Optimal. Leaf size=77 \[ \frac {1}{2} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{16} \left (89-14 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1083}{32} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 77, 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 = {1265, 846, 793,
635, 212} \begin {gather*} \frac {1}{2} \sqrt {x^4+5 x^2+3} x^4+\frac {3}{16} \left (89-14 x^2\right ) \sqrt {x^4+5 x^2+3}-\frac {1083}{32} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 793
Rule 846
Rule 1265
Rubi steps
\begin {align*} \int \frac {x^5 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2 (2+3 x)}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^4 \sqrt {3+5 x^2+x^4}+\frac {1}{6} \text {Subst}\left (\int \frac {\left (-18-\frac {63 x}{2}\right ) x}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{16} \left (89-14 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1083}{32} \text {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{16} \left (89-14 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1083}{16} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {5+2 x^2}{\sqrt {3+5 x^2+x^4}}\right )\\ &=\frac {1}{2} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{16} \left (89-14 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1083}{32} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 59, normalized size = 0.77 \begin {gather*} \frac {1}{16} \sqrt {3+5 x^2+x^4} \left (267-42 x^2+8 x^4\right )+\frac {1083}{32} \log \left (-5-2 x^2+2 \sqrt {3+5 x^2+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 70, normalized size = 0.91
method | result | size |
risch | \(\frac {\left (8 x^{4}-42 x^{2}+267\right ) \sqrt {x^{4}+5 x^{2}+3}}{16}-\frac {1083 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{32}\) | \(48\) |
trager | \(\left (\frac {1}{2} x^{4}-\frac {21}{8} x^{2}+\frac {267}{16}\right ) \sqrt {x^{4}+5 x^{2}+3}-\frac {1083 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right )}{32}\) | \(51\) |
default | \(\frac {x^{4} \sqrt {x^{4}+5 x^{2}+3}}{2}-\frac {21 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{8}+\frac {267 \sqrt {x^{4}+5 x^{2}+3}}{16}-\frac {1083 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{32}\) | \(70\) |
elliptic | \(\frac {x^{4} \sqrt {x^{4}+5 x^{2}+3}}{2}-\frac {21 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{8}+\frac {267 \sqrt {x^{4}+5 x^{2}+3}}{16}-\frac {1083 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{32}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 73, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{4} - \frac {21}{8} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} + \frac {267}{16} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {1083}{32} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 51, normalized size = 0.66 \begin {gather*} \frac {1}{16} \, {\left (8 \, x^{4} - 42 \, x^{2} + 267\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {1083}{32} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \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 {x^{5} \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 [A]
time = 4.15, size = 53, normalized size = 0.69 \begin {gather*} \frac {1}{16} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, x^{2} - 21\right )} x^{2} + 267\right )} + \frac {1083}{32} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \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 = -0.01 \begin {gather*} \int \frac {x^5\,\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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