Optimal. Leaf size=79 \[ -\frac{9}{416} (2 x+3)^{13/2}+\frac{15}{32} (2 x+3)^{11/2}-\frac{359}{144} (2 x+3)^{9/2}+\frac{93}{16} (2 x+3)^{7/2}-\frac{213}{32} (2 x+3)^{5/2}+\frac{325}{96} (2 x+3)^{3/2} \]
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Rubi [A] time = 0.0225127, 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} \[ -\frac{9}{416} (2 x+3)^{13/2}+\frac{15}{32} (2 x+3)^{11/2}-\frac{359}{144} (2 x+3)^{9/2}+\frac{93}{16} (2 x+3)^{7/2}-\frac{213}{32} (2 x+3)^{5/2}+\frac{325}{96} (2 x+3)^{3/2} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (5-x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2 \, dx &=\int \left (\frac{325}{32} \sqrt{3+2 x}-\frac{1065}{32} (3+2 x)^{3/2}+\frac{651}{16} (3+2 x)^{5/2}-\frac{359}{16} (3+2 x)^{7/2}+\frac{165}{32} (3+2 x)^{9/2}-\frac{9}{32} (3+2 x)^{11/2}\right ) \, dx\\ &=\frac{325}{96} (3+2 x)^{3/2}-\frac{213}{32} (3+2 x)^{5/2}+\frac{93}{16} (3+2 x)^{7/2}-\frac{359}{144} (3+2 x)^{9/2}+\frac{15}{32} (3+2 x)^{11/2}-\frac{9}{416} (3+2 x)^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0151189, size = 38, normalized size = 0.48 \[ -\frac{1}{117} (2 x+3)^{3/2} \left (81 x^5-270 x^4-1109 x^3-1332 x^2-648 x-132\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 35, normalized size = 0.4 \begin{align*} -{\frac{81\,{x}^{5}-270\,{x}^{4}-1109\,{x}^{3}-1332\,{x}^{2}-648\,x-132}{117} \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 = 0.975527, size = 74, normalized size = 0.94 \begin{align*} -\frac{9}{416} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{15}{32} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{359}{144} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{93}{16} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{213}{32} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{325}{96} \,{\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 = 1.67844, size = 122, normalized size = 1.54 \begin{align*} -\frac{1}{117} \,{\left (162 \, x^{6} - 297 \, x^{5} - 3028 \, x^{4} - 5991 \, x^{3} - 5292 \, x^{2} - 2208 \, x - 396\right )} \sqrt{2 \, x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.82986, size = 70, normalized size = 0.89 \begin{align*} - \frac{9 \left (2 x + 3\right )^{\frac{13}{2}}}{416} + \frac{15 \left (2 x + 3\right )^{\frac{11}{2}}}{32} - \frac{359 \left (2 x + 3\right )^{\frac{9}{2}}}{144} + \frac{93 \left (2 x + 3\right )^{\frac{7}{2}}}{16} - \frac{213 \left (2 x + 3\right )^{\frac{5}{2}}}{32} + \frac{325 \left (2 x + 3\right )^{\frac{3}{2}}}{96} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11675, size = 74, normalized size = 0.94 \begin{align*} -\frac{9}{416} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{15}{32} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{359}{144} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{93}{16} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{213}{32} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{325}{96} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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