Optimal. Leaf size=79 \[ -\frac{1}{32} (2 x+3)^{9/2}+\frac{165}{224} (2 x+3)^{7/2}-\frac{359}{80} (2 x+3)^{5/2}+\frac{217}{16} (2 x+3)^{3/2}-\frac{1065}{32} \sqrt{2 x+3}-\frac{325}{32 \sqrt{2 x+3}} \]
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Rubi [A] time = 0.0328533, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {771} \[ -\frac{1}{32} (2 x+3)^{9/2}+\frac{165}{224} (2 x+3)^{7/2}-\frac{359}{80} (2 x+3)^{5/2}+\frac{217}{16} (2 x+3)^{3/2}-\frac{1065}{32} \sqrt{2 x+3}-\frac{325}{32 \sqrt{2 x+3}} \]
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 )^2}{(3+2 x)^{3/2}} \, dx &=\int \left (\frac{325}{32 (3+2 x)^{3/2}}-\frac{1065}{32 \sqrt{3+2 x}}+\frac{651}{16} \sqrt{3+2 x}-\frac{359}{16} (3+2 x)^{3/2}+\frac{165}{32} (3+2 x)^{5/2}-\frac{9}{32} (3+2 x)^{7/2}\right ) \, dx\\ &=-\frac{325}{32 \sqrt{3+2 x}}-\frac{1065}{32} \sqrt{3+2 x}+\frac{217}{16} (3+2 x)^{3/2}-\frac{359}{80} (3+2 x)^{5/2}+\frac{165}{224} (3+2 x)^{7/2}-\frac{1}{32} (3+2 x)^{9/2}\\ \end{align*}
Mathematica [A] time = 0.0162905, size = 38, normalized size = 0.48 \[ -\frac{35 x^5-150 x^4-431 x^3-632 x^2+432 x+1996}{35 \sqrt{2 x+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 35, normalized size = 0.4 \begin{align*} -{\frac{35\,{x}^{5}-150\,{x}^{4}-431\,{x}^{3}-632\,{x}^{2}+432\,x+1996}{35}{\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 = 0.991004, size = 74, normalized size = 0.94 \begin{align*} -\frac{1}{32} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{165}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{359}{80} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{217}{16} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1065}{32} \, \sqrt{2 \, x + 3} - \frac{325}{32 \, \sqrt{2 \, x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7479, size = 101, normalized size = 1.28 \begin{align*} -\frac{35 \, x^{5} - 150 \, x^{4} - 431 \, x^{3} - 632 \, x^{2} + 432 \, x + 1996}{35 \, \sqrt{2 \, x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 27.979, size = 68, normalized size = 0.86 \begin{align*} - \frac{\left (2 x + 3\right )^{\frac{9}{2}}}{32} + \frac{165 \left (2 x + 3\right )^{\frac{7}{2}}}{224} - \frac{359 \left (2 x + 3\right )^{\frac{5}{2}}}{80} + \frac{217 \left (2 x + 3\right )^{\frac{3}{2}}}{16} - \frac{1065 \sqrt{2 x + 3}}{32} - \frac{325}{32 \sqrt{2 x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14232, size = 74, normalized size = 0.94 \begin{align*} -\frac{1}{32} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{165}{224} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{359}{80} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{217}{16} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1065}{32} \, \sqrt{2 \, x + 3} - \frac{325}{32 \, \sqrt{2 \, x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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