Integrand size = 22, antiderivative size = 279 \[ \int \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=-\frac {23 x \left (5+\sqrt {13}+2 x^2\right )}{15 \sqrt {3+5 x^2+x^4}}+\frac {1}{15} x \left (25+9 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {23 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{15 \sqrt {3+5 x^2+x^4}}+\frac {\sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]
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Time = 0.06 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1190, 1203, 1113, 1149} \[ \int \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {\sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {x^4+5 x^2+3}}+\frac {23 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{15 \sqrt {x^4+5 x^2+3}}-\frac {23 x \left (2 x^2+\sqrt {13}+5\right )}{15 \sqrt {x^4+5 x^2+3}}+\frac {1}{15} x \left (9 x^2+25\right ) \sqrt {x^4+5 x^2+3} \]
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Rule 1113
Rule 1149
Rule 1190
Rule 1203
Rubi steps \begin{align*} \text {integral}& = \frac {1}{15} x \left (25+9 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{15} \int \frac {15-46 x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {1}{15} x \left (25+9 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {46}{15} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx+\int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {23 x \left (5+\sqrt {13}+2 x^2\right )}{15 \sqrt {3+5 x^2+x^4}}+\frac {1}{15} x \left (25+9 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {23 \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 )}{15 \sqrt {3+5 x^2+x^4}}+\frac {\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 = 5.06 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.82 \[ \int \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {2 x \left (75+152 x^2+70 x^4+9 x^6\right )-23 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+i \sqrt {2} \left (-130+23 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{30 \sqrt {3+5 x^2+x^4}} \]
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Time = 1.74 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {x \left (9 x^{2}+25\right ) \sqrt {x^{4}+5 x^{2}+3}}{15}+\frac {6 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {552 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(216\) |
default | \(\frac {5 x \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {6 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {552 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}+\frac {3 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{5}\) | \(226\) |
elliptic | \(\frac {5 x \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {6 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {552 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}+\frac {3 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{5}\) | \(226\) |
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Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.46 \[ \int \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=-\frac {46 \, {\left (\sqrt {13} \sqrt {2} x - 5 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (51 \, \sqrt {13} \sqrt {2} x - 205 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - 4 \, {\left (9 \, x^{4} + 25 \, x^{2} - 46\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{60 \, x} \]
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\[ \int \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int \left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}\, dx \]
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\[ \int \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )} \,d x } \]
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\[ \int \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} + 2\right )} \,d x } \]
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Timed out. \[ \int \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\int \left (3\,x^2+2\right )\,\sqrt {x^4+5\,x^2+3} \,d x \]
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