Optimal. Leaf size=44 \[ \frac {1}{2} \left (x^4+5\right )^{3/2}+\frac {5}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {1}{2} \sqrt {x^4+5} x^2 \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1248, 641, 195, 215} \[ \frac {1}{2} \sqrt {x^4+5} x^2+\frac {1}{2} \left (x^4+5\right )^{3/2}+\frac {5}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 641
Rule 1248
Rubi steps
\begin {align*} \int x \left (2+3 x^2\right ) \sqrt {5+x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (2+3 x) \sqrt {5+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \left (5+x^4\right )^{3/2}+\operatorname {Subst}\left (\int \sqrt {5+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^2 \sqrt {5+x^4}+\frac {1}{2} \left (5+x^4\right )^{3/2}+\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^2 \sqrt {5+x^4}+\frac {1}{2} \left (5+x^4\right )^{3/2}+\frac {5}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 36, normalized size = 0.82 \[ \frac {5}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {1}{2} \sqrt {x^4+5} \left (x^4+x^2+5\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 34, normalized size = 0.77 \[ \frac {1}{2} \, {\left (x^{4} + x^{2} + 5\right )} \sqrt {x^{4} + 5} - \frac {5}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 38, normalized size = 0.86 \[ \frac {1}{2} \, \sqrt {x^{4} + 5} x^{2} + \frac {1}{2} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} - \frac {5}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 34, normalized size = 0.77 \[ \frac {\sqrt {x^{4}+5}\, x^{2}}{2}+\frac {5 \arcsinh \left (\frac {\sqrt {5}\, x^{2}}{5}\right )}{2}+\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.32, size = 67, normalized size = 1.52 \[ \frac {1}{2} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} + \frac {5 \, \sqrt {x^{4} + 5}}{2 \, x^{2} {\left (\frac {x^{4} + 5}{x^{4}} - 1\right )}} + \frac {5}{4} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) - \frac {5}{4} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 32, normalized size = 0.73 \[ \frac {5\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{2}+\sqrt {x^4+5}\,\left (\frac {x^4}{2}+\frac {x^2}{2}+\frac {5}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.08, size = 53, normalized size = 1.20 \[ \frac {x^{6}}{2 \sqrt {x^{4} + 5}} + \frac {5 x^{2}}{2 \sqrt {x^{4} + 5}} + \frac {\left (x^{4} + 5\right )^{\frac {3}{2}}}{2} + \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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