Optimal. Leaf size=77 \[ -\frac {x^2 \left (33+47 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {133}{26} \sqrt {3+5 x^2+x^4}-\frac {41}{4} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right ) \]
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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, 832, 654,
635, 212} \begin {gather*} -\frac {\left (47 x^2+33\right ) x^2}{13 \sqrt {x^4+5 x^2+3}}+\frac {133}{26} \sqrt {x^4+5 x^2+3}-\frac {41}{4} \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 654
Rule 832
Rule 1265
Rubi steps
\begin {align*} \int \frac {x^5 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2 (2+3 x)}{\left (3+5 x+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {x^2 \left (33+47 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {1}{13} \text {Subst}\left (\int \frac {33+\frac {133 x}{2}}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^2 \left (33+47 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {133}{26} \sqrt {3+5 x^2+x^4}-\frac {41}{4} \text {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^2 \left (33+47 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {133}{26} \sqrt {3+5 x^2+x^4}-\frac {41}{2} \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 {x^2 \left (33+47 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {133}{26} \sqrt {3+5 x^2+x^4}-\frac {41}{4} \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 = 0.18, size = 59, normalized size = 0.77 \begin {gather*} \frac {399+599 x^2+39 x^4}{26 \sqrt {3+5 x^2+x^4}}+\frac {41}{4} \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 = 0.11, size = 91, normalized size = 1.18
method | result | size |
risch | \(\frac {39 x^{4}+599 x^{2}+399}{26 \sqrt {x^{4}+5 x^{2}+3}}-\frac {41 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}\) | \(48\) |
trager | \(\frac {39 x^{4}+599 x^{2}+399}{26 \sqrt {x^{4}+5 x^{2}+3}}+\frac {41 \ln \left (-2 x^{2}+2 \sqrt {x^{4}+5 x^{2}+3}-5\right )}{4}\) | \(52\) |
default | \(\frac {3 x^{4}}{2 \sqrt {x^{4}+5 x^{2}+3}}+\frac {41 x^{2}}{4 \sqrt {x^{4}+5 x^{2}+3}}-\frac {133}{8 \sqrt {x^{4}+5 x^{2}+3}}+\frac {\frac {665 x^{2}}{52}+\frac {3325}{104}}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {41 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}\) | \(91\) |
elliptic | \(\frac {3 x^{4}}{2 \sqrt {x^{4}+5 x^{2}+3}}+\frac {41 x^{2}}{4 \sqrt {x^{4}+5 x^{2}+3}}-\frac {133}{8 \sqrt {x^{4}+5 x^{2}+3}}+\frac {\frac {665 x^{2}}{52}+\frac {3325}{104}}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {41 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 73, normalized size = 0.95 \begin {gather*} \frac {3 \, x^{4}}{2 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {599 \, x^{2}}{26 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {399}{26 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} - \frac {41}{4} \, \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 = 0.38, size = 86, normalized size = 1.12 \begin {gather*} \frac {1811 \, x^{4} + 9055 \, x^{2} + 1066 \, {\left (x^{4} + 5 \, x^{2} + 3\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) + 4 \, {\left (39 \, x^{4} + 599 \, x^{2} + 399\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} + 5433}{104 \, {\left (x^{4} + 5 \, x^{2} + 3\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 )}{\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 [A]
time = 4.43, size = 52, normalized size = 0.68 \begin {gather*} \frac {{\left (39 \, x^{2} + 599\right )} x^{2} + 399}{26 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {41}{4} \, \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 )}{{\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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