Optimal. Leaf size=53 \[ -\frac{3}{40} (2 x+3)^{5/2}+\frac{47}{24} (2 x+3)^{3/2}-\frac{109}{8} \sqrt{2 x+3}-\frac{65}{8 \sqrt{2 x+3}} \]
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Rubi [A] time = 0.015937, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {771} \[ -\frac{3}{40} (2 x+3)^{5/2}+\frac{47}{24} (2 x+3)^{3/2}-\frac{109}{8} \sqrt{2 x+3}-\frac{65}{8 \sqrt{2 x+3}} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx &=\int \left (\frac{65}{8 (3+2 x)^{3/2}}-\frac{109}{8 \sqrt{3+2 x}}+\frac{47}{8} \sqrt{3+2 x}-\frac{3}{8} (3+2 x)^{3/2}\right ) \, dx\\ &=-\frac{65}{8 \sqrt{3+2 x}}-\frac{109}{8} \sqrt{3+2 x}+\frac{47}{24} (3+2 x)^{3/2}-\frac{3}{40} (3+2 x)^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0120587, size = 28, normalized size = 0.53 \[ -\frac{9 x^3-77 x^2+117 x+501}{15 \sqrt{2 x+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{9\,{x}^{3}-77\,{x}^{2}+117\,x+501}{15}{\frac{1}{\sqrt{3+2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02658, size = 50, normalized size = 0.94 \begin{align*} -\frac{3}{40} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{47}{24} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{109}{8} \, \sqrt{2 \, x + 3} - \frac{65}{8 \, \sqrt{2 \, x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75775, size = 70, normalized size = 1.32 \begin{align*} -\frac{9 \, x^{3} - 77 \, x^{2} + 117 \, x + 501}{15 \, \sqrt{2 \, x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.0716, size = 46, normalized size = 0.87 \begin{align*} - \frac{3 \left (2 x + 3\right )^{\frac{5}{2}}}{40} + \frac{47 \left (2 x + 3\right )^{\frac{3}{2}}}{24} - \frac{109 \sqrt{2 x + 3}}{8} - \frac{65}{8 \sqrt{2 x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09219, size = 50, normalized size = 0.94 \begin{align*} -\frac{3}{40} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{47}{24} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{109}{8} \, \sqrt{2 \, x + 3} - \frac{65}{8 \, \sqrt{2 \, x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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