Integrand size = 25, antiderivative size = 286 \[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=-\frac {11 x \left (5+\sqrt {13}+2 x^2\right )}{26 \sqrt {3+5 x^2+x^4}}+\frac {x \left (8+11 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {11 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{26 \sqrt {3+5 x^2+x^4}}-\frac {4 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{13 \sqrt {3+5 x^2+x^4}} \]
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Time = 0.07 (sec) , antiderivative size = 286, 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 = {1289, 1203, 1113, 1149} \[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=-\frac {4 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{13 \sqrt {x^4+5 x^2+3}}+\frac {11 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{26 \sqrt {x^4+5 x^2+3}}-\frac {11 x \left (2 x^2+\sqrt {13}+5\right )}{26 \sqrt {x^4+5 x^2+3}}+\frac {x \left (11 x^2+8\right )}{13 \sqrt {x^4+5 x^2+3}} \]
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Rule 1113
Rule 1149
Rule 1203
Rule 1289
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (8+11 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {1}{13} \int \frac {-8-11 x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {x \left (8+11 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}-\frac {8}{13} \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx-\frac {11}{13} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {11 x \left (5+\sqrt {13}+2 x^2\right )}{26 \sqrt {3+5 x^2+x^4}}+\frac {x \left (8+11 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {11 \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 )}{26 \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 )}{13 \sqrt {3+5 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.25 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {4 x \left (8+11 x^2\right )-11 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+i \sqrt {2} \left (-39+11 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{52 \sqrt {3+5 x^2+x^4}} \]
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Time = 2.62 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {x \left (11 x^{2}+8\right )}{13 \sqrt {x^{4}+5 x^{2}+3}}-\frac {48 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {396 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(216\) |
elliptic | \(-\frac {2 \left (-\frac {11}{26} x^{3}-\frac {4}{13} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {48 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {396 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(217\) |
default | \(-\frac {6 \left (-\frac {5}{26} x^{3}-\frac {3}{13} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {48 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {396 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {4 \left (\frac {1}{13} x^{3}+\frac {5}{26} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}\) | \(240\) |
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Time = 0.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {11 \, {\left (\sqrt {13} \sqrt {6} \sqrt {3} {\left (x^{4} + 5 \, x^{2} + 3\right )} - 5 \, \sqrt {6} \sqrt {3} {\left (x^{4} + 5 \, x^{2} + 3\right )}\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (3 \, \sqrt {13} \sqrt {6} \sqrt {3} {\left (x^{4} + 5 \, x^{2} + 3\right )} - 95 \, \sqrt {6} \sqrt {3} {\left (x^{4} + 5 \, x^{2} + 3\right )}\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {13} - 5}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 36 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (11 \, x^{3} + 8 \, x\right )}}{468 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}} \]
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\[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {x^{2} \cdot \left (3 x^{2} + 2\right )}{\left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (3 \, x^{2} + 2\right )} x^{2}}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (3 \, x^{2} + 2\right )} x^{2}}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {x^2\,\left (3\,x^2+2\right )}{{\left (x^4+5\,x^2+3\right )}^{3/2}} \,d x \]
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