Optimal. Leaf size=35 \[ \frac {1}{4} \left (3 x^2+4\right ) \sqrt {x^4+5}-\frac {15}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
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Rubi [A] time = 0.03, 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 = {1252, 780, 215} \[ \frac {1}{4} \left (3 x^2+4\right ) \sqrt {x^4+5}-\frac {15}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 780
Rule 1252
Rubi steps
\begin {align*} \int \frac {x^3 \left (2+3 x^2\right )}{\sqrt {5+x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x (2+3 x)}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{4} \left (4+3 x^2\right ) \sqrt {5+x^4}-\frac {15}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{4} \left (4+3 x^2\right ) \sqrt {5+x^4}-\frac {15}{4} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 0.97 \[ \frac {1}{4} \left (\left (3 x^2+4\right ) \sqrt {x^4+5}-15 \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 33, normalized size = 0.94 \[ \frac {1}{4} \, \sqrt {x^{4} + 5} {\left (3 \, x^{2} + 4\right )} + \frac {15}{4} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 33, normalized size = 0.94 \[ \frac {1}{4} \, \sqrt {x^{4} + 5} {\left (3 \, x^{2} + 4\right )} + \frac {15}{4} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.91 \[ \frac {3 \sqrt {x^{4}+5}\, x^{2}}{4}-\frac {15 \arcsinh \left (\frac {\sqrt {5}\, x^{2}}{5}\right )}{4}+\sqrt {x^{4}+5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.16, size = 65, normalized size = 1.86 \[ \sqrt {x^{4} + 5} + \frac {15 \, \sqrt {x^{4} + 5}}{4 \, x^{2} {\left (\frac {x^{4} + 5}{x^{4}} - 1\right )}} - \frac {15}{8} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) + \frac {15}{8} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 27, normalized size = 0.77 \[ \sqrt {x^4+5}\,\left (\frac {3\,x^2}{4}+1\right )-\frac {15\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.04, size = 53, normalized size = 1.51 \[ \frac {3 x^{6}}{4 \sqrt {x^{4} + 5}} + \frac {15 x^{2}}{4 \sqrt {x^{4} + 5}} + \sqrt {x^{4} + 5} - \frac {15 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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