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 = 0.0129593, antiderivative size = 38, normalized size of antiderivative = 1., 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 266
Rule 43
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*}
Mathematica [A] time = 0.0063945, 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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Maple [A] time = 0.003, size = 22, normalized size = 0.6 \begin{align*}{\frac{3\,{x}^{4}-20\,{x}^{2}+200}{15}\sqrt{{x}^{2}+5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43064, size = 46, normalized size = 1.21 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.425337, size = 58, normalized size = 1.53 \begin{align*} \frac{1}{15} \,{\left (3 \, x^{4} - 20 \, x^{2} + 200\right )} \sqrt{x^{2} + 5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.24128, size = 39, normalized size = 1.03 \begin{align*} \frac{x^{4} \sqrt{x^{2} + 5}}{5} - \frac{4 x^{2} \sqrt{x^{2} + 5}}{3} + \frac{40 \sqrt{x^{2} + 5}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08353, size = 38, normalized size = 1. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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