Optimal. Leaf size=42 \[ \frac{1}{20} \left (4 x^2+12 x+9\right )^{5/2}-\frac{3}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} \]
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Rubi [A] time = 0.008506, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {640, 609} \[ \frac{1}{20} \left (4 x^2+12 x+9\right )^{5/2}-\frac{3}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 640
Rule 609
Rubi steps
\begin{align*} \int x \left (9+12 x+4 x^2\right )^{3/2} \, dx &=\frac{1}{20} \left (9+12 x+4 x^2\right )^{5/2}-\frac{3}{2} \int \left (9+12 x+4 x^2\right )^{3/2} \, dx\\ &=-\frac{3}{16} (3+2 x) \left (9+12 x+4 x^2\right )^{3/2}+\frac{1}{20} \left (9+12 x+4 x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0116702, size = 37, normalized size = 0.88 \[ \frac{x^2 \sqrt{(2 x+3)^2} \left (16 x^3+90 x^2+180 x+135\right )}{20 x+30} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 37, normalized size = 0.9 \begin{align*}{\frac{{x}^{2} \left ( 16\,{x}^{3}+90\,{x}^{2}+180\,x+135 \right ) }{10\, \left ( 3+2\,x \right ) ^{3}} \left ( \left ( 3+2\,x \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.71987, size = 59, normalized size = 1.4 \begin{align*} \frac{1}{20} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} - \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} x - \frac{9}{16} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65142, size = 50, normalized size = 1.19 \begin{align*} \frac{8}{5} \, x^{5} + 9 \, x^{4} + 18 \, x^{3} + \frac{27}{2} \, x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (\left (2 x + 3\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26485, size = 72, normalized size = 1.71 \begin{align*} \frac{8}{5} \, x^{5} \mathrm{sgn}\left (2 \, x + 3\right ) + 9 \, x^{4} \mathrm{sgn}\left (2 \, x + 3\right ) + 18 \, x^{3} \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{27}{2} \, x^{2} \mathrm{sgn}\left (2 \, x + 3\right ) - \frac{243}{80} \, \mathrm{sgn}\left (2 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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