Integrand size = 25, antiderivative size = 307 \[ \int \frac {x^4 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {43 x \left (5+\sqrt {13}+2 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {x^3 \left (8+11 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}-\frac {11}{13} x \sqrt {3+5 x^2+x^4}-\frac {43 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {11 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{13 \sqrt {3+5 x^2+x^4}} \]
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Time = 0.08 (sec) , antiderivative size = 307, 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 = {1289, 1293, 1203, 1113, 1149} \[ \int \frac {x^4 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {11 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{13 \sqrt {x^4+5 x^2+3}}-\frac {43 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{13 \sqrt {x^4+5 x^2+3}}-\frac {11}{13} \sqrt {x^4+5 x^2+3} x+\frac {43 \left (2 x^2+\sqrt {13}+5\right ) x}{13 \sqrt {x^4+5 x^2+3}}+\frac {\left (11 x^2+8\right ) x^3}{13 \sqrt {x^4+5 x^2+3}} \]
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Rule 1113
Rule 1149
Rule 1203
Rule 1289
Rule 1293
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \left (8+11 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {1}{13} \int \frac {x^2 \left (-24-33 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {x^3 \left (8+11 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}-\frac {11}{13} x \sqrt {3+5 x^2+x^4}-\frac {1}{39} \int \frac {-99-258 x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {x^3 \left (8+11 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}-\frac {11}{13} x \sqrt {3+5 x^2+x^4}+\frac {33}{13} \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx+\frac {86}{13} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {43 x \left (5+\sqrt {13}+2 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {x^3 \left (8+11 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}-\frac {11}{13} x \sqrt {3+5 x^2+x^4}-\frac {43 \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 )}{13 \sqrt {3+5 x^2+x^4}}+\frac {11 \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 )}{13 \sqrt {3+5 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.23 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.71 \[ \int \frac {x^4 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {-2 x \left (33+47 x^2\right )+43 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-182+43 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{26 \sqrt {3+5 x^2+x^4}} \]
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Time = 4.03 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {x \left (47 x^{2}+33\right )}{13 \sqrt {x^{4}+5 x^{2}+3}}+\frac {198 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {3096 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(216\) |
elliptic | \(-\frac {2 \left (\frac {47}{26} x^{3}+\frac {33}{26} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}+\frac {198 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {3096 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(217\) |
default | \(-\frac {6 \left (\frac {19}{26} x^{3}+\frac {15}{26} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}+\frac {198 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {3096 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {4 \left (-\frac {5}{26} x^{3}-\frac {3}{13} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}\) | \(240\) |
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Time = 0.08 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.60 \[ \int \frac {x^4 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {86 \, {\left (\sqrt {13} \sqrt {2} {\left (x^{5} + 5 \, x^{3} + 3 \, x\right )} - 5 \, \sqrt {2} {\left (x^{5} + 5 \, x^{3} + 3 \, x\right )}\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - 5 \, {\left (15 \, \sqrt {13} \sqrt {2} {\left (x^{5} + 5 \, x^{3} + 3 \, x\right )} - 97 \, \sqrt {2} {\left (x^{5} + 5 \, x^{3} + 3 \, x\right )}\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 4 \, {\left (39 \, x^{4} + 397 \, x^{2} + 258\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{52 \, {\left (x^{5} + 5 \, x^{3} + 3 \, x\right )}} \]
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\[ \int \frac {x^4 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {x^{4} \cdot \left (3 x^{2} + 2\right )}{\left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^4 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (3 \, x^{2} + 2\right )} x^{4}}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^4 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (3 \, x^{2} + 2\right )} x^{4}}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {x^4\,\left (3\,x^2+2\right )}{{\left (x^4+5\,x^2+3\right )}^{3/2}} \,d x \]
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