Optimal. Leaf size=79 \[ -\frac {1}{32} (2 x+3)^{9/2}+\frac {165}{224} (2 x+3)^{7/2}-\frac {359}{80} (2 x+3)^{5/2}+\frac {217}{16} (2 x+3)^{3/2}-\frac {1065}{32} \sqrt {2 x+3}-\frac {325}{32 \sqrt {2 x+3}} \]
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Rubi [A] time = 0.03, antiderivative size = 79, 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 {1}{32} (2 x+3)^{9/2}+\frac {165}{224} (2 x+3)^{7/2}-\frac {359}{80} (2 x+3)^{5/2}+\frac {217}{16} (2 x+3)^{3/2}-\frac {1065}{32} \sqrt {2 x+3}-\frac {325}{32 \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 )^2}{(3+2 x)^{3/2}} \, dx &=\int \left (\frac {325}{32 (3+2 x)^{3/2}}-\frac {1065}{32 \sqrt {3+2 x}}+\frac {651}{16} \sqrt {3+2 x}-\frac {359}{16} (3+2 x)^{3/2}+\frac {165}{32} (3+2 x)^{5/2}-\frac {9}{32} (3+2 x)^{7/2}\right ) \, dx\\ &=-\frac {325}{32 \sqrt {3+2 x}}-\frac {1065}{32} \sqrt {3+2 x}+\frac {217}{16} (3+2 x)^{3/2}-\frac {359}{80} (3+2 x)^{5/2}+\frac {165}{224} (3+2 x)^{7/2}-\frac {1}{32} (3+2 x)^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {35 x^5-150 x^4-431 x^3-632 x^2+432 x+1996}{35 \sqrt {2 x+3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 58, normalized size = 0.73 \begin {gather*} \frac {-35 (2 x+3)^5+825 (2 x+3)^4-5026 (2 x+3)^3+15190 (2 x+3)^2-37275 (2 x+3)-11375}{1120 \sqrt {2 x+3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 34, normalized size = 0.43 \begin {gather*} -\frac {35 \, x^{5} - 150 \, x^{4} - 431 \, x^{3} - 632 \, x^{2} + 432 \, x + 1996}{35 \, \sqrt {2 \, x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 55, normalized size = 0.70 \begin {gather*} -\frac {1}{32} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {165}{224} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {359}{80} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {217}{16} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {1065}{32} \, \sqrt {2 \, x + 3} - \frac {325}{32 \, \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 = 35, normalized size = 0.44 \begin {gather*} -\frac {35 x^{5}-150 x^{4}-431 x^{3}-632 x^{2}+432 x +1996}{35 \sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 55, normalized size = 0.70 \begin {gather*} -\frac {1}{32} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {165}{224} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {359}{80} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {217}{16} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {1065}{32} \, \sqrt {2 \, x + 3} - \frac {325}{32 \, \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 = 55, normalized size = 0.70 \begin {gather*} \frac {217\,{\left (2\,x+3\right )}^{3/2}}{16}-\frac {1065\,\sqrt {2\,x+3}}{32}-\frac {325}{32\,\sqrt {2\,x+3}}-\frac {359\,{\left (2\,x+3\right )}^{5/2}}{80}+\frac {165\,{\left (2\,x+3\right )}^{7/2}}{224}-\frac {{\left (2\,x+3\right )}^{9/2}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 35.24, size = 68, normalized size = 0.86 \begin {gather*} - \frac {\left (2 x + 3\right )^{\frac {9}{2}}}{32} + \frac {165 \left (2 x + 3\right )^{\frac {7}{2}}}{224} - \frac {359 \left (2 x + 3\right )^{\frac {5}{2}}}{80} + \frac {217 \left (2 x + 3\right )^{\frac {3}{2}}}{16} - \frac {1065 \sqrt {2 x + 3}}{32} - \frac {325}{32 \sqrt {2 x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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