Optimal. Leaf size=356 \[ -\frac {4210}{429} \sqrt {x^4+5 x^2+3} x+\frac {176723 \left (2 x^2+\sqrt {13}+5\right ) x}{4290 \sqrt {x^4+5 x^2+3}}+\frac {2105 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{143 \sqrt {x^4+5 x^2+3}}-\frac {176723 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{4290 \sqrt {x^4+5 x^2+3}}+\frac {1}{143} \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2} x^5-\frac {1}{429} \left (272 x^2+283\right ) \sqrt {x^4+5 x^2+3} x^5+\frac {1251}{715} \sqrt {x^4+5 x^2+3} x^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1273, 1279, 1189, 1099, 1135} \[ \frac {1}{143} \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2} x^5-\frac {1}{429} \left (272 x^2+283\right ) \sqrt {x^4+5 x^2+3} x^5+\frac {1251}{715} \sqrt {x^4+5 x^2+3} x^3-\frac {4210}{429} \sqrt {x^4+5 x^2+3} x+\frac {176723 \left (2 x^2+\sqrt {13}+5\right ) x}{4290 \sqrt {x^4+5 x^2+3}}+\frac {2105 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{143 \sqrt {x^4+5 x^2+3}}-\frac {176723 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{4290 \sqrt {x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1099
Rule 1135
Rule 1189
Rule 1273
Rule 1279
Rubi steps
\begin {align*} \int x^4 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx &=\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*}
________________________________________________________________________________________
Mathematica [C] time = 0.27, size = 249, normalized size = 0.70 \[ \frac {-i \sqrt {2} \left (176723 \sqrt {13}-757315\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+176723 i \sqrt {2} \left (\sqrt {13}-5\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+4 x \left (495 x^{14}+6015 x^{12}+24635 x^{10}+39650 x^8+29003 x^6+3055 x^4-93991 x^2-63150\right )}{8580 \sqrt {x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (3 \, x^{10} + 17 \, x^{8} + 19 \, x^{6} + 6 \, x^{4}\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 294, normalized size = 0.83 \[ \frac {3 \sqrt {x^{4}+5 x^{2}+3}\, x^{11}}{13}+\frac {236 \sqrt {x^{4}+5 x^{2}+3}\, x^{9}}{143}+\frac {1090 \sqrt {x^{4}+5 x^{2}+3}\, x^{7}}{429}+\frac {356 \sqrt {x^{4}+5 x^{2}+3}\, x^{5}}{429}+\frac {1251 \sqrt {x^{4}+5 x^{2}+3}\, x^{3}}{715}-\frac {4210 \sqrt {x^{4}+5 x^{2}+3}\, x}{429}+\frac {25260 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{143 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2120676 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )+\EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{715 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (\sqrt {13}+5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________