Optimal. Leaf size=35 \[ \frac {-2-3 x^2}{2 \sqrt {5+x^4}}+\frac {3}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1266, 792, 221}
\begin {gather*} \frac {3}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {3 x^2+2}{2 \sqrt {x^4+5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 792
Rule 1266
Rubi steps
\begin {align*} \int \frac {x^3 \left (2+3 x^2\right )}{\left (5+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x (2+3 x)}{\left (5+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {2+3 x^2}{2 \sqrt {5+x^4}}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {2+3 x^2}{2 \sqrt {5+x^4}}+\frac {3}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 39, normalized size = 1.11 \begin {gather*} \frac {-2-3 x^2}{2 \sqrt {5+x^4}}+\frac {3}{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 = 0.15, size = 34, normalized size = 0.97
method | result | size |
risch | \(-\frac {3 x^{2}+2}{2 \sqrt {x^{4}+5}}+\frac {3 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}\) | \(29\) |
trager | \(-\frac {3 x^{2}+2}{2 \sqrt {x^{4}+5}}+\frac {3 \ln \left (x^{2}+\sqrt {x^{4}+5}\right )}{2}\) | \(32\) |
default | \(-\frac {3 x^{2}}{2 \sqrt {x^{4}+5}}+\frac {3 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}-\frac {1}{\sqrt {x^{4}+5}}\) | \(34\) |
elliptic | \(-\frac {3 x^{2}}{2 \sqrt {x^{4}+5}}+\frac {3 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}-\frac {1}{\sqrt {x^{4}+5}}\) | \(34\) |
meijerg | \(\frac {-\frac {3 \sqrt {\pi }\, x^{2} \sqrt {5}}{10 \sqrt {1+\frac {x^{4}}{5}}}+\frac {3 \sqrt {\pi }\, \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}}{\sqrt {\pi }}+\frac {\sqrt {5}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1+\frac {x^{4}}{5}}}\right )}{5 \sqrt {\pi }}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 54, normalized size = 1.54 \begin {gather*} -\frac {3 \, x^{2}}{2 \, \sqrt {x^{4} + 5}} - \frac {1}{\sqrt {x^{4} + 5}} + \frac {3}{4} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) - \frac {3}{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 = 0.38, size = 52, normalized size = 1.49 \begin {gather*} -\frac {3 \, x^{4} + 3 \, {\left (x^{4} + 5\right )} \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) + \sqrt {x^{4} + 5} {\left (3 \, x^{2} + 2\right )} + 15}{2 \, {\left (x^{4} + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.42, size = 39, normalized size = 1.11 \begin {gather*} - \frac {3 x^{2}}{2 \sqrt {x^{4} + 5}} + \frac {3 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{2} - \frac {1}{\sqrt {x^{4} + 5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.32, size = 33, normalized size = 0.94 \begin {gather*} -\frac {3 \, x^{2} + 2}{2 \, \sqrt {x^{4} + 5}} - \frac {3}{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 = 0.84, size = 82, normalized size = 2.34 \begin {gather*} \frac {3\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{2}-\frac {\sqrt {5}\,\left (2+\sqrt {5}\,3{}\mathrm {i}\right )\,\sqrt {x^4+5}\,1{}\mathrm {i}}{20\,\left (-x^2+\sqrt {5}\,1{}\mathrm {i}\right )}+\frac {\sqrt {5}\,\left (-2+\sqrt {5}\,3{}\mathrm {i}\right )\,\sqrt {x^4+5}\,1{}\mathrm {i}}{20\,\left (x^2+\sqrt {5}\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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