Optimal. Leaf size=31 \[ \frac{1}{405} \left (4-9 x^2\right )^{5/2}-\frac{4}{243} \left (4-9 x^2\right )^{3/2} \]
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Rubi [A] time = 0.0139225, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{1}{405} \left (4-9 x^2\right )^{5/2}-\frac{4}{243} \left (4-9 x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \sqrt{4-9 x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{4-9 x} x \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{4}{9} \sqrt{4-9 x}-\frac{1}{9} (4-9 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{4}{243} \left (4-9 x^2\right )^{3/2}+\frac{1}{405} \left (4-9 x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0105125, size = 22, normalized size = 0.71 \[ -\frac{\left (4-9 x^2\right )^{3/2} \left (27 x^2+8\right )}{1215} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 29, normalized size = 0.9 \begin{align*}{\frac{ \left ( -2+3\,x \right ) \left ( 2+3\,x \right ) \left ( 27\,{x}^{2}+8 \right ) }{1215}\sqrt{-9\,{x}^{2}+4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.39839, size = 35, normalized size = 1.13 \begin{align*} -\frac{1}{45} \,{\left (-9 \, x^{2} + 4\right )}^{\frac{3}{2}} x^{2} - \frac{8}{1215} \,{\left (-9 \, x^{2} + 4\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.56603, size = 66, normalized size = 2.13 \begin{align*} \frac{1}{1215} \,{\left (243 \, x^{4} - 36 \, x^{2} - 32\right )} \sqrt{-9 \, x^{2} + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.592645, size = 44, normalized size = 1.42 \begin{align*} \frac{x^{4} \sqrt{4 - 9 x^{2}}}{5} - \frac{4 x^{2} \sqrt{4 - 9 x^{2}}}{135} - \frac{32 \sqrt{4 - 9 x^{2}}}{1215} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06412, size = 43, normalized size = 1.39 \begin{align*} \frac{1}{405} \,{\left (9 \, x^{2} - 4\right )}^{2} \sqrt{-9 \, x^{2} + 4} - \frac{4}{243} \,{\left (-9 \, x^{2} + 4\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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