Optimal. Leaf size=53 \[ -\frac {1}{8} (2 x+3)^{3/2}+\frac {47}{8} \sqrt {2 x+3}+\frac {109}{8 \sqrt {2 x+3}}-\frac {65}{24 (2 x+3)^{3/2}} \]
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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 {1}{8} (2 x+3)^{3/2}+\frac {47}{8} \sqrt {2 x+3}+\frac {109}{8 \sqrt {2 x+3}}-\frac {65}{24 (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+2 x)^{5/2}} \, dx &=\int \left (\frac {65}{8 (3+2 x)^{5/2}}-\frac {109}{8 (3+2 x)^{3/2}}+\frac {47}{8 \sqrt {3+2 x}}-\frac {3}{8} \sqrt {3+2 x}\right ) \, dx\\ &=-\frac {65}{24 (3+2 x)^{3/2}}+\frac {109}{8 \sqrt {3+2 x}}+\frac {47}{8} \sqrt {3+2 x}-\frac {1}{8} (3+2 x)^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.53 \begin {gather*} -\frac {3 x^3-57 x^2-273 x-263}{3 (2 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 40, normalized size = 0.75 \begin {gather*} \frac {-3 (2 x+3)^3+141 (2 x+3)^2+327 (2 x+3)-65}{24 (2 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 36, normalized size = 0.68 \begin {gather*} -\frac {{\left (3 \, x^{3} - 57 \, x^{2} - 273 \, x - 263\right )} \sqrt {2 \, x + 3}}{3 \, {\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 = 0.20, size = 33, normalized size = 0.62 \begin {gather*} -\frac {1}{8} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {47}{8} \, \sqrt {2 \, x + 3} + \frac {327 \, x + 458}{12 \, {\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 = 0.00, size = 25, normalized size = 0.47 \begin {gather*} -\frac {3 x^{3}-57 x^{2}-273 x -263}{3 \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 = 0.58, size = 33, normalized size = 0.62 \begin {gather*} -\frac {1}{8} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {47}{8} \, \sqrt {2 \, x + 3} + \frac {327 \, x + 458}{12 \, {\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 = 0.05, size = 38, normalized size = 0.72 \begin {gather*} \frac {654\,x+141\,{\left (2\,x+3\right )}^2-3\,{\left (2\,x+3\right )}^3+916}{\sqrt {2\,x+3}\,\left (48\,x+72\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.66, size = 102, normalized size = 1.92 \begin {gather*} - \frac {3 x^{3}}{6 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}} + \frac {57 x^{2}}{6 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}} + \frac {273 x}{6 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}} + \frac {263}{6 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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