Optimal. Leaf size=356 \[ \frac {176723 x \left (5+\sqrt {13}+2 x^2\right )}{4290 \sqrt {3+5 x^2+x^4}}-\frac {4210}{429} x \sqrt {3+5 x^2+x^4}+\frac {1251}{715} x^3 \sqrt {3+5 x^2+x^4}-\frac {1}{429} x^5 \left (283+272 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {176723 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{4290 \sqrt {3+5 x^2+x^4}}+\frac {2105 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{143 \sqrt {3+5 x^2+x^4}} \]
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Rubi [A]
time = 0.17, 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 = {1287, 1293,
1203, 1113, 1149} \begin {gather*} \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 (\text {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 (\text {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 \end {gather*}
Antiderivative was successfully verified.
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Rule 1113
Rule 1149
Rule 1203
Rule 1287
Rule 1293
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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 9.02, size = 249, normalized size = 0.70 \begin {gather*} \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 \sinh ^{-1}\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} F\left (i \sinh ^{-1}\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}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.07, size = 294, normalized size = 0.83
method | result | size |
risch | \(\frac {x \left (495 x^{10}+3540 x^{8}+5450 x^{6}+1780 x^{4}+3753 x^{2}-21050\right ) \sqrt {x^{4}+5 x^{2}+3}}{2145}-\frac {2120676 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{715 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}+\frac {25260 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{143 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}\) | \(236\) |
default | \(\frac {3 x^{11} \sqrt {x^{4}+5 x^{2}+3}}{13}+\frac {236 x^{9} \sqrt {x^{4}+5 x^{2}+3}}{143}+\frac {1090 x^{7} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {356 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {1251 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{715}-\frac {4210 x \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {25260 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{143 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2120676 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{715 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(294\) |
elliptic | \(\frac {3 x^{11} \sqrt {x^{4}+5 x^{2}+3}}{13}+\frac {236 x^{9} \sqrt {x^{4}+5 x^{2}+3}}{143}+\frac {1090 x^{7} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {356 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {1251 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{715}-\frac {4210 x \sqrt {x^{4}+5 x^{2}+3}}{429}+\frac {25260 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{143 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {2120676 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{715 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(294\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \cdot \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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