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 = 0.01, antiderivative size = 25, 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} \[ \frac {1}{3} \left (x^2+4\right )^{3/2}-4 \sqrt {x^2+4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.41, size = 14, normalized size = 0.56 \[ \frac {1}{3} \, \sqrt {x^{2} + 4} {\left (x^{2} - 8\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 19, normalized size = 0.76 \[ \frac {1}{3} \, {\left (x^{2} + 4\right )}^{\frac {3}{2}} - 4 \, \sqrt {x^{2} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.60 \[ \frac {\sqrt {x^{2}+4}\, \left (x^{2}-8\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 22, normalized size = 0.88 \[ \frac {1}{3} \, \sqrt {x^{2} + 4} x^{2} - \frac {8}{3} \, \sqrt {x^{2} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 14, normalized size = 0.56 \[ \frac {\sqrt {x^2+4}\,\left (x^2-8\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 24, normalized size = 0.96 \[ \frac {x^{2} \sqrt {x^{2} + 4}}{3} - \frac {8 \sqrt {x^{2} + 4}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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