Optimal. Leaf size=58 \[ -\frac {\sqrt {5+x^4}}{10 x^4}-\frac {3 \sqrt {5+x^4}}{10 x^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )}{10 \sqrt {5}} \]
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Rubi [A]
time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1266, 849, 821,
272, 65, 213} \begin {gather*} -\frac {\sqrt {x^4+5}}{10 x^4}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )}{10 \sqrt {5}}-\frac {3 \sqrt {x^4+5}}{10 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 213
Rule 272
Rule 821
Rule 849
Rule 1266
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x^5 \sqrt {5+x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {2+3 x}{x^3 \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {5+x^4}}{10 x^4}-\frac {1}{20} \text {Subst}\left (\int \frac {-30+2 x}{x^2 \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {5+x^4}}{10 x^4}-\frac {3 \sqrt {5+x^4}}{10 x^2}-\frac {1}{10} \text {Subst}\left (\int \frac {1}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {5+x^4}}{10 x^4}-\frac {3 \sqrt {5+x^4}}{10 x^2}-\frac {1}{20} \text {Subst}\left (\int \frac {1}{x \sqrt {5+x}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {5+x^4}}{10 x^4}-\frac {3 \sqrt {5+x^4}}{10 x^2}-\frac {1}{10} \text {Subst}\left (\int \frac {1}{-5+x^2} \, dx,x,\sqrt {5+x^4}\right )\\ &=-\frac {\sqrt {5+x^4}}{10 x^4}-\frac {3 \sqrt {5+x^4}}{10 x^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )}{10 \sqrt {5}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 55, normalized size = 0.95 \begin {gather*} \frac {1}{50} \left (-\frac {5 \left (1+3 x^2\right ) \sqrt {5+x^4}}{x^4}-2 \sqrt {5} \tanh ^{-1}\left (\frac {x^2-\sqrt {5+x^4}}{\sqrt {5}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 43, normalized size = 0.74
method | result | size |
default | \(-\frac {\sqrt {x^{4}+5}}{10 x^{4}}+\frac {\sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{50}-\frac {3 \sqrt {x^{4}+5}}{10 x^{2}}\) | \(43\) |
elliptic | \(-\frac {\sqrt {x^{4}+5}}{10 x^{4}}+\frac {\sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{50}-\frac {3 \sqrt {x^{4}+5}}{10 x^{2}}\) | \(43\) |
risch | \(-\frac {3 x^{6}+x^{4}+15 x^{2}+5}{10 x^{4} \sqrt {x^{4}+5}}+\frac {\sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{50}\) | \(46\) |
trager | \(-\frac {\left (3 x^{2}+1\right ) \sqrt {x^{4}+5}}{10 x^{4}}-\frac {\RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-5\right )-\sqrt {x^{4}+5}}{x^{2}}\right )}{50}\) | \(51\) |
meijerg | \(\frac {\sqrt {5}\, \left (\frac {5 \sqrt {\pi }\, \left (\frac {4 x^{4}}{5}+8\right )}{8 x^{4}}-\frac {5 \sqrt {\pi }\, \sqrt {1+\frac {x^{4}}{5}}}{x^{4}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{4}}{5}}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+4 \ln \left (x \right )-\ln \left (5\right )\right ) \sqrt {\pi }}{2}-\frac {5 \sqrt {\pi }}{x^{4}}\right )}{50 \sqrt {\pi }}-\frac {3 \sqrt {5}\, \sqrt {1+\frac {x^{4}}{5}}}{10 x^{2}}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 59, normalized size = 1.02 \begin {gather*} -\frac {1}{100} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{\sqrt {5} + \sqrt {x^{4} + 5}}\right ) - \frac {3 \, \sqrt {x^{4} + 5}}{10 \, x^{2}} - \frac {\sqrt {x^{4} + 5}}{10 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 50, normalized size = 0.86 \begin {gather*} \frac {\sqrt {5} x^{4} \log \left (\frac {\sqrt {5} + \sqrt {x^{4} + 5}}{x^{2}}\right ) - 15 \, x^{4} - 5 \, \sqrt {x^{4} + 5} {\left (3 \, x^{2} + 1\right )}}{50 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.66, size = 88, normalized size = 1.52 \begin {gather*} \frac {\sqrt {5} \left (- \frac {\log {\left (\sqrt {\frac {x^{4}}{5} + 1} - 1 \right )}}{4} + \frac {\log {\left (\sqrt {\frac {x^{4}}{5} + 1} + 1 \right )}}{4} - \frac {1}{4 \left (\sqrt {\frac {x^{4}}{5} + 1} + 1\right )} - \frac {1}{4 \left (\sqrt {\frac {x^{4}}{5} + 1} - 1\right )}\right )}{25} - \frac {3 \sqrt {5} \sqrt {5 x^{4} + 25}}{50 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs.
\(2 (43) = 86\).
time = 5.76, size = 114, normalized size = 1.97 \begin {gather*} -\frac {1}{50} \, \sqrt {5} \log \left (-\frac {x^{2} + \sqrt {5} - \sqrt {x^{4} + 5}}{x^{2} - \sqrt {5} - \sqrt {x^{4} + 5}}\right ) + \frac {{\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{3} + 15 \, {\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{2} + 5 \, x^{2} - 5 \, \sqrt {x^{4} + 5} - 75}{5 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{2} - 5\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.69, size = 43, normalized size = 0.74 \begin {gather*} \frac {\sqrt {5}\,\mathrm {atanh}\left (\frac {\sqrt {5}\,\sqrt {x^4+5}}{5}\right )}{50}-\frac {3\,\sqrt {x^4+5}}{10\,x^2}-\frac {\sqrt {x^4+5}}{10\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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