Optimal. Leaf size=25 \[ \frac{1}{3} \left (x^2+4\right )^{3/2}-4 \sqrt{x^2+4} \]
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Rubi [A] time = 0.0111117, antiderivative size = 25, 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} \[ \frac{1}{3} \left (x^2+4\right )^{3/2}-4 \sqrt{x^2+4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{4+x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{4+x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{4}{\sqrt{4+x}}+\sqrt{4+x}\right ) \, dx,x,x^2\right )\\ &=-4 \sqrt{4+x^2}+\frac{1}{3} \left (4+x^2\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0044458, size = 18, normalized size = 0.72 \[ \frac{1}{3} \left (x^2-8\right ) \sqrt{x^2+4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 15, normalized size = 0.6 \begin{align*}{\frac{{x}^{2}-8}{3}\sqrt{{x}^{2}+4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40537, size = 30, normalized size = 1.2 \begin{align*} \frac{1}{3} \, \sqrt{x^{2} + 4} x^{2} - \frac{8}{3} \, \sqrt{x^{2} + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73206, size = 39, normalized size = 1.56 \begin{align*} \frac{1}{3} \, \sqrt{x^{2} + 4}{\left (x^{2} - 8\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.341551, size = 24, normalized size = 0.96 \begin{align*} \frac{x^{2} \sqrt{x^{2} + 4}}{3} - \frac{8 \sqrt{x^{2} + 4}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04427, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{3} \,{\left (x^{2} + 4\right )}^{\frac{3}{2}} - 4 \, \sqrt{x^{2} + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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