Optimal. Leaf size=53 \[ \frac {9}{8} (1-2 x)^{5/2}-\frac {103}{8} (1-2 x)^{3/2}+\frac {707}{8} \sqrt {1-2 x}+\frac {539}{8 \sqrt {1-2 x}} \]
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Rubi [A] time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \[ \frac {9}{8} (1-2 x)^{5/2}-\frac {103}{8} (1-2 x)^{3/2}+\frac {707}{8} \sqrt {1-2 x}+\frac {539}{8 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2 (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {539}{8 (1-2 x)^{3/2}}-\frac {707}{8 \sqrt {1-2 x}}+\frac {309}{8} \sqrt {1-2 x}-\frac {45}{8} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac {539}{8 \sqrt {1-2 x}}+\frac {707}{8} \sqrt {1-2 x}-\frac {103}{8} (1-2 x)^{3/2}+\frac {9}{8} (1-2 x)^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.47 \[ \frac {-9 x^3-38 x^2-132 x+144}{\sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 30, normalized size = 0.57 \[ \frac {{\left (9 \, x^{3} + 38 \, x^{2} + 132 \, x - 144\right )} \sqrt {-2 \, x + 1}}{2 \, x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 44, normalized size = 0.83 \[ \frac {9}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {103}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {707}{8} \, \sqrt {-2 \, x + 1} + \frac {539}{8 \, \sqrt {-2 \, x + 1}} \]
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 \[ -\frac {9 x^{3}+38 x^{2}+132 x -144}{\sqrt {-2 x +1}} \]
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 \[ \frac {9}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {103}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {707}{8} \, \sqrt {-2 \, x + 1} + \frac {539}{8 \, \sqrt {-2 \, x + 1}} \]
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 \[ \frac {539}{8\,\sqrt {1-2\,x}}+\frac {707\,\sqrt {1-2\,x}}{8}-\frac {103\,{\left (1-2\,x\right )}^{3/2}}{8}+\frac {9\,{\left (1-2\,x\right )}^{5/2}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.69, size = 46, normalized size = 0.87 \[ \frac {9 \left (1 - 2 x\right )^{\frac {5}{2}}}{8} - \frac {103 \left (1 - 2 x\right )^{\frac {3}{2}}}{8} + \frac {707 \sqrt {1 - 2 x}}{8} + \frac {539}{8 \sqrt {1 - 2 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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