+\frac {6 \sqrt [4]{3 x^4+2}}{x}+3 \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{3 x^4+2}}\right )-3 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{3 x^4+2}}\right )-\frac {2 \left (3 x^4+2\right )^{5/4}}{5 x^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 264
Rule 277
Rule 298
Rule 331
Rule 510
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx &=\int \left (\frac {4 \sqrt [4]{2+3 x^4}}{x^6}-\frac {6 \sqrt [4]{2+3 x^4}}{x^2}+\frac {x^2 \sqrt [4]{2+3 x^4}}{1+x^4}+\frac {10 x^2 \sqrt [4]{2+3 x^4}}{1+2 x^4}\right ) \, dx\\ &=4 \int \frac {\sqrt [4]{2+3 x^4}}{x^6} \, dx-6 \int \frac {\sqrt [4]{2+3 x^4}}{x^2} \, dx+10 \int \frac {x^2 \sqrt [4]{2+3 x^4}}{1+2 x^4} \, dx+\int \frac {x^2 \sqrt [4]{2+3 x^4}}{1+x^4} \, dx\\ &=\frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-x^4,-\frac {3 x^4}{2}\right )-18 \int \frac {x^2}{\left (2+3 x^4\right )^{3/4}} \, dx\\ &=\frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-x^4,-\frac {3 x^4}{2}\right )-18 \operatorname {Subst}\left (\int \frac {x^2}{1-3 x^4} \, dx,x,\frac {x}{\sqrt [4]{2+3 x^4}}\right )\\ &=\frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-x^4,-\frac {3 x^4}{2}\right )-\left (3 \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+3 x^4}}\right )+\left (3 \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+3 x^4}}\right )\\ &=\frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-x^4,-\frac {3 x^4}{2}\right )+3 \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2+3 x^4}}\right )-3 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2+3 x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.21, size = 94, normalized size = 0.61 \begin {gather*} \frac {x^3 \left (5 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {x^4}{3 x^4+2}\right )-\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {x^4}{3 x^4+2}\right )\right )}{3 \left (3 x^4+2\right )^{3/4}}+\left (\frac {24}{5 x}-\frac {4}{5 x^5}\right ) \sqrt [4]{3 x^4+2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 154, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{3 x^4+2}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{3 x^4+2}}\right )+\frac {4 \sqrt [4]{3 x^4+2} \left (6 x^4-1\right )}{5 x^5}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{3 x^4+2}}{\sqrt {3 x^4+2}-x^2}\right )}{2 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{3 x^4+2}}{\sqrt {3 x^4+2}+x^2}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 5.78, size = 724, normalized size = 4.70 \begin {gather*} \frac {100 \, \sqrt {2} x^{5} \arctan \left (-\frac {4 \, x^{8} + 4 \, x^{4} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (4 \, x^{7} + 3 \, x^{3}\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (2 \, x^{6} + x^{2}\right )} \sqrt {3 \, x^{4} + 2} - {\left (4 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x^{5} + \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} - \sqrt {2} {\left (4 \, x^{8} + x^{4} - 1\right )} + 2 \, {\left (2 \, x^{7} + x^{3}\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {2 \, x^{4} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {3 \, x^{4} + 2} x^{2} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}} + 1}{8 \, x^{8} + 4 \, x^{4} - 1}\right ) - 100 \, \sqrt {2} x^{5} \arctan \left (-\frac {4 \, x^{8} + 4 \, x^{4} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (4 \, x^{7} + 3 \, x^{3}\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (2 \, x^{6} + x^{2}\right )} \sqrt {3 \, x^{4} + 2} - {\left (4 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x^{5} - \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} + \sqrt {2} {\left (4 \, x^{8} + x^{4} - 1\right )} + 2 \, {\left (2 \, x^{7} + x^{3}\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {2 \, x^{4} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {3 \, x^{4} + 2} x^{2} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}} + 1}{8 \, x^{8} + 4 \, x^{4} - 1}\right ) - 25 \, \sqrt {2} x^{5} \log \left (\frac {4 \, {\left (2 \, x^{4} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {3 \, x^{4} + 2} x^{2} + \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1\right )}}{2 \, x^{4} + 1}\right ) + 25 \, \sqrt {2} x^{5} \log \left (\frac {4 \, {\left (2 \, x^{4} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {3 \, x^{4} + 2} x^{2} - \sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1\right )}}{2 \, x^{4} + 1}\right ) + 20 \, x^{5} \arctan \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) + 20 \, x^{5} \log \left (-\frac {2 \, x^{4} - {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + \sqrt {3 \, x^{4} + 2} x^{2} - {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{x^{4} + 1}\right ) + 64 \, {\left (6 \, x^{4} - 1\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{80 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 221, normalized size = 1.44 \begin {gather*} -\frac {5}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \frac {5}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \frac {5}{8} \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {3 \, x^{4} + 2}}{x^{2}} + 1\right ) + \frac {5}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {3 \, x^{4} + 2}}{x^{2}} + 1\right ) - \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} {\left (\frac {2}{x^{4}} + 3\right )}}{5 \, x} + \frac {6 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} - \frac {1}{2} \, \arctan \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{4} \, \log \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.55, size = 987, normalized size = 6.41
result too large to display
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*} \int \frac {{\left (x^{8} + 6 \, x^{4} + 4\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (2 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )} x^{6}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {{\left (3\,x^4+2\right )}^{1/4}\,\left (x^8+6\,x^4+4\right )}{x^6\,\left (x^4+1\right )\,\left (2\,x^4+1\right )} \,d x \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 [4]{3 x^{4} + 2} \left (x^{8} + 6 x^{4} + 4\right )}{x^{6} \left (x^{4} + 1\right ) \left (2 x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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