Optimal. Leaf size=31 \[ \frac{1}{80} \left (4 x^2+9\right )^{5/2}-\frac{3}{16} \left (4 x^2+9\right )^{3/2} \]
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Rubi [A] time = 0.014846, 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} \[ \frac{1}{80} \left (4 x^2+9\right )^{5/2}-\frac{3}{16} \left (4 x^2+9\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \sqrt{9+4 x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \sqrt{9+4 x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{9}{4} \sqrt{9+4 x}+\frac{1}{4} (9+4 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{3}{16} \left (9+4 x^2\right )^{3/2}+\frac{1}{80} \left (9+4 x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.006506, size = 22, normalized size = 0.71 \[ \frac{1}{40} \left (2 x^2-3\right ) \left (4 x^2+9\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 0.6 \begin{align*}{\frac{2\,{x}^{2}-3}{40} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.28448, size = 35, normalized size = 1.13 \begin{align*} \frac{1}{20} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} x^{2} - \frac{3}{40} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66133, size = 58, normalized size = 1.87 \begin{align*} \frac{1}{40} \,{\left (8 \, x^{4} + 6 \, x^{2} - 27\right )} \sqrt{4 \, x^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.584487, size = 44, normalized size = 1.42 \begin{align*} \frac{x^{4} \sqrt{4 x^{2} + 9}}{5} + \frac{3 x^{2} \sqrt{4 x^{2} + 9}}{20} - \frac{27 \sqrt{4 x^{2} + 9}}{40} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.61329, size = 31, normalized size = 1. \begin{align*} \frac{1}{80} \,{\left (4 \, x^{2} + 9\right )}^{\frac{5}{2}} - \frac{3}{16} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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