Optimal. Leaf size=53 \[ -\frac{1}{8} (2 x+3)^{3/2}+\frac{47}{8} \sqrt{2 x+3}+\frac{109}{8 \sqrt{2 x+3}}-\frac{65}{24 (2 x+3)^{3/2}} \]
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Rubi [A] time = 0.0162295, 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} \[ -\frac{1}{8} (2 x+3)^{3/2}+\frac{47}{8} \sqrt{2 x+3}+\frac{109}{8 \sqrt{2 x+3}}-\frac{65}{24 (2 x+3)^{3/2}} \]
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)^{5/2}} \, dx &=\int \left (\frac{65}{8 (3+2 x)^{5/2}}-\frac{109}{8 (3+2 x)^{3/2}}+\frac{47}{8 \sqrt{3+2 x}}-\frac{3}{8} \sqrt{3+2 x}\right ) \, dx\\ &=-\frac{65}{24 (3+2 x)^{3/2}}+\frac{109}{8 \sqrt{3+2 x}}+\frac{47}{8} \sqrt{3+2 x}-\frac{1}{8} (3+2 x)^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0122776, size = 28, normalized size = 0.53 \[ -\frac{3 x^3-57 x^2-273 x-263}{3 (2 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{3\,{x}^{3}-57\,{x}^{2}-273\,x-263}{3} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983456, size = 45, normalized size = 0.85 \begin{align*} -\frac{1}{8} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{47}{8} \, \sqrt{2 \, x + 3} + \frac{327 \, x + 458}{12 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74869, size = 95, normalized size = 1.79 \begin{align*} -\frac{{\left (3 \, x^{3} - 57 \, x^{2} - 273 \, x - 263\right )} \sqrt{2 \, x + 3}}{3 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.687744, size = 102, normalized size = 1.92 \begin{align*} - \frac{3 x^{3}}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} + \frac{57 x^{2}}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} + \frac{273 x}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} + \frac{263}{6 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11615, size = 45, normalized size = 0.85 \begin{align*} -\frac{1}{8} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{47}{8} \, \sqrt{2 \, x + 3} + \frac{327 \, x + 458}{12 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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