Optimal. Leaf size=322 \[ -\frac {1924 x \left (5+\sqrt {13}+2 x^2\right )}{105 \sqrt {3+5 x^2+x^4}}+\frac {13}{3} x \sqrt {3+5 x^2+x^4}-\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {962 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{105 \sqrt {3+5 x^2+x^4}}-\frac {13 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]
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Rubi [A]
time = 0.19, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {13 \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 )}{\sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {x^4+5 x^2+3}}+\frac {962 \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 ) 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 )}{105 \sqrt {x^4+5 x^2+3}}+\frac {13}{3} \sqrt {x^4+5 x^2+3} x-\frac {1924 \left (2 x^2+\sqrt {13}+5\right ) x}{105 \sqrt {x^4+5 x^2+3}}+\frac {1}{21} \left (7 x^2+11\right ) \sqrt {x^4+5 x^2+3} x^5-\frac {26}{35} \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 ) \sqrt {3+5 x^2+x^4} \, dx &=\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{63} \int \frac {x^4 \left (-117-234 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1}{315} \int \frac {x^2 \left (-2106-4095 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=\frac {13}{3} x \sqrt {3+5 x^2+x^4}-\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{945} \int \frac {-12285-34632 x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=\frac {13}{3} x \sqrt {3+5 x^2+x^4}-\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}-13 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx-\frac {3848}{105} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {1924 x \left (5+\sqrt {13}+2 x^2\right )}{105 \sqrt {3+5 x^2+x^4}}+\frac {13}{3} x \sqrt {3+5 x^2+x^4}-\frac {26}{35} x^3 \sqrt {3+5 x^2+x^4}+\frac {1}{21} x^5 \left (11+7 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {962 \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 ) 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 )}{105 \sqrt {3+5 x^2+x^4}}-\frac {13 \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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.93, size = 237, normalized size = 0.74 \begin {gather*} \frac {2730 x+4082 x^3+460 x^5+604 x^7+460 x^9+70 x^{11}-1924 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 )+13 i \sqrt {2} \left (-635+148 \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 )}{210 \sqrt {3+5 x^2+x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 260, normalized size = 0.81
method | result | size |
risch | \(\frac {x \left (35 x^{6}+55 x^{4}-78 x^{2}+455\right ) \sqrt {x^{4}+5 x^{2}+3}}{105}+\frac {46176 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {78 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}\) | \(226\) |
default | \(\frac {x^{7} \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {11 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{21}-\frac {26 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{35}+\frac {13 x \sqrt {x^{4}+5 x^{2}+3}}{3}-\frac {78 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {46176 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(260\) |
elliptic | \(\frac {x^{7} \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {11 x^{5} \sqrt {x^{4}+5 x^{2}+3}}{21}-\frac {26 x^{3} \sqrt {x^{4}+5 x^{2}+3}}{35}+\frac {13 x \sqrt {x^{4}+5 x^{2}+3}}{3}-\frac {78 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}+\frac {46176 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}\) | \(260\) |
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 ) \sqrt {x^{4} + 5 x^{2} + 3}\, 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 )\,\sqrt {x^4+5\,x^2+3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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