Optimal. Leaf size=46 \[ \frac {3 x^2+2}{10 \sqrt {x^4+5}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )}{5 \sqrt {5}} \]
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Rubi [A] time = 0.04, antiderivative size = 46, 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 = {1252, 823, 12, 266, 63, 207} \[ \frac {3 x^2+2}{10 \sqrt {x^4+5}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )}{5 \sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 207
Rule 266
Rule 823
Rule 1252
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x \left (5+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {2+3 x}{x \left (5+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {2+3 x^2}{10 \sqrt {5+x^4}}-\frac {1}{50} \operatorname {Subst}\left (\int -\frac {10}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {2+3 x^2}{10 \sqrt {5+x^4}}+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {2+3 x^2}{10 \sqrt {5+x^4}}+\frac {1}{10} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {5+x}} \, dx,x,x^4\right )\\ &=\frac {2+3 x^2}{10 \sqrt {5+x^4}}+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{-5+x^2} \, dx,x,\sqrt {5+x^4}\right )\\ &=\frac {2+3 x^2}{10 \sqrt {5+x^4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )}{5 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 1.00 \[ \frac {1}{50} \left (\frac {5 \left (3 x^2+2\right )}{\sqrt {x^4+5}}-2 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 61, normalized size = 1.33 \[ \frac {15 \, x^{4} + 2 \, \sqrt {5} {\left (x^{4} + 5\right )} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{x^{2}}\right ) + 5 \, \sqrt {x^{4} + 5} {\left (3 \, x^{2} + 2\right )} + 75}{50 \, {\left (x^{4} + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 61, normalized size = 1.33 \[ \frac {1}{25} \, \sqrt {5} \log \left (x^{2} + \sqrt {5} - \sqrt {x^{4} + 5}\right ) - \frac {1}{25} \, \sqrt {5} \log \left (-x^{2} + \sqrt {5} + \sqrt {x^{4} + 5}\right ) + \frac {3 \, x^{2} + 2}{10 \, \sqrt {x^{4} + 5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 40, normalized size = 0.87 \[ \frac {3 x^{2}}{10 \sqrt {x^{4}+5}}-\frac {\sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{25}+\frac {1}{5 \sqrt {x^{4}+5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 56, normalized size = 1.22 \[ \frac {3 \, x^{2}}{10 \, \sqrt {x^{4} + 5}} + \frac {1}{50} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{\sqrt {5} + \sqrt {x^{4} + 5}}\right ) + \frac {1}{5 \, \sqrt {x^{4} + 5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 40, normalized size = 0.87 \[ \frac {1}{5\,\sqrt {x^4+5}}-\frac {\sqrt {5}\,\mathrm {atanh}\left (\frac {\sqrt {5}\,\sqrt {x^4+5}}{5}\right )}{25}+\frac {3\,x^2}{10\,\sqrt {x^4+5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 19.62, size = 212, normalized size = 4.61 \[ \frac {2 x^{4} \log {\left (x^{4} \right )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} - \frac {4 x^{4} \log {\left (\sqrt {\frac {x^{4}}{5} + 1} + 1 \right )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} - \frac {2 x^{4} \log {\relax (5 )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} + \frac {3 x^{2}}{10 \sqrt {x^{4} + 5}} + \frac {4 \sqrt {5} \sqrt {x^{4} + 5}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} + \frac {10 \log {\left (x^{4} \right )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} - \frac {20 \log {\left (\sqrt {\frac {x^{4}}{5} + 1} + 1 \right )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} - \frac {10 \log {\relax (5 )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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