Optimal. Leaf size=53 \[ -\frac {3}{56} (2 x+3)^{7/2}+\frac {47}{40} (2 x+3)^{5/2}-\frac {109}{24} (2 x+3)^{3/2}+\frac {65}{8} \sqrt {2 x+3} \]
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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} \[ -\frac {3}{56} (2 x+3)^{7/2}+\frac {47}{40} (2 x+3)^{5/2}-\frac {109}{24} (2 x+3)^{3/2}+\frac {65}{8} \sqrt {2 x+3} \]
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 )}{\sqrt {3+2 x}} \, dx &=\int \left (\frac {65}{8 \sqrt {3+2 x}}-\frac {109}{8} \sqrt {3+2 x}+\frac {47}{8} (3+2 x)^{3/2}-\frac {3}{8} (3+2 x)^{5/2}\right ) \, dx\\ &=\frac {65}{8} \sqrt {3+2 x}-\frac {109}{24} (3+2 x)^{3/2}+\frac {47}{40} (3+2 x)^{5/2}-\frac {3}{56} (3+2 x)^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.53 \[ -\frac {1}{105} \sqrt {2 x+3} \left (45 x^3-291 x^2-223 x-381\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 24, normalized size = 0.45 \[ -\frac {1}{105} \, {\left (45 \, x^{3} - 291 \, x^{2} - 223 \, x - 381\right )} \sqrt {2 \, x + 3} \]
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 \[ -\frac {3}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {47}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {109}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {65}{8} \, \sqrt {2 \, x + 3} \]
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 \[ -\frac {\left (45 x^{3}-291 x^{2}-223 x -381\right ) \sqrt {2 x +3}}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 37, normalized size = 0.70 \[ -\frac {3}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {47}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {109}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {65}{8} \, \sqrt {2 \, x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 37, normalized size = 0.70 \[ \frac {65\,\sqrt {2\,x+3}}{8}-\frac {109\,{\left (2\,x+3\right )}^{3/2}}{24}+\frac {47\,{\left (2\,x+3\right )}^{5/2}}{40}-\frac {3\,{\left (2\,x+3\right )}^{7/2}}{56} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 41.28, size = 46, normalized size = 0.87 \[ - \frac {3 \left (2 x + 3\right )^{\frac {7}{2}}}{56} + \frac {47 \left (2 x + 3\right )^{\frac {5}{2}}}{40} - \frac {109 \left (2 x + 3\right )^{\frac {3}{2}}}{24} + \frac {65 \sqrt {2 x + 3}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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