Optimal. Leaf size=79 \[ -\frac {26411}{160} (1-2 x)^{5/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {27}{32} (1-2 x)^{15/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 79, 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 {27}{32} (1-2 x)^{15/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {26411}{160} (1-2 x)^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx &=\int \left (\frac {26411}{32} (1-2 x)^{3/2}-\frac {57281}{32} (1-2 x)^{5/2}+\frac {24843}{16} (1-2 x)^{7/2}-\frac {10773}{16} (1-2 x)^{9/2}+\frac {4671}{32} (1-2 x)^{11/2}-\frac {405}{32} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {26411}{160} (1-2 x)^{5/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {27}{32} (1-2 x)^{15/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {(1-2 x)^{5/2} \left (66592+230000 x+410320 x^2+424440 x^3+240570 x^4+57915 x^5\right )}{2145} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 56, normalized size = 0.71
method | result | size |
gosper | \(-\frac {\left (57915 x^{5}+240570 x^{4}+424440 x^{3}+410320 x^{2}+230000 x +66592\right ) \left (1-2 x \right )^{\frac {5}{2}}}{2145}\) | \(35\) |
trager | \(\left (-108 x^{7}-\frac {4428}{13} x^{6}-\frac {52893}{143} x^{5}-\frac {36818}{429} x^{4}+\frac {59368}{429} x^{3}+\frac {81104}{715} x^{2}+\frac {36368}{2145} x -\frac {66592}{2145}\right ) \sqrt {1-2 x}\) | \(44\) |
risch | \(\frac {\left (231660 x^{7}+730620 x^{6}+793395 x^{5}+184090 x^{4}-296840 x^{3}-243312 x^{2}-36368 x +66592\right ) \left (-1+2 x \right )}{2145 \sqrt {1-2 x}}\) | \(50\) |
derivativedivides | \(-\frac {26411 \left (1-2 x \right )^{\frac {5}{2}}}{160}+\frac {8183 \left (1-2 x \right )^{\frac {7}{2}}}{32}-\frac {8281 \left (1-2 x \right )^{\frac {9}{2}}}{48}+\frac {10773 \left (1-2 x \right )^{\frac {11}{2}}}{176}-\frac {4671 \left (1-2 x \right )^{\frac {13}{2}}}{416}+\frac {27 \left (1-2 x \right )^{\frac {15}{2}}}{32}\) | \(56\) |
default | \(-\frac {26411 \left (1-2 x \right )^{\frac {5}{2}}}{160}+\frac {8183 \left (1-2 x \right )^{\frac {7}{2}}}{32}-\frac {8281 \left (1-2 x \right )^{\frac {9}{2}}}{48}+\frac {10773 \left (1-2 x \right )^{\frac {11}{2}}}{176}-\frac {4671 \left (1-2 x \right )^{\frac {13}{2}}}{416}+\frac {27 \left (1-2 x \right )^{\frac {15}{2}}}{32}\) | \(56\) |
meijerg | \(-\frac {18 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {368 \sqrt {\pi }}{35}-\frac {46 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{35}}{\sqrt {\pi }}-\frac {423 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{4 \sqrt {\pi }}+\frac {\frac {1152 \sqrt {\pi }}{385}-\frac {9 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{385}}{\sqrt {\pi }}-\frac {3969 \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (147840 x^{6}-94080 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {72 \sqrt {\pi }}{1001}-\frac {9 \sqrt {\pi }\, \left (1537536 x^{7}-946176 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{128128}}{\sqrt {\pi }}\) | \(275\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 55, normalized size = 0.70 \begin {gather*} \frac {27}{32} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {4671}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {10773}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {8281}{48} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {8183}{32} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {26411}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.29, size = 44, normalized size = 0.56 \begin {gather*} -\frac {1}{2145} \, {\left (231660 \, x^{7} + 730620 \, x^{6} + 793395 \, x^{5} + 184090 \, x^{4} - 296840 \, x^{3} - 243312 \, x^{2} - 36368 \, x + 66592\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 9.70, size = 70, normalized size = 0.89 \begin {gather*} \frac {27 \left (1 - 2 x\right )^{\frac {15}{2}}}{32} - \frac {4671 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} + \frac {10773 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} - \frac {8281 \left (1 - 2 x\right )^{\frac {9}{2}}}{48} + \frac {8183 \left (1 - 2 x\right )^{\frac {7}{2}}}{32} - \frac {26411 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 97, normalized size = 1.23 \begin {gather*} -\frac {27}{32} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {4671}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {10773}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {8281}{48} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {8183}{32} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {26411}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {8183\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {26411\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {8281\,{\left (1-2\,x\right )}^{9/2}}{48}+\frac {10773\,{\left (1-2\,x\right )}^{11/2}}{176}-\frac {4671\,{\left (1-2\,x\right )}^{13/2}}{416}+\frac {27\,{\left (1-2\,x\right )}^{15/2}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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