Optimal. Leaf size=38 \[ \frac {1}{5} \left (x^2+5\right )^{5/2}-\frac {10}{3} \left (x^2+5\right )^{3/2}+25 \sqrt {x^2+5} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {1}{5} \left (x^2+5\right )^{5/2}-\frac {10}{3} \left (x^2+5\right )^{3/2}+25 \sqrt {x^2+5} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {5+x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {5+x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {25}{\sqrt {5+x}}-10 \sqrt {5+x}+(5+x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=25 \sqrt {5+x^2}-\frac {10}{3} \left (5+x^2\right )^{3/2}+\frac {1}{5} \left (5+x^2\right )^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.66 \[ \frac {1}{15} \sqrt {x^2+5} \left (3 x^4-20 x^2+200\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 21, normalized size = 0.55 \[ \frac {1}{15} \, {\left (3 \, x^{4} - 20 \, x^{2} + 200\right )} \sqrt {x^{2} + 5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 28, normalized size = 0.74 \[ \frac {1}{5} \, {\left (x^{2} + 5\right )}^{\frac {5}{2}} - \frac {10}{3} \, {\left (x^{2} + 5\right )}^{\frac {3}{2}} + 25 \, \sqrt {x^{2} + 5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 0.58 \[ \frac {\sqrt {x^{2}+5}\, \left (3 x^{4}-20 x^{2}+200\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 34, normalized size = 0.89 \[ \frac {1}{5} \, \sqrt {x^{2} + 5} x^{4} - \frac {4}{3} \, \sqrt {x^{2} + 5} x^{2} + \frac {40}{3} \, \sqrt {x^{2} + 5} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 20, normalized size = 0.53 \[ \sqrt {x^2+5}\,\left (\frac {x^4}{5}-\frac {4\,x^2}{3}+\frac {40}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 39, normalized size = 1.03 \[ \frac {x^{4} \sqrt {x^{2} + 5}}{5} - \frac {4 x^{2} \sqrt {x^{2} + 5}}{3} + \frac {40 \sqrt {x^{2} + 5}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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