Optimal. Leaf size=309 \[ \frac {19 x \left (5+\sqrt {13}+2 x^2\right )}{234 \sqrt {3+5 x^2+x^4}}-\frac {7+8 x^2}{39 x \sqrt {3+5 x^2+x^4}}-\frac {19 \sqrt {3+5 x^2+x^4}}{117 x}-\frac {19 \sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{234 \sqrt {3+5 x^2+x^4}}-\frac {4 \sqrt {\frac {2}{3 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{39 \sqrt {3+5 x^2+x^4}} \]
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Rubi [A]
time = 0.12, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1291, 1295,
1203, 1113, 1149} \begin {gather*} -\frac {4 \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 )}{39 \sqrt {x^4+5 x^2+3}}-\frac {19 \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 )}{234 \sqrt {x^4+5 x^2+3}}+\frac {19 x \left (2 x^2+\sqrt {13}+5\right )}{234 \sqrt {x^4+5 x^2+3}}-\frac {19 \sqrt {x^4+5 x^2+3}}{117 x}-\frac {8 x^2+7}{39 x \sqrt {x^4+5 x^2+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1113
Rule 1149
Rule 1203
Rule 1291
Rule 1295
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x^2 \left (3+5 x^2+x^4\right )^{3/2}} \, dx &=-\frac {7+8 x^2}{39 x \sqrt {3+5 x^2+x^4}}-\frac {1}{39} \int \frac {-19+8 x^2}{x^2 \sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {7+8 x^2}{39 x \sqrt {3+5 x^2+x^4}}-\frac {19 \sqrt {3+5 x^2+x^4}}{117 x}+\frac {1}{117} \int \frac {-24+19 x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {7+8 x^2}{39 x \sqrt {3+5 x^2+x^4}}-\frac {19 \sqrt {3+5 x^2+x^4}}{117 x}+\frac {19}{117} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx-\frac {8}{39} \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=\frac {19 x \left (5+\sqrt {13}+2 x^2\right )}{234 \sqrt {3+5 x^2+x^4}}-\frac {7+8 x^2}{39 x \sqrt {3+5 x^2+x^4}}-\frac {19 \sqrt {3+5 x^2+x^4}}{117 x}-\frac {19 \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 )}{234 \sqrt {3+5 x^2+x^4}}-\frac {4 \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 )}{39 \sqrt {3+5 x^2+x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.16, size = 228, normalized size = 0.74 \begin {gather*} \frac {-4 \left (78+119 x^2+19 x^4\right )+19 i \sqrt {2} \left (-5+\sqrt {13}\right ) x \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 (-143+19 \sqrt {13}\right ) x \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 )}{468 x \sqrt {3+5 x^2+x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.08, size = 257, normalized size = 0.83
method | result | size |
risch | \(-\frac {19 x^{4}+119 x^{2}+78}{117 x \sqrt {x^{4}+5 x^{2}+3}}-\frac {76 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {16 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}\) | \(223\) |
elliptic | \(-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{9 x}-\frac {2 \left (-\frac {7}{234} x^{3}-\frac {11}{234} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {76 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {16 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}\) | \(234\) |
default | \(-\frac {6 \left (-\frac {19}{78} x -\frac {5}{78} x^{3}\right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {16 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {76 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {2 \sqrt {x^{4}+5 x^{2}+3}}{9 x}-\frac {4 \left (\frac {19}{234} x^{3}+\frac {40}{117} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}\) | \(257\) |
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 \frac {3 x^{2} + 2}{x^{2} \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 \frac {3\,x^2+2}{x^2\,{\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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