Integrand size = 25, antiderivative size = 356 \[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {176723 x \left (5+\sqrt {13}+2 x^2\right )}{4290 \sqrt {3+5 x^2+x^4}}-\frac {4210}{429} x \sqrt {3+5 x^2+x^4}+\frac {1251}{715} x^3 \sqrt {3+5 x^2+x^4}-\frac {1}{429} x^5 \left (283+272 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {176723 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{4290 \sqrt {3+5 x^2+x^4}}+\frac {2105 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{143 \sqrt {3+5 x^2+x^4}} \]
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Time = 0.13 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1287, 1293, 1203, 1113, 1149} \[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {2105 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{143 \sqrt {x^4+5 x^2+3}}-\frac {176723 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{4290 \sqrt {x^4+5 x^2+3}}-\frac {4210}{429} \sqrt {x^4+5 x^2+3} x+\frac {176723 \left (2 x^2+\sqrt {13}+5\right ) x}{4290 \sqrt {x^4+5 x^2+3}}+\frac {1}{143} \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2} x^5-\frac {1}{429} \left (272 x^2+283\right ) \sqrt {x^4+5 x^2+3} x^5+\frac {1251}{715} \sqrt {x^4+5 x^2+3} x^3 \]
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Rule 1113
Rule 1149
Rule 1203
Rule 1287
Rule 1293
Rubi steps \begin{align*} \text {integral}& = \frac {1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{143} \int x^4 \left (-69-272 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx \\ & = -\frac {1}{429} x^5 \left (283+272 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {\int \frac {x^4 \left (16674+26271 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx}{3003} \\ & = \frac {1251}{715} x^3 \sqrt {3+5 x^2+x^4}-\frac {1}{429} x^5 \left (283+272 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {\int \frac {x^2 \left (236439+442050 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx}{15015} \\ & = -\frac {4210}{429} x \sqrt {3+5 x^2+x^4}+\frac {1251}{715} x^3 \sqrt {3+5 x^2+x^4}-\frac {1}{429} x^5 \left (283+272 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {\int \frac {1326150+3711183 x^2}{\sqrt {3+5 x^2+x^4}} \, dx}{45045} \\ & = -\frac {4210}{429} x \sqrt {3+5 x^2+x^4}+\frac {1251}{715} x^3 \sqrt {3+5 x^2+x^4}-\frac {1}{429} x^5 \left (283+272 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {4210}{143} \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx+\frac {176723 \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx}{2145} \\ & = \frac {176723 x \left (5+\sqrt {13}+2 x^2\right )}{4290 \sqrt {3+5 x^2+x^4}}-\frac {4210}{429} x \sqrt {3+5 x^2+x^4}+\frac {1251}{715} x^3 \sqrt {3+5 x^2+x^4}-\frac {1}{429} x^5 \left (283+272 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {176723 \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 )}{4290 \sqrt {3+5 x^2+x^4}}+\frac {2105 \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 )}{143 \sqrt {3+5 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 9.17 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.70 \[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {4 x \left (-63150-93991 x^2+3055 x^4+29003 x^6+39650 x^8+24635 x^{10}+6015 x^{12}+495 x^{14}\right )+176723 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-757315+176723 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{8580 \sqrt {3+5 x^2+x^4}} \]
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Time = 4.21 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {x \left (495 x^{10}+3540 x^{8}+5450 x^{6}+1780 x^{4}+3753 x^{2}-21050\right ) \sqrt {x^{4}+5 x^{2}+3}}{2145}+\frac {25260 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{143 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2120676 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{715 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(236\) |
default | \(\frac {3 x^{11} \sqrt {x^{4}+5 x^{2}+3}}{13}+\frac {236 x^{9} \sqrt {x^{4}+5 x^{2}+3}}{143}+\frac {1090 x^{7} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {356 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {1251 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{715}-\frac {4210 x \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {25260 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{143 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2120676 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{715 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(294\) |
elliptic | \(\frac {3 x^{11} \sqrt {x^{4}+5 x^{2}+3}}{13}+\frac {236 x^{9} \sqrt {x^{4}+5 x^{2}+3}}{143}+\frac {1090 x^{7} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {356 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {1251 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{715}-\frac {4210 x \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {25260 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{143 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2120676 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{715 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(294\) |
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Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.42 \[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {176723 \, {\left (\sqrt {13} \sqrt {2} x - 5 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (155673 \, \sqrt {13} \sqrt {2} x - 988865 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 4 \, {\left (495 \, x^{12} + 3540 \, x^{10} + 5450 \, x^{8} + 1780 \, x^{6} + 3753 \, x^{4} - 21050 \, x^{2} + 176723\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{8580 \, x} \]
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\[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int x^{4} \cdot \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}\, dx \]
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\[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )} x^{4} \,d x } \]
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\[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int x^4\,\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2} \,d x \]
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