Optimal. Leaf size=53 \[ \frac {539}{8 \sqrt {1-2 x}}+\frac {707}{8} \sqrt {1-2 x}-\frac {103}{8} (1-2 x)^{3/2}+\frac {9}{8} (1-2 x)^{5/2} \]
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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 = {78}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
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 = 0.02, size = 25, normalized size = 0.47 \begin {gather*} \frac {144-132 x-38 x^2-9 x^3}{\sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 38, normalized size = 0.72
method | result | size |
gosper | \(-\frac {9 x^{3}+38 x^{2}+132 x -144}{\sqrt {1-2 x}}\) | \(25\) |
risch | \(-\frac {9 x^{3}+38 x^{2}+132 x -144}{\sqrt {1-2 x}}\) | \(25\) |
trager | \(\frac {\left (9 x^{3}+38 x^{2}+132 x -144\right ) \sqrt {1-2 x}}{-1+2 x}\) | \(31\) |
derivativedivides | \(-\frac {103 \left (1-2 x \right )^{\frac {3}{2}}}{8}+\frac {9 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {539}{8 \sqrt {1-2 x}}+\frac {707 \sqrt {1-2 x}}{8}\) | \(38\) |
default | \(-\frac {103 \left (1-2 x \right )^{\frac {3}{2}}}{8}+\frac {9 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {539}{8 \sqrt {1-2 x}}+\frac {707 \sqrt {1-2 x}}{8}\) | \(38\) |
meijerg | \(-\frac {12 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-56 \sqrt {\pi }+\frac {7 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {87 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{4 \sqrt {\pi }}+\frac {-18 \sqrt {\pi }+\frac {9 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{64 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 37, normalized size = 0.70 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.20, size = 30, normalized size = 0.57 \begin {gather*} \frac {{\left (9 \, x^{3} + 38 \, x^{2} + 132 \, x - 144\right )} \sqrt {-2 \, x + 1}}{2 \, x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.63, size = 46, normalized size = 0.87 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 44, normalized size = 0.83 \begin {gather*} \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}} \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 {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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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