Integrand size = 25, antiderivative size = 309 \[ \int \frac {2+3 x^2}{x^2 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {19 x \left (5+\sqrt {13}+2 x^2\right )}{234 \sqrt {3+5 x^2+x^4}}-\frac {7+8 x^2}{39 x \sqrt {3+5 x^2+x^4}}-\frac {19 \sqrt {3+5 x^2+x^4}}{117 x}-\frac {19 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{234 \sqrt {3+5 x^2+x^4}}-\frac {4 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{39 \sqrt {3+5 x^2+x^4}} \]
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Time = 0.08 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1291, 1295, 1203, 1113, 1149} \[ \int \frac {2+3 x^2}{x^2 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=-\frac {4 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{39 \sqrt {x^4+5 x^2+3}}-\frac {19 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{234 \sqrt {x^4+5 x^2+3}}+\frac {19 x \left (2 x^2+\sqrt {13}+5\right )}{234 \sqrt {x^4+5 x^2+3}}-\frac {19 \sqrt {x^4+5 x^2+3}}{117 x}-\frac {8 x^2+7}{39 x \sqrt {x^4+5 x^2+3}} \]
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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 \sqrt {3+5 x^2+x^4}}-\frac {1}{39} \int \frac {-19+8 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {7+8 x^2}{39 x \sqrt {3+5 x^2+x^4}}-\frac {19 \sqrt {3+5 x^2+x^4}}{117 x}+\frac {1}{117} \int \frac {-24+19 x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {7+8 x^2}{39 x \sqrt {3+5 x^2+x^4}}-\frac {19 \sqrt {3+5 x^2+x^4}}{117 x}+\frac {19}{117} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx-\frac {8}{39} \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {19 x \left (5+\sqrt {13}+2 x^2\right )}{234 \sqrt {3+5 x^2+x^4}}-\frac {7+8 x^2}{39 x \sqrt {3+5 x^2+x^4}}-\frac {19 \sqrt {3+5 x^2+x^4}}{117 x}-\frac {19 \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 )}{234 \sqrt {3+5 x^2+x^4}}-\frac {4 \sqrt {\frac {2}{3 \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 ) 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 )}{39 \sqrt {3+5 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.26 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.74 \[ \int \frac {2+3 x^2}{x^2 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {-4 \left (78+119 x^2+19 x^4\right )+19 i \sqrt {2} \left (-5+\sqrt {13}\right ) x \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-143+19 \sqrt {13}\right ) x \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{468 x \sqrt {3+5 x^2+x^4}} \]
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Time = 1.51 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {19 x^{4}+119 x^{2}+78}{117 x \sqrt {x^{4}+5 x^{2}+3}}-\frac {16 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {76 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(223\) |
elliptic | \(-\frac {2 \left (-\frac {7}{234} x^{3}-\frac {11}{234} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{9 x}-\frac {16 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {76 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(234\) |
default | \(-\frac {6 \left (-\frac {19}{78} x -\frac {5}{78} x^{3}\right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {16 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {76 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {4 \left (\frac {19}{234} x^{3}+\frac {40}{117} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{9 x}\) | \(257\) |
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Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.62 \[ \int \frac {2+3 x^2}{x^2 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=-\frac {19 \, {\left (\sqrt {13} \sqrt {6} \sqrt {3} {\left (x^{5} + 5 \, x^{3} + 3 \, x\right )} - 5 \, \sqrt {6} \sqrt {3} {\left (x^{5} + 5 \, x^{3} + 3 \, x\right )}\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (43 \, \sqrt {13} \sqrt {6} \sqrt {3} {\left (x^{5} + 5 \, x^{3} + 3 \, x\right )} + 25 \, \sqrt {6} \sqrt {3} {\left (x^{5} + 5 \, x^{3} + 3 \, x\right )}\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 36 \, {\left (19 \, x^{4} + 119 \, x^{2} + 78\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{4212 \, {\left (x^{5} + 5 \, x^{3} + 3 \, x\right )}} \]
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\[ \int \frac {2+3 x^2}{x^2 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {3 x^{2} + 2}{x^{2} \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {2+3 x^2}{x^2 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {3 \, x^{2} + 2}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
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\[ \int \frac {2+3 x^2}{x^2 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {3 \, x^{2} + 2}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {2+3 x^2}{x^2 \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {3\,x^2+2}{x^2\,{\left (x^4+5\,x^2+3\right )}^{3/2}} \,d x \]
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