Optimal. Leaf size=105 \[ -\frac {3}{128} (2 x+3)^{9/2}+\frac {81}{128} (2 x+3)^{7/2}-\frac {3519}{640} (2 x+3)^{5/2}+\frac {10475}{384} (2 x+3)^{3/2}-\frac {17201}{128} \sqrt {2 x+3}-\frac {16005}{128 \sqrt {2 x+3}}+\frac {7925}{384 (2 x+3)^{3/2}}-\frac {325}{128 (2 x+3)^{5/2}} \]
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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 {3}{128} (2 x+3)^{9/2}+\frac {81}{128} (2 x+3)^{7/2}-\frac {3519}{640} (2 x+3)^{5/2}+\frac {10475}{384} (2 x+3)^{3/2}-\frac {17201}{128} \sqrt {2 x+3}-\frac {16005}{128 \sqrt {2 x+3}}+\frac {7925}{384 (2 x+3)^{3/2}}-\frac {325}{128 (2 x+3)^{5/2}} \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)^{7/2}} \, dx &=\int \left (\frac {1625}{128 (3+2 x)^{7/2}}-\frac {7925}{128 (3+2 x)^{5/2}}+\frac {16005}{128 (3+2 x)^{3/2}}-\frac {17201}{128 \sqrt {3+2 x}}+\frac {10475}{128} \sqrt {3+2 x}-\frac {3519}{128} (3+2 x)^{3/2}+\frac {567}{128} (3+2 x)^{5/2}-\frac {27}{128} (3+2 x)^{7/2}\right ) \, dx\\ &=-\frac {325}{128 (3+2 x)^{5/2}}+\frac {7925}{384 (3+2 x)^{3/2}}-\frac {16005}{128 \sqrt {3+2 x}}-\frac {17201}{128} \sqrt {3+2 x}+\frac {10475}{384} (3+2 x)^{3/2}-\frac {3519}{640} (3+2 x)^{5/2}+\frac {81}{128} (3+2 x)^{7/2}-\frac {3}{128} (3+2 x)^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.46 \begin {gather*} -\frac {45 x^7-135 x^6-702 x^5-1940 x^4+3195 x^3+41805 x^2+85070 x+51162}{15 (2 x+3)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 76, normalized size = 0.72 \begin {gather*} \frac {-45 (2 x+3)^7+1215 (2 x+3)^6-10557 (2 x+3)^5+52375 (2 x+3)^4-258015 (2 x+3)^3-240075 (2 x+3)^2+39625 (2 x+3)-4875}{1920 (2 x+3)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 61, normalized size = 0.58 \begin {gather*} -\frac {{\left (45 \, x^{7} - 135 \, x^{6} - 702 \, x^{5} - 1940 \, x^{4} + 3195 \, x^{3} + 41805 \, x^{2} + 85070 \, x + 51162\right )} \sqrt {2 \, x + 3}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 {3}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {81}{128} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {3519}{640} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {10475}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {17201}{128} \, \sqrt {2 \, x + 3} - \frac {5 \, {\left (9603 \, {\left (2 \, x + 3\right )}^{2} - 3170 \, x - 4560\right )}}{384 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 45, normalized size = 0.43 \begin {gather*} -\frac {45 x^{7}-135 x^{6}-702 x^{5}-1940 x^{4}+3195 x^{3}+41805 x^{2}+85070 x +51162}{15 \left (2 x +3\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 69, normalized size = 0.66 \begin {gather*} -\frac {3}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {81}{128} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {3519}{640} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {10475}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {17201}{128} \, \sqrt {2 \, x + 3} - \frac {5 \, {\left (9603 \, {\left (2 \, x + 3\right )}^{2} - 3170 \, x - 4560\right )}}{384 \, {\left (2 \, x + 3\right )}^{\frac {5}{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}{192}-\frac {16005\,{\left (2\,x+3\right )}^2}{128}+\frac {475}{8}}{{\left (2\,x+3\right )}^{5/2}}-\frac {17201\,\sqrt {2\,x+3}}{128}+\frac {10475\,{\left (2\,x+3\right )}^{3/2}}{384}-\frac {3519\,{\left (2\,x+3\right )}^{5/2}}{640}+\frac {81\,{\left (2\,x+3\right )}^{7/2}}{128}-\frac {3\,{\left (2\,x+3\right )}^{9/2}}{128} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 62.14, size = 94, normalized size = 0.90 \begin {gather*} - \frac {3 \left (2 x + 3\right )^{\frac {9}{2}}}{128} + \frac {81 \left (2 x + 3\right )^{\frac {7}{2}}}{128} - \frac {3519 \left (2 x + 3\right )^{\frac {5}{2}}}{640} + \frac {10475 \left (2 x + 3\right )^{\frac {3}{2}}}{384} - \frac {17201 \sqrt {2 x + 3}}{128} - \frac {16005}{128 \sqrt {2 x + 3}} + \frac {7925}{384 \left (2 x + 3\right )^{\frac {3}{2}}} - \frac {325}{128 \left (2 x + 3\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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