Integrand size = 25, antiderivative size = 298 \[ \int \frac {x^4 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx=\frac {419 x \left (5+\sqrt {13}+2 x^2\right )}{30 \sqrt {3+5 x^2+x^4}}-\frac {10}{3} x \sqrt {3+5 x^2+x^4}+\frac {3}{5} x^3 \sqrt {3+5 x^2+x^4}-\frac {419 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{30 \sqrt {3+5 x^2+x^4}}+\frac {5 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {3+5 x^2+x^4}} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1293, 1203, 1113, 1149} \[ \int \frac {x^4 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx=\frac {5 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right ),\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {419 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\arctan \left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{30 \sqrt {x^4+5 x^2+3}}-\frac {10}{3} \sqrt {x^4+5 x^2+3} x+\frac {419 \left (2 x^2+\sqrt {13}+5\right ) x}{30 \sqrt {x^4+5 x^2+3}}+\frac {3}{5} \sqrt {x^4+5 x^2+3} x^3 \]
[In]
[Out]
Rule 1113
Rule 1149
Rule 1203
Rule 1293
Rubi steps \begin{align*} \text {integral}& = \frac {3}{5} x^3 \sqrt {3+5 x^2+x^4}-\frac {1}{5} \int \frac {x^2 \left (27+50 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {10}{3} x \sqrt {3+5 x^2+x^4}+\frac {3}{5} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{15} \int \frac {150+419 x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = -\frac {10}{3} x \sqrt {3+5 x^2+x^4}+\frac {3}{5} x^3 \sqrt {3+5 x^2+x^4}+10 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx+\frac {419}{15} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx \\ & = \frac {419 x \left (5+\sqrt {13}+2 x^2\right )}{30 \sqrt {3+5 x^2+x^4}}-\frac {10}{3} x \sqrt {3+5 x^2+x^4}+\frac {3}{5} x^3 \sqrt {3+5 x^2+x^4}-\frac {419 \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 )}{30 \sqrt {3+5 x^2+x^4}}+\frac {5 \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 = 10.23 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.77 \[ \int \frac {x^4 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx=\frac {4 x \left (-150-223 x^2-5 x^4+9 x^6\right )+419 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (-1795+419 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right ),\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{60 \sqrt {3+5 x^2+x^4}} \]
[In]
[Out]
Time = 3.89 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {x \left (9 x^{2}-50\right ) \sqrt {x^{4}+5 x^{2}+3}}{15}+\frac {60 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {5028 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(216\) |
default | \(\frac {3 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{5}-\frac {10 x \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {60 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {5028 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(226\) |
elliptic | \(\frac {3 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{5}-\frac {10 x \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {60 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {5028 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{5 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(226\) |
[In]
[Out]
none
Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.43 \[ \int \frac {x^4 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx=\frac {419 \, {\left (\sqrt {13} \sqrt {2} x - 5 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) - {\left (369 \, \sqrt {13} \sqrt {2} x - 2345 \, \sqrt {2} x\right )} \sqrt {\sqrt {13} - 5} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} - 5}}{2 \, x}\right )\,|\,\frac {5}{6} \, \sqrt {13} + \frac {19}{6}) + 4 \, {\left (9 \, x^{4} - 50 \, x^{2} + 419\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{60 \, x} \]
[In]
[Out]
\[ \int \frac {x^4 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx=\int \frac {x^{4} \cdot \left (3 x^{2} + 2\right )}{\sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
[In]
[Out]
\[ \int \frac {x^4 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx=\int { \frac {{\left (3 \, x^{2} + 2\right )} x^{4}}{\sqrt {x^{4} + 5 \, x^{2} + 3}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^4 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx=\int { \frac {{\left (3 \, x^{2} + 2\right )} x^{4}}{\sqrt {x^{4} + 5 \, x^{2} + 3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^4 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx=\int \frac {x^4\,\left (3\,x^2+2\right )}{\sqrt {x^4+5\,x^2+3}} \,d x \]
[In]
[Out]