Optimal. Leaf size=53 \[ -\frac {539}{24} (1-2 x)^{3/2}+\frac {707}{40} (1-2 x)^{5/2}-\frac {309}{56} (1-2 x)^{7/2}+\frac {5}{8} (1-2 x)^{9/2} \]
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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 {5}{8} (1-2 x)^{9/2}-\frac {309}{56} (1-2 x)^{7/2}+\frac {707}{40} (1-2 x)^{5/2}-\frac {539}{24} (1-2 x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x) \, dx &=\int \left (\frac {539}{8} \sqrt {1-2 x}-\frac {707}{8} (1-2 x)^{3/2}+\frac {309}{8} (1-2 x)^{5/2}-\frac {45}{8} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {539}{24} (1-2 x)^{3/2}+\frac {707}{40} (1-2 x)^{5/2}-\frac {309}{56} (1-2 x)^{7/2}+\frac {5}{8} (1-2 x)^{9/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {1}{105} (1-2 x)^{3/2} \left (1016+1788 x+1530 x^2+525 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 38, normalized size = 0.72
method | result | size |
gosper | \(-\frac {\left (525 x^{3}+1530 x^{2}+1788 x +1016\right ) \left (1-2 x \right )^{\frac {3}{2}}}{105}\) | \(25\) |
trager | \(\left (10 x^{4}+\frac {169}{7} x^{3}+\frac {682}{35} x^{2}+\frac {244}{105} x -\frac {1016}{105}\right ) \sqrt {1-2 x}\) | \(29\) |
risch | \(-\frac {\left (1050 x^{4}+2535 x^{3}+2046 x^{2}+244 x -1016\right ) \left (-1+2 x \right )}{105 \sqrt {1-2 x}}\) | \(35\) |
derivativedivides | \(-\frac {539 \left (1-2 x \right )^{\frac {3}{2}}}{24}+\frac {707 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {309 \left (1-2 x \right )^{\frac {7}{2}}}{56}+\frac {5 \left (1-2 x \right )^{\frac {9}{2}}}{8}\) | \(38\) |
default | \(-\frac {539 \left (1-2 x \right )^{\frac {3}{2}}}{24}+\frac {707 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {309 \left (1-2 x \right )^{\frac {7}{2}}}{56}+\frac {5 \left (1-2 x \right )^{\frac {9}{2}}}{8}\) | \(38\) |
meijerg | \(\frac {4 \sqrt {\pi }-2 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {7 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{\sqrt {\pi }}+\frac {\frac {58 \sqrt {\pi }}{35}-\frac {29 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{140}}{\sqrt {\pi }}-\frac {45 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{32 \sqrt {\pi }}\) | \(129\) |
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 {5}{8} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {309}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {707}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {539}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.52, size = 29, normalized size = 0.55 \begin {gather*} \frac {1}{105} \, {\left (1050 \, x^{4} + 2535 \, x^{3} + 2046 \, x^{2} + 244 \, x - 1016\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.61, size = 46, normalized size = 0.87 \begin {gather*} \frac {5 \left (1 - 2 x\right )^{\frac {9}{2}}}{8} - \frac {309 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} + \frac {707 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} - \frac {539 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 58, normalized size = 1.09 \begin {gather*} \frac {5}{8} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {309}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {707}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {539}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \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 {707\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {539\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {309\,{\left (1-2\,x\right )}^{7/2}}{56}+\frac {5\,{\left (1-2\,x\right )}^{9/2}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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