Optimal. Leaf size=105 \[ -\frac{27 (2 x+3)^{17/2}}{2176}+\frac{189}{640} (2 x+3)^{15/2}-\frac{3519 (2 x+3)^{13/2}}{1664}+\frac{10475 (2 x+3)^{11/2}}{1408}-\frac{17201 (2 x+3)^{9/2}}{1152}+\frac{16005}{896} (2 x+3)^{7/2}-\frac{1585}{128} (2 x+3)^{5/2}+\frac{1625}{384} (2 x+3)^{3/2} \]
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Rubi [A] time = 0.0341499, antiderivative size = 105, 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} \[ -\frac{27 (2 x+3)^{17/2}}{2176}+\frac{189}{640} (2 x+3)^{15/2}-\frac{3519 (2 x+3)^{13/2}}{1664}+\frac{10475 (2 x+3)^{11/2}}{1408}-\frac{17201 (2 x+3)^{9/2}}{1152}+\frac{16005}{896} (2 x+3)^{7/2}-\frac{1585}{128} (2 x+3)^{5/2}+\frac{1625}{384} (2 x+3)^{3/2} \]
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 )^3 \, dx &=\int \left (\frac{1625}{128} \sqrt{3+2 x}-\frac{7925}{128} (3+2 x)^{3/2}+\frac{16005}{128} (3+2 x)^{5/2}-\frac{17201}{128} (3+2 x)^{7/2}+\frac{10475}{128} (3+2 x)^{9/2}-\frac{3519}{128} (3+2 x)^{11/2}+\frac{567}{128} (3+2 x)^{13/2}-\frac{27}{128} (3+2 x)^{15/2}\right ) \, dx\\ &=\frac{1625}{384} (3+2 x)^{3/2}-\frac{1585}{128} (3+2 x)^{5/2}+\frac{16005}{896} (3+2 x)^{7/2}-\frac{17201 (3+2 x)^{9/2}}{1152}+\frac{10475 (3+2 x)^{11/2}}{1408}-\frac{3519 (3+2 x)^{13/2}}{1664}+\frac{189}{640} (3+2 x)^{15/2}-\frac{27 (3+2 x)^{17/2}}{2176}\\ \end{align*}
Mathematica [A] time = 0.018437, size = 48, normalized size = 0.46 \[ -\frac{(2 x+3)^{3/2} \left (1216215 x^7-1702701 x^6-20968794 x^5-47286540 x^4-50880095 x^3-29756385 x^2-9013014 x-1197186\right )}{765765} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 45, normalized size = 0.4 \begin{align*} -{\frac{1216215\,{x}^{7}-1702701\,{x}^{6}-20968794\,{x}^{5}-47286540\,{x}^{4}-50880095\,{x}^{3}-29756385\,{x}^{2}-9013014\,x-1197186}{765765} \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 = 0.962201, size = 99, normalized size = 0.94 \begin{align*} -\frac{27}{2176} \,{\left (2 \, x + 3\right )}^{\frac{17}{2}} + \frac{189}{640} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} - \frac{3519}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{10475}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{17201}{1152} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{16005}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{1585}{128} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{1625}{384} \,{\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 = 1.8537, size = 208, normalized size = 1.98 \begin{align*} -\frac{1}{765765} \,{\left (2432430 \, x^{8} + 243243 \, x^{7} - 47045691 \, x^{6} - 157479462 \, x^{5} - 243619810 \, x^{4} - 212153055 \, x^{3} - 107295183 \, x^{2} - 29433414 \, x - 3591558\right )} \sqrt{2 \, x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.42181, size = 94, normalized size = 0.9 \begin{align*} - \frac{27 \left (2 x + 3\right )^{\frac{17}{2}}}{2176} + \frac{189 \left (2 x + 3\right )^{\frac{15}{2}}}{640} - \frac{3519 \left (2 x + 3\right )^{\frac{13}{2}}}{1664} + \frac{10475 \left (2 x + 3\right )^{\frac{11}{2}}}{1408} - \frac{17201 \left (2 x + 3\right )^{\frac{9}{2}}}{1152} + \frac{16005 \left (2 x + 3\right )^{\frac{7}{2}}}{896} - \frac{1585 \left (2 x + 3\right )^{\frac{5}{2}}}{128} + \frac{1625 \left (2 x + 3\right )^{\frac{3}{2}}}{384} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10085, size = 99, normalized size = 0.94 \begin{align*} -\frac{27}{2176} \,{\left (2 \, x + 3\right )}^{\frac{17}{2}} + \frac{189}{640} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} - \frac{3519}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{10475}{1408} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{17201}{1152} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{16005}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - \frac{1585}{128} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{1625}{384} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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