Integrand size = 25, antiderivative size = 322 \[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=-\frac {1924 x \left (5+\sqrt {13}+2 x^2\right )}{105 \sqrt {3+5 x^2+x^4}}+\frac {13}{3} x \sqrt {3+5 x^2+x^4}-\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {962 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{105 \sqrt {3+5 x^2+x^4}}-\frac {13 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]
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Time = 0.16 (sec) , antiderivative size = 322, 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 = {1287, 1293, 1203, 1113, 1149} \[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=-\frac {13 \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {x^4+5 x^2+3}}+\frac {962 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{105 \sqrt {x^4+5 x^2+3}}+\frac {13}{3} \sqrt {x^4+5 x^2+3} x-\frac {1924 \left (2 x^2+\sqrt {13}+5\right ) x}{105 \sqrt {x^4+5 x^2+3}}+\frac {1}{21} \left (7 x^2+11\right ) \sqrt {x^4+5 x^2+3} x^5-\frac {26}{35} \sqrt {x^4+5 x^2+3} x^3 \]
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Rule 1113
Rule 1149
Rule 1203
Rule 1287
Rule 1293
Rubi steps \begin{align*} \text {integral}& = \frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{63} \int \frac {x^4 \left (-117-234 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1}{315} \int \frac {x^2 \left (-2106-4095 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {13}{3} x \sqrt {3+5 x^2+x^4}-\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{945} \int \frac {-12285-34632 x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {13}{3} x \sqrt {3+5 x^2+x^4}-\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}-13 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx-\frac {3848}{105} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {1924 x \left (5+\sqrt {13}+2 x^2\right )}{105 \sqrt {3+5 x^2+x^4}}+\frac {13}{3} x \sqrt {3+5 x^2+x^4}-\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {962 \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 ) 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 )}{105 \sqrt {3+5 x^2+x^4}}-\frac {13 \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 {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.06 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.74 \[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {2730 x+4082 x^3+460 x^5+604 x^7+460 x^9+70 x^{11}-1924 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+13 i \sqrt {2} \left (-635+148 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{210 \sqrt {3+5 x^2+x^4}} \]
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Time = 5.58 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.70
method | result | size |
risch | \(\frac {x \left (35 x^{6}+55 x^{4}-78 x^{2}+455\right ) \sqrt {x^{4}+5 x^{2}+3}}{105}-\frac {78 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {46176 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(226\) |
default | \(\frac {x^{7} \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {11 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{21}-\frac {26 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{35}+\frac {13 x \sqrt {x^{4}+5 x^{2}+3}}{3}-\frac {78 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {46176 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(260\) |
elliptic | \(\frac {x^{7} \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {11 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{21}-\frac {26 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{35}+\frac {13 x \sqrt {x^{4}+5 x^{2}+3}}{3}-\frac {78 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {46176 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(260\) |
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Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.43 \[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=-\frac {3848 \, {\left (\sqrt {13} \sqrt {2} x - 5 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - 13 \, {\left (261 \, \sqrt {13} \sqrt {2} x - 1655 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - 4 \, {\left (35 \, x^{8} + 55 \, x^{6} - 78 \, x^{4} + 455 \, x^{2} - 3848\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{420 \, x} \]
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\[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int x^{4} \cdot \left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}\, dx \]
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\[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )} x^{4} \,d x } \]
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\[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int x^4\,\left (3\,x^2+2\right )\,\sqrt {x^4+5\,x^2+3} \,d x \]
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