Optimal. Leaf size=105 \[ \frac {1294139}{128 \sqrt {1-2 x}}+\frac {3916031}{128} \sqrt {1-2 x}-\frac {1692705}{128} (1-2 x)^{3/2}+\frac {731619}{128} (1-2 x)^{5/2}-\frac {225855}{128} (1-2 x)^{7/2}+\frac {45549}{128} (1-2 x)^{9/2}-\frac {59049 (1-2 x)^{11/2}}{1408}+\frac {3645 (1-2 x)^{13/2}}{1664} \]
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Rubi [A]
time = 0.01, antiderivative size = 105, 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 {3645 (1-2 x)^{13/2}}{1664}-\frac {59049 (1-2 x)^{11/2}}{1408}+\frac {45549}{128} (1-2 x)^{9/2}-\frac {225855}{128} (1-2 x)^{7/2}+\frac {731619}{128} (1-2 x)^{5/2}-\frac {1692705}{128} (1-2 x)^{3/2}+\frac {3916031}{128} \sqrt {1-2 x}+\frac {1294139}{128 \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)^6 (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {1294139}{128 (1-2 x)^{3/2}}-\frac {3916031}{128 \sqrt {1-2 x}}+\frac {5078115}{128} \sqrt {1-2 x}-\frac {3658095}{128} (1-2 x)^{3/2}+\frac {1580985}{128} (1-2 x)^{5/2}-\frac {409941}{128} (1-2 x)^{7/2}+\frac {59049}{128} (1-2 x)^{9/2}-\frac {3645}{128} (1-2 x)^{11/2}\right ) \, dx\\ &=\frac {1294139}{128 \sqrt {1-2 x}}+\frac {3916031}{128} \sqrt {1-2 x}-\frac {1692705}{128} (1-2 x)^{3/2}+\frac {731619}{128} (1-2 x)^{5/2}-\frac {225855}{128} (1-2 x)^{7/2}+\frac {45549}{128} (1-2 x)^{9/2}-\frac {59049 (1-2 x)^{11/2}}{1408}+\frac {3645 (1-2 x)^{13/2}}{1664}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.46 \begin {gather*} \frac {4539904-4512448 x-2109792 x^2-1663632 x^3-1230120 x^4-687420 x^5-243486 x^6-40095 x^7}{143 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 74, normalized size = 0.70
method | result | size |
gosper | \(-\frac {40095 x^{7}+243486 x^{6}+687420 x^{5}+1230120 x^{4}+1663632 x^{3}+2109792 x^{2}+4512448 x -4539904}{143 \sqrt {1-2 x}}\) | \(45\) |
risch | \(-\frac {40095 x^{7}+243486 x^{6}+687420 x^{5}+1230120 x^{4}+1663632 x^{3}+2109792 x^{2}+4512448 x -4539904}{143 \sqrt {1-2 x}}\) | \(45\) |
trager | \(\frac {\left (40095 x^{7}+243486 x^{6}+687420 x^{5}+1230120 x^{4}+1663632 x^{3}+2109792 x^{2}+4512448 x -4539904\right ) \sqrt {1-2 x}}{-143+286 x}\) | \(52\) |
derivativedivides | \(-\frac {1692705 \left (1-2 x \right )^{\frac {3}{2}}}{128}+\frac {731619 \left (1-2 x \right )^{\frac {5}{2}}}{128}-\frac {225855 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {45549 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {59049 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {3645 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {1294139}{128 \sqrt {1-2 x}}+\frac {3916031 \sqrt {1-2 x}}{128}\) | \(74\) |
default | \(-\frac {1692705 \left (1-2 x \right )^{\frac {3}{2}}}{128}+\frac {731619 \left (1-2 x \right )^{\frac {5}{2}}}{128}-\frac {225855 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {45549 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {59049 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {3645 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {1294139}{128 \sqrt {1-2 x}}+\frac {3916031 \sqrt {1-2 x}}{128}\) | \(74\) |
meijerg | \(-\frac {192 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-2048 \sqrt {\pi }+\frac {256 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {2340 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-9504 \sqrt {\pi }+\frac {297 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{4 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {9045 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{4 \sqrt {\pi }}+\frac {-\frac {29376 \sqrt {\pi }}{7}+\frac {459 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{112 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {16767 \left (\frac {1024 \sqrt {\pi }}{231}-\frac {\sqrt {\pi }\, \left (-2688 x^{6}-1792 x^{5}-1280 x^{4}-1024 x^{3}-1024 x^{2}-2048 x +2048\right )}{462 \sqrt {1-2 x}}\right )}{64 \sqrt {\pi }}+\frac {-\frac {19440 \sqrt {\pi }}{143}+\frac {1215 \sqrt {\pi }\, \left (-67584 x^{7}-43008 x^{6}-28672 x^{5}-20480 x^{4}-16384 x^{3}-16384 x^{2}-32768 x +32768\right )}{292864 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(324\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 73, normalized size = 0.70 \begin {gather*} \frac {3645}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {59049}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {45549}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {225855}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {731619}{128} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1692705}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {3916031}{128} \, \sqrt {-2 \, x + 1} + \frac {1294139}{128 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.94, size = 51, normalized size = 0.49 \begin {gather*} \frac {{\left (40095 \, x^{7} + 243486 \, x^{6} + 687420 \, x^{5} + 1230120 \, x^{4} + 1663632 \, x^{3} + 2109792 \, x^{2} + 4512448 \, x - 4539904\right )} \sqrt {-2 \, x + 1}}{143 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 27.02, size = 94, normalized size = 0.90 \begin {gather*} \frac {3645 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {59049 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} + \frac {45549 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} - \frac {225855 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} + \frac {731619 \left (1 - 2 x\right )^{\frac {5}{2}}}{128} - \frac {1692705 \left (1 - 2 x\right )^{\frac {3}{2}}}{128} + \frac {3916031 \sqrt {1 - 2 x}}{128} + \frac {1294139}{128 \sqrt {1 - 2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.32, size = 108, normalized size = 1.03 \begin {gather*} \frac {3645}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {59049}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {45549}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {225855}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {731619}{128} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {1692705}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {3916031}{128} \, \sqrt {-2 \, x + 1} + \frac {1294139}{128 \, \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 = 73, normalized size = 0.70 \begin {gather*} \frac {1294139}{128\,\sqrt {1-2\,x}}+\frac {3916031\,\sqrt {1-2\,x}}{128}-\frac {1692705\,{\left (1-2\,x\right )}^{3/2}}{128}+\frac {731619\,{\left (1-2\,x\right )}^{5/2}}{128}-\frac {225855\,{\left (1-2\,x\right )}^{7/2}}{128}+\frac {45549\,{\left (1-2\,x\right )}^{9/2}}{128}-\frac {59049\,{\left (1-2\,x\right )}^{11/2}}{1408}+\frac {3645\,{\left (1-2\,x\right )}^{13/2}}{1664} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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