Optimal. Leaf size=53 \[ -\frac{4}{567} (3 x+2)^{7/2}+\frac{8}{135} (3 x+2)^{5/2}-\frac{10}{81} (3 x+2)^{3/2}-\frac{4}{81} \sqrt{3 x+2} \]
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Rubi [A] time = 0.0208054, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1593, 772} \[ -\frac{4}{567} (3 x+2)^{7/2}+\frac{8}{135} (3 x+2)^{5/2}-\frac{10}{81} (3 x+2)^{3/2}-\frac{4}{81} \sqrt{3 x+2} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 772
Rubi steps
\begin{align*} \int \frac{x-2 x^3}{\sqrt{2+3 x}} \, dx &=\int \frac{x \left (1-2 x^2\right )}{\sqrt{2+3 x}} \, dx\\ &=\int \left (-\frac{2}{27 \sqrt{2+3 x}}-\frac{5}{9} \sqrt{2+3 x}+\frac{4}{9} (2+3 x)^{3/2}-\frac{2}{27} (2+3 x)^{5/2}\right ) \, dx\\ &=-\frac{4}{81} \sqrt{2+3 x}-\frac{10}{81} (2+3 x)^{3/2}+\frac{8}{135} (2+3 x)^{5/2}-\frac{4}{567} (2+3 x)^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0250221, size = 28, normalized size = 0.53 \[ -\frac{2 \sqrt{3 x+2} \left (270 x^3-216 x^2-123 x+164\right )}{2835} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 25, normalized size = 0.5 \begin{align*} -{\frac{540\,{x}^{3}-432\,{x}^{2}-246\,x+328}{2835}\sqrt{2+3\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0009, size = 50, normalized size = 0.94 \begin{align*} -\frac{4}{567} \,{\left (3 \, x + 2\right )}^{\frac{7}{2}} + \frac{8}{135} \,{\left (3 \, x + 2\right )}^{\frac{5}{2}} - \frac{10}{81} \,{\left (3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{81} \, \sqrt{3 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44073, size = 77, normalized size = 1.45 \begin{align*} -\frac{2}{2835} \,{\left (270 \, x^{3} - 216 \, x^{2} - 123 \, x + 164\right )} \sqrt{3 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.6686, size = 46, normalized size = 0.87 \begin{align*} - \frac{4 \left (3 x + 2\right )^{\frac{7}{2}}}{567} + \frac{8 \left (3 x + 2\right )^{\frac{5}{2}}}{135} - \frac{10 \left (3 x + 2\right )^{\frac{3}{2}}}{81} - \frac{4 \sqrt{3 x + 2}}{81} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1018, size = 50, normalized size = 0.94 \begin{align*} -\frac{4}{567} \,{\left (3 \, x + 2\right )}^{\frac{7}{2}} + \frac{8}{135} \,{\left (3 \, x + 2\right )}^{\frac{5}{2}} - \frac{10}{81} \,{\left (3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{81} \, \sqrt{3 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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