Integrand size = 25, antiderivative size = 326 \[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=-\frac {133 x \left (5+\sqrt {13}+2 x^2\right )}{1053 \sqrt {3+5 x^2+x^4}}-\frac {7+8 x^2}{39 x^3 \sqrt {3+5 x^2+x^4}}-\frac {5 \sqrt {3+5 x^2+x^4}}{351 x^3}+\frac {266 \sqrt {3+5 x^2+x^4}}{1053 x}+\frac {133 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{1053 \sqrt {3+5 x^2+x^4}}-\frac {5 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{351 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]
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Time = 0.11 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1291, 1295, 1203, 1113, 1149} \[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=-\frac {5 \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{351 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {x^4+5 x^2+3}}+\frac {133 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{1053 \sqrt {x^4+5 x^2+3}}-\frac {133 x \left (2 x^2+\sqrt {13}+5\right )}{1053 \sqrt {x^4+5 x^2+3}}+\frac {266 \sqrt {x^4+5 x^2+3}}{1053 x}-\frac {5 \sqrt {x^4+5 x^2+3}}{351 x^3}-\frac {8 x^2+7}{39 x^3 \sqrt {x^4+5 x^2+3}} \]
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Rule 1113
Rule 1149
Rule 1203
Rule 1291
Rule 1295
Rubi steps \begin{align*} \text {integral}& = -\frac {7+8 x^2}{39 x^3 \sqrt {3+5 x^2+x^4}}-\frac {1}{39} \int \frac {-5+24 x^2}{x^4 \sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {7+8 x^2}{39 x^3 \sqrt {3+5 x^2+x^4}}-\frac {5 \sqrt {3+5 x^2+x^4}}{351 x^3}+\frac {1}{351} \int \frac {-266-5 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {7+8 x^2}{39 x^3 \sqrt {3+5 x^2+x^4}}-\frac {5 \sqrt {3+5 x^2+x^4}}{351 x^3}+\frac {266 \sqrt {3+5 x^2+x^4}}{1053 x}-\frac {\int \frac {15+266 x^2}{\sqrt {3+5 x^2+x^4}} \, dx}{1053} \\ & = -\frac {7+8 x^2}{39 x^3 \sqrt {3+5 x^2+x^4}}-\frac {5 \sqrt {3+5 x^2+x^4}}{351 x^3}+\frac {266 \sqrt {3+5 x^2+x^4}}{1053 x}-\frac {5}{351} \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx-\frac {266 \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx}{1053} \\ & = -\frac {133 x \left (5+\sqrt {13}+2 x^2\right )}{1053 \sqrt {3+5 x^2+x^4}}-\frac {7+8 x^2}{39 x^3 \sqrt {3+5 x^2+x^4}}-\frac {5 \sqrt {3+5 x^2+x^4}}{351 x^3}+\frac {266 \sqrt {3+5 x^2+x^4}}{1053 x}+\frac {133 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{1053 \sqrt {3+5 x^2+x^4}}-\frac {5 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{351 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.72 \[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {-468+1014 x^2+2630 x^4+532 x^6-133 i \sqrt {2} \left (-5+\sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+i \sqrt {2} \left (-650+133 \sqrt {13}\right ) x^3 \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{2106 x^3 \sqrt {3+5 x^2+x^4}} \]
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Time = 2.98 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.70
method | result | size |
risch | \(\frac {266 x^{6}+1315 x^{4}+507 x^{2}-234}{1053 x^{3} \sqrt {x^{4}+5 x^{2}+3}}-\frac {10 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {1064 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(228\) |
elliptic | \(-\frac {2 \left (\frac {11}{702} x^{3}+\frac {17}{351} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{27 x^{3}}+\frac {23 \sqrt {x^{4}+5 x^{2}+3}}{81 x}-\frac {10 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {1064 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(251\) |
default | \(-\frac {6 \left (\frac {19}{234} x^{3}+\frac {40}{117} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}+\frac {23 \sqrt {x^{4}+5 x^{2}+3}}{81 x}-\frac {10 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {1064 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{117 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {4 \left (-\frac {40}{351} x^{3}-\frac {343}{702} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{27 x^{3}}\) | \(274\) |
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Time = 0.08 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.63 \[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {266 \, {\left (\sqrt {13} \sqrt {6} \sqrt {3} {\left (x^{7} + 5 \, x^{5} + 3 \, x^{3}\right )} - 5 \, \sqrt {6} \sqrt {3} {\left (x^{7} + 5 \, x^{5} + 3 \, x^{3}\right )}\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (251 \, \sqrt {13} \sqrt {6} \sqrt {3} {\left (x^{7} + 5 \, x^{5} + 3 \, x^{3}\right )} - 1405 \, \sqrt {6} \sqrt {3} {\left (x^{7} + 5 \, x^{5} + 3 \, x^{3}\right )}\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 36 \, {\left (266 \, x^{6} + 1315 \, x^{4} + 507 \, x^{2} - 234\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{37908 \, {\left (x^{7} + 5 \, x^{5} + 3 \, x^{3}\right )}} \]
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\[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {3 x^{2} + 2}{x^{4} \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {3 \, x^{2} + 2}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
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\[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {3 \, x^{2} + 2}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {2+3 x^2}{x^4 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {3\,x^2+2}{x^4\,{\left (x^4+5\,x^2+3\right )}^{3/2}} \,d x \]
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