Optimal. Leaf size=51 \[ \frac {1}{2} x^4 \sqrt {5+x^4}-\frac {1}{2} \left (10-x^2\right ) \sqrt {5+x^4}-\frac {5}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1266, 847, 794,
221} \begin {gather*} \frac {1}{2} \sqrt {x^4+5} x^4-\frac {5}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {1}{2} \left (10-x^2\right ) \sqrt {x^4+5} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 794
Rule 847
Rule 1266
Rubi steps
\begin {align*} \int \frac {x^5 \left (2+3 x^2\right )}{\sqrt {5+x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2 (2+3 x)}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^4 \sqrt {5+x^4}+\frac {1}{6} \text {Subst}\left (\int \frac {x (-30+6 x)}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^4 \sqrt {5+x^4}-\frac {1}{2} \left (10-x^2\right ) \sqrt {5+x^4}-\frac {5}{2} \text {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^4 \sqrt {5+x^4}-\frac {1}{2} \left (10-x^2\right ) \sqrt {5+x^4}-\frac {5}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 40, normalized size = 0.78 \begin {gather*} \frac {1}{2} \sqrt {5+x^4} \left (-10+x^2+x^4\right )-\frac {5}{2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {5+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 39, normalized size = 0.76
method | result | size |
risch | \(\frac {\left (x^{4}+x^{2}-10\right ) \sqrt {x^{4}+5}}{2}-\frac {5 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}\) | \(30\) |
trager | \(\left (\frac {1}{2} x^{4}+\frac {1}{2} x^{2}-5\right ) \sqrt {x^{4}+5}-\frac {5 \ln \left (x^{2}+\sqrt {x^{4}+5}\right )}{2}\) | \(36\) |
default | \(\frac {\sqrt {x^{4}+5}\, \left (x^{4}-10\right )}{2}+\frac {x^{2} \sqrt {x^{4}+5}}{2}-\frac {5 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}\) | \(39\) |
elliptic | \(\frac {x^{4} \sqrt {x^{4}+5}}{2}-5 \sqrt {x^{4}+5}+\frac {x^{2} \sqrt {x^{4}+5}}{2}-\frac {5 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}\) | \(46\) |
meijerg | \(\frac {15 \sqrt {5}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {4 x^{4}}{5}+8\right ) \sqrt {1+\frac {x^{4}}{5}}}{6}\right )}{4 \sqrt {\pi }}+\frac {\frac {\sqrt {\pi }\, x^{2} \sqrt {5}\, \sqrt {1+\frac {x^{4}}{5}}}{2}-\frac {5 \sqrt {\pi }\, \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}}{\sqrt {\pi }}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 76, normalized size = 1.49 \begin {gather*} \frac {1}{2} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} - \frac {15}{2} \, \sqrt {x^{4} + 5} + \frac {5 \, \sqrt {x^{4} + 5}}{2 \, x^{2} {\left (\frac {x^{4} + 5}{x^{4}} - 1\right )}} - \frac {5}{4} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) + \frac {5}{4} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 34, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, {\left (x^{4} + x^{2} - 10\right )} \sqrt {x^{4} + 5} + \frac {5}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.70, size = 66, normalized size = 1.29 \begin {gather*} \frac {x^{6}}{2 \sqrt {x^{4} + 5}} + \frac {x^{4} \sqrt {x^{4} + 5}}{2} + \frac {5 x^{2}}{2 \sqrt {x^{4} + 5}} - 5 \sqrt {x^{4} + 5} - \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.04, size = 37, normalized size = 0.73 \begin {gather*} \frac {1}{2} \, \sqrt {x^{4} + 5} {\left ({\left (x^{2} + 1\right )} x^{2} - 10\right )} + \frac {5}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 32, normalized size = 0.63 \begin {gather*} \sqrt {x^4+5}\,\left (\frac {x^4}{2}+\frac {x^2}{2}-5\right )-\frac {5\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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