Integrand size = 25, antiderivative size = 278 \[ \int \frac {2+3 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx=\frac {x \left (5+\sqrt {13}+2 x^2\right )}{3 \sqrt {3+5 x^2+x^4}}-\frac {2 \sqrt {3+5 x^2+x^4}}{3 x}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{3 \sqrt {3+5 x^2+x^4}}+\frac {\sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {3+5 x^2+x^4}} \]
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Time = 0.06 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1295, 1203, 1113, 1149} \[ \int \frac {2+3 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx=\frac {\sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{3 \sqrt {x^4+5 x^2+3}}+\frac {x \left (2 x^2+\sqrt {13}+5\right )}{3 \sqrt {x^4+5 x^2+3}}-\frac {2 \sqrt {x^4+5 x^2+3}}{3 x} \]
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Rule 1113
Rule 1149
Rule 1203
Rule 1295
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {3+5 x^2+x^4}}{3 x}-\frac {1}{3} \int \frac {-9-2 x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {2 \sqrt {3+5 x^2+x^4}}{3 x}+\frac {2}{3} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx+3 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {x \left (5+\sqrt {13}+2 x^2\right )}{3 \sqrt {3+5 x^2+x^4}}-\frac {2 \sqrt {3+5 x^2+x^4}}{3 x}-\frac {\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 )}{3 \sqrt {3+5 x^2+x^4}}+\frac {\sqrt {\frac {3}{2 \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 )}{\sqrt {3+5 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.23 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.81 \[ \int \frac {2+3 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx=\frac {-4 \left (3+5 x^2+x^4\right )+i \sqrt {2} \left (-5+\sqrt {13}\right ) x \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (4+\sqrt {13}\right ) x \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{6 x \sqrt {3+5 x^2+x^4}} \]
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Time = 1.40 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {18 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{3 x}-\frac {24 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(211\) |
risch | \(\frac {18 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{3 x}-\frac {24 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(211\) |
elliptic | \(\frac {18 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{3 x}-\frac {24 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(211\) |
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Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.45 \[ \int \frac {2+3 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx=-\frac {2 \, {\left (\sqrt {13} \sqrt {6} \sqrt {3} x - 5 \, \sqrt {6} \sqrt {3} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + {\left (7 \, \sqrt {13} \sqrt {6} \sqrt {3} x + 55 \, \sqrt {6} \sqrt {3} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 72 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{108 \, x} \]
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\[ \int \frac {2+3 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx=\int \frac {3 x^{2} + 2}{x^{2} \sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
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\[ \int \frac {2+3 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3} x^{2}} \,d x } \]
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\[ \int \frac {2+3 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + 5 \, x^{2} + 3} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {2+3 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx=\int \frac {3\,x^2+2}{x^2\,\sqrt {x^4+5\,x^2+3}} \,d x \]
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