Optimal. Leaf size=53 \[ -\frac {3}{40} (2 x+3)^{5/2}+\frac {47}{24} (2 x+3)^{3/2}-\frac {109}{8} \sqrt {2 x+3}-\frac {65}{8 \sqrt {2 x+3}} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \begin {gather*} -\frac {3}{40} (2 x+3)^{5/2}+\frac {47}{24} (2 x+3)^{3/2}-\frac {109}{8} \sqrt {2 x+3}-\frac {65}{8 \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+2 x)^{3/2}} \, dx &=\int \left (\frac {65}{8 (3+2 x)^{3/2}}-\frac {109}{8 \sqrt {3+2 x}}+\frac {47}{8} \sqrt {3+2 x}-\frac {3}{8} (3+2 x)^{3/2}\right ) \, dx\\ &=-\frac {65}{8 \sqrt {3+2 x}}-\frac {109}{8} \sqrt {3+2 x}+\frac {47}{24} (3+2 x)^{3/2}-\frac {3}{40} (3+2 x)^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.53 \begin {gather*} -\frac {9 x^3-77 x^2+117 x+501}{15 \sqrt {2 x+3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 40, normalized size = 0.75 \begin {gather*} \frac {-9 (2 x+3)^3+235 (2 x+3)^2-1635 (2 x+3)-975}{120 \sqrt {2 x+3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 24, normalized size = 0.45 \begin {gather*} -\frac {9 \, x^{3} - 77 \, x^{2} + 117 \, x + 501}{15 \, \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 = 37, normalized size = 0.70 \begin {gather*} -\frac {3}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {47}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {109}{8} \, \sqrt {2 \, x + 3} - \frac {65}{8 \, \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 = 25, normalized size = 0.47 \begin {gather*} -\frac {9 x^{3}-77 x^{2}+117 x +501}{15 \sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 37, normalized size = 0.70 \begin {gather*} -\frac {3}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {47}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {109}{8} \, \sqrt {2 \, x + 3} - \frac {65}{8 \, \sqrt {2 \, x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {47\,{\left (2\,x+3\right )}^{3/2}}{24}-\frac {109\,\sqrt {2\,x+3}}{8}-\frac {65}{8\,\sqrt {2\,x+3}}-\frac {3\,{\left (2\,x+3\right )}^{5/2}}{40} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.15, size = 46, normalized size = 0.87 \begin {gather*} - \frac {3 \left (2 x + 3\right )^{\frac {5}{2}}}{40} + \frac {47 \left (2 x + 3\right )^{\frac {3}{2}}}{24} - \frac {109 \sqrt {2 x + 3}}{8} - \frac {65}{8 \sqrt {2 x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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