Optimal. Leaf size=105 \[ -\frac {27 (2 x+3)^{11/2}}{1408}+\frac {63}{128} (2 x+3)^{9/2}-\frac {3519}{896} (2 x+3)^{7/2}+\frac {2095}{128} (2 x+3)^{5/2}-\frac {17201}{384} (2 x+3)^{3/2}+\frac {16005}{128} \sqrt {2 x+3}+\frac {7925}{128 \sqrt {2 x+3}}-\frac {1625}{384 (2 x+3)^{3/2}} \]
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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 {27 (2 x+3)^{11/2}}{1408}+\frac {63}{128} (2 x+3)^{9/2}-\frac {3519}{896} (2 x+3)^{7/2}+\frac {2095}{128} (2 x+3)^{5/2}-\frac {17201}{384} (2 x+3)^{3/2}+\frac {16005}{128} \sqrt {2 x+3}+\frac {7925}{128 \sqrt {2 x+3}}-\frac {1625}{384 (2 x+3)^{3/2}} \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}{(3+2 x)^{5/2}} \, dx &=\int \left (\frac {1625}{128 (3+2 x)^{5/2}}-\frac {7925}{128 (3+2 x)^{3/2}}+\frac {16005}{128 \sqrt {3+2 x}}-\frac {17201}{128} \sqrt {3+2 x}+\frac {10475}{128} (3+2 x)^{3/2}-\frac {3519}{128} (3+2 x)^{5/2}+\frac {567}{128} (3+2 x)^{7/2}-\frac {27}{128} (3+2 x)^{9/2}\right ) \, dx\\ &=-\frac {1625}{384 (3+2 x)^{3/2}}+\frac {7925}{128 \sqrt {3+2 x}}+\frac {16005}{128} \sqrt {3+2 x}-\frac {17201}{384} (3+2 x)^{3/2}+\frac {2095}{128} (3+2 x)^{5/2}-\frac {3519}{896} (3+2 x)^{7/2}+\frac {63}{128} (3+2 x)^{9/2}-\frac {27 (3+2 x)^{11/2}}{1408}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.46 \begin {gather*} -\frac {567 x^7-1323 x^6-9666 x^5-21360 x^4-17663 x^3-42003 x^2-184566 x-181486}{231 (2 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 76, normalized size = 0.72 \begin {gather*} \frac {-567 (2 x+3)^7+14553 (2 x+3)^6-116127 (2 x+3)^5+483945 (2 x+3)^4-1324477 (2 x+3)^3+3697155 (2 x+3)^2+1830675 (2 x+3)-125125}{29568 (2 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 56, normalized size = 0.53 \begin {gather*} -\frac {{\left (567 \, x^{7} - 1323 \, x^{6} - 9666 \, x^{5} - 21360 \, x^{4} - 17663 \, x^{3} - 42003 \, x^{2} - 184566 \, x - 181486\right )} \sqrt {2 \, x + 3}}{231 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 69, normalized size = 0.66 \begin {gather*} -\frac {27}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {63}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {3519}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {2095}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {17201}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {16005}{128} \, \sqrt {2 \, x + 3} + \frac {25 \, {\left (951 \, x + 1394\right )}}{192 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} \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 {567 x^{7}-1323 x^{6}-9666 x^{5}-21360 x^{4}-17663 x^{3}-42003 x^{2}-184566 x -181486}{231 \left (2 x +3\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 69, normalized size = 0.66 \begin {gather*} -\frac {27}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {63}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {3519}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {2095}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {17201}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {16005}{128} \, \sqrt {2 \, x + 3} + \frac {25 \, {\left (951 \, x + 1394\right )}}{192 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 68, normalized size = 0.65 \begin {gather*} \frac {\frac {7925\,x}{64}+\frac {17425}{96}}{{\left (2\,x+3\right )}^{3/2}}+\frac {16005\,\sqrt {2\,x+3}}{128}-\frac {17201\,{\left (2\,x+3\right )}^{3/2}}{384}+\frac {2095\,{\left (2\,x+3\right )}^{5/2}}{128}-\frac {3519\,{\left (2\,x+3\right )}^{7/2}}{896}+\frac {63\,{\left (2\,x+3\right )}^{9/2}}{128}-\frac {27\,{\left (2\,x+3\right )}^{11/2}}{1408} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 56.09, size = 94, normalized size = 0.90 \begin {gather*} - \frac {27 \left (2 x + 3\right )^{\frac {11}{2}}}{1408} + \frac {63 \left (2 x + 3\right )^{\frac {9}{2}}}{128} - \frac {3519 \left (2 x + 3\right )^{\frac {7}{2}}}{896} + \frac {2095 \left (2 x + 3\right )^{\frac {5}{2}}}{128} - \frac {17201 \left (2 x + 3\right )^{\frac {3}{2}}}{384} + \frac {16005 \sqrt {2 x + 3}}{128} + \frac {7925}{128 \sqrt {2 x + 3}} - \frac {1625}{384 \left (2 x + 3\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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