Integrand size = 25, antiderivative size = 305 \[ \int x^2 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {1247 x \left (5+\sqrt {13}+2 x^2\right )}{210 \sqrt {3+5 x^2+x^4}}-\frac {4}{3} x \sqrt {3+5 x^2+x^4}+\frac {1}{35} x^3 \left (29+15 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1247 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{210 \sqrt {3+5 x^2+x^4}}+\frac {2 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {3+5 x^2+x^4}} \]
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Time = 0.11 (sec) , antiderivative size = 305, 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 = {1287, 1293, 1203, 1113, 1149} \[ \int x^2 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {2 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {1247 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{210 \sqrt {x^4+5 x^2+3}}-\frac {4}{3} \sqrt {x^4+5 x^2+3} x+\frac {1247 \left (2 x^2+\sqrt {13}+5\right ) x}{210 \sqrt {x^4+5 x^2+3}}+\frac {1}{35} \left (15 x^2+29\right ) \sqrt {x^4+5 x^2+3} x^3 \]
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Rule 1113
Rule 1149
Rule 1203
Rule 1287
Rule 1293
Rubi steps \begin{align*} \text {integral}& = \frac {1}{35} x^3 \left (29+15 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{35} \int \frac {x^2 \left (-51-140 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {4}{3} x \sqrt {3+5 x^2+x^4}+\frac {1}{35} x^3 \left (29+15 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1}{105} \int \frac {-420-1247 x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {4}{3} x \sqrt {3+5 x^2+x^4}+\frac {1}{35} x^3 \left (29+15 x^2\right ) \sqrt {3+5 x^2+x^4}+4 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx+\frac {1247}{105} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {1247 x \left (5+\sqrt {13}+2 x^2\right )}{210 \sqrt {3+5 x^2+x^4}}-\frac {4}{3} x \sqrt {3+5 x^2+x^4}+\frac {1}{35} x^3 \left (29+15 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1247 \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 )}{210 \sqrt {3+5 x^2+x^4}}+\frac {2 \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 )}{\sqrt {3+5 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.09 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.77 \[ \int x^2 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {4 x \left (-420-439 x^2+430 x^4+312 x^6+45 x^8\right )+1247 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-5395+1247 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{420 \sqrt {3+5 x^2+x^4}} \]
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Time = 2.00 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {x \left (45 x^{4}+87 x^{2}-140\right ) \sqrt {x^{4}+5 x^{2}+3}}{105}+\frac {24 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {14964 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(221\) |
default | \(\frac {3 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{7}+\frac {29 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{35}-\frac {4 x \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {24 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {14964 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(243\) |
elliptic | \(\frac {3 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{7}+\frac {29 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{35}-\frac {4 x \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {24 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {14964 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(243\) |
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Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.44 \[ \int x^2 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {1247 \, {\left (\sqrt {13} \sqrt {2} x - 5 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (1107 \, \sqrt {13} \sqrt {2} x - 6935 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 4 \, {\left (45 \, x^{6} + 87 \, x^{4} - 140 \, x^{2} + 1247\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{420 \, x} \]
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\[ \int x^2 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int x^{2} \cdot \left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}\, dx \]
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\[ \int x^2 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )} x^{2} \,d x } \]
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\[ \int x^2 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int x^2\,\left (3\,x^2+2\right )\,\sqrt {x^4+5\,x^2+3} \,d x \]
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