Optimal. Leaf size=79 \[ -\frac {9}{352} (2 x+3)^{11/2}+\frac {55}{96} (2 x+3)^{9/2}-\frac {359}{112} (2 x+3)^{7/2}+\frac {651}{80} (2 x+3)^{5/2}-\frac {355}{32} (2 x+3)^{3/2}+\frac {325}{32} \sqrt {2 x+3} \]
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Rubi [A] time = 0.02, 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 {9}{352} (2 x+3)^{11/2}+\frac {55}{96} (2 x+3)^{9/2}-\frac {359}{112} (2 x+3)^{7/2}+\frac {651}{80} (2 x+3)^{5/2}-\frac {355}{32} (2 x+3)^{3/2}+\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}{\sqrt {3+2 x}} \, dx &=\int \left (\frac {325}{32 \sqrt {3+2 x}}-\frac {1065}{32} \sqrt {3+2 x}+\frac {651}{16} (3+2 x)^{3/2}-\frac {359}{16} (3+2 x)^{5/2}+\frac {165}{32} (3+2 x)^{7/2}-\frac {9}{32} (3+2 x)^{9/2}\right ) \, dx\\ &=\frac {325}{32} \sqrt {3+2 x}-\frac {355}{32} (3+2 x)^{3/2}+\frac {651}{80} (3+2 x)^{5/2}-\frac {359}{112} (3+2 x)^{7/2}+\frac {55}{96} (3+2 x)^{9/2}-\frac {9}{352} (3+2 x)^{11/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {\sqrt {2 x+3} \left (945 x^5-3500 x^4-12645 x^3-15354 x^2-6252 x-4344\right )}{1155} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 71, normalized size = 0.90 \begin {gather*} \frac {-945 (2 x+3)^{11/2}+21175 (2 x+3)^{9/2}-118470 (2 x+3)^{7/2}+300762 (2 x+3)^{5/2}-410025 (2 x+3)^{3/2}+375375 \sqrt {2 x+3}}{36960} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 34, normalized size = 0.43 \begin {gather*} -\frac {1}{1155} \, {\left (945 \, x^{5} - 3500 \, x^{4} - 12645 \, x^{3} - 15354 \, x^{2} - 6252 \, x - 4344\right )} \sqrt {2 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 55, normalized size = 0.70 \begin {gather*} -\frac {9}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {55}{96} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {359}{112} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {651}{80} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {355}{32} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \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 = 0.00, size = 35, normalized size = 0.44 \begin {gather*} -\frac {\left (945 x^{5}-3500 x^{4}-12645 x^{3}-15354 x^{2}-6252 x -4344\right ) \sqrt {2 x +3}}{1155} \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 {9}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {55}{96} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {359}{112} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {651}{80} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {355}{32} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \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 {325\,\sqrt {2\,x+3}}{32}-\frac {355\,{\left (2\,x+3\right )}^{3/2}}{32}+\frac {651\,{\left (2\,x+3\right )}^{5/2}}{80}-\frac {359\,{\left (2\,x+3\right )}^{7/2}}{112}+\frac {55\,{\left (2\,x+3\right )}^{9/2}}{96}-\frac {9\,{\left (2\,x+3\right )}^{11/2}}{352} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 85.81, size = 70, normalized size = 0.89 \begin {gather*} - \frac {9 \left (2 x + 3\right )^{\frac {11}{2}}}{352} + \frac {55 \left (2 x + 3\right )^{\frac {9}{2}}}{96} - \frac {359 \left (2 x + 3\right )^{\frac {7}{2}}}{112} + \frac {651 \left (2 x + 3\right )^{\frac {5}{2}}}{80} - \frac {355 \left (2 x + 3\right )^{\frac {3}{2}}}{32} + \frac {325 \sqrt {2 x + 3}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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