Optimal. Leaf size=105 \[ -\frac {9}{640} (2 x+3)^{15/2}+\frac {567 (2 x+3)^{13/2}}{1664}-\frac {3519 (2 x+3)^{11/2}}{1408}+\frac {10475 (2 x+3)^{9/2}}{1152}-\frac {17201}{896} (2 x+3)^{7/2}+\frac {3201}{128} (2 x+3)^{5/2}-\frac {7925}{384} (2 x+3)^{3/2}+\frac {1625}{128} \sqrt {2 x+3} \]
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Rubi [A] time = 0.03, antiderivative size = 105, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {9}{640} (2 x+3)^{15/2}+\frac {567 (2 x+3)^{13/2}}{1664}-\frac {3519 (2 x+3)^{11/2}}{1408}+\frac {10475 (2 x+3)^{9/2}}{1152}-\frac {17201}{896} (2 x+3)^{7/2}+\frac {3201}{128} (2 x+3)^{5/2}-\frac {7925}{384} (2 x+3)^{3/2}+\frac {1625}{128} \sqrt {2 x+3} \end {gather*}
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}{\sqrt {3+2 x}} \, dx &=\int \left (\frac {1625}{128 \sqrt {3+2 x}}-\frac {7925}{128} \sqrt {3+2 x}+\frac {16005}{128} (3+2 x)^{3/2}-\frac {17201}{128} (3+2 x)^{5/2}+\frac {10475}{128} (3+2 x)^{7/2}-\frac {3519}{128} (3+2 x)^{9/2}+\frac {567}{128} (3+2 x)^{11/2}-\frac {27}{128} (3+2 x)^{13/2}\right ) \, dx\\ &=\frac {1625}{128} \sqrt {3+2 x}-\frac {7925}{384} (3+2 x)^{3/2}+\frac {3201}{128} (3+2 x)^{5/2}-\frac {17201}{896} (3+2 x)^{7/2}+\frac {10475 (3+2 x)^{9/2}}{1152}-\frac {3519 (3+2 x)^{11/2}}{1408}+\frac {567 (3+2 x)^{13/2}}{1664}-\frac {9}{640} (3+2 x)^{15/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.46 \begin {gather*} -\frac {\sqrt {2 x+3} \left (81081 x^7-130977 x^6-1407294 x^5-3109960 x^4-3285105 x^3-1924641 x^2-535098 x-196506\right )}{45045} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 93, normalized size = 0.89 \begin {gather*} \frac {-81081 (2 x+3)^{15/2}+1964655 (2 x+3)^{13/2}-14410305 (2 x+3)^{11/2}+52427375 (2 x+3)^{9/2}-110688435 (2 x+3)^{7/2}+144189045 (2 x+3)^{5/2}-118993875 (2 x+3)^{3/2}+73198125 \sqrt {2 x+3}}{5765760} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 44, normalized size = 0.42 \begin {gather*} -\frac {1}{45045} \, {\left (81081 \, x^{7} - 130977 \, x^{6} - 1407294 \, x^{5} - 3109960 \, x^{4} - 3285105 \, x^{3} - 1924641 \, x^{2} - 535098 \, x - 196506\right )} \sqrt {2 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 73, normalized size = 0.70 \begin {gather*} -\frac {9}{640} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {567}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {3519}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {10475}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {17201}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {3201}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {7925}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {1625}{128} \, \sqrt {2 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 0.43 \begin {gather*} -\frac {\left (81081 x^{7}-130977 x^{6}-1407294 x^{5}-3109960 x^{4}-3285105 x^{3}-1924641 x^{2}-535098 x -196506\right ) \sqrt {2 x +3}}{45045} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 73, normalized size = 0.70 \begin {gather*} -\frac {9}{640} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {567}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {3519}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {10475}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {17201}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {3201}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {7925}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {1625}{128} \, \sqrt {2 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 73, normalized size = 0.70 \begin {gather*} \frac {1625\,\sqrt {2\,x+3}}{128}-\frac {7925\,{\left (2\,x+3\right )}^{3/2}}{384}+\frac {3201\,{\left (2\,x+3\right )}^{5/2}}{128}-\frac {17201\,{\left (2\,x+3\right )}^{7/2}}{896}+\frac {10475\,{\left (2\,x+3\right )}^{9/2}}{1152}-\frac {3519\,{\left (2\,x+3\right )}^{11/2}}{1408}+\frac {567\,{\left (2\,x+3\right )}^{13/2}}{1664}-\frac {9\,{\left (2\,x+3\right )}^{15/2}}{640} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 124.64, size = 94, normalized size = 0.90 \begin {gather*} - \frac {9 \left (2 x + 3\right )^{\frac {15}{2}}}{640} + \frac {567 \left (2 x + 3\right )^{\frac {13}{2}}}{1664} - \frac {3519 \left (2 x + 3\right )^{\frac {11}{2}}}{1408} + \frac {10475 \left (2 x + 3\right )^{\frac {9}{2}}}{1152} - \frac {17201 \left (2 x + 3\right )^{\frac {7}{2}}}{896} + \frac {3201 \left (2 x + 3\right )^{\frac {5}{2}}}{128} - \frac {7925 \left (2 x + 3\right )^{\frac {3}{2}}}{384} + \frac {1625 \sqrt {2 x + 3}}{128} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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