Optimal. Leaf size=105 \[ \frac {2033647}{384 (1-2 x)^{3/2}}-\frac {6206585}{128 \sqrt {1-2 x}}-\frac {8117095}{128} \sqrt {1-2 x}+\frac {1965635}{128} (1-2 x)^{3/2}-\frac {514017}{128} (1-2 x)^{5/2}+\frac {672003}{896} (1-2 x)^{7/2}-\frac {10845}{128} (1-2 x)^{9/2}+\frac {6075 (1-2 x)^{11/2}}{1408} \]
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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90}
\begin {gather*} \frac {6075 (1-2 x)^{11/2}}{1408}-\frac {10845}{128} (1-2 x)^{9/2}+\frac {672003}{896} (1-2 x)^{7/2}-\frac {514017}{128} (1-2 x)^{5/2}+\frac {1965635}{128} (1-2 x)^{3/2}-\frac {8117095}{128} \sqrt {1-2 x}-\frac {6206585}{128 \sqrt {1-2 x}}+\frac {2033647}{384 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {2033647}{128 (1-2 x)^{5/2}}-\frac {6206585}{128 (1-2 x)^{3/2}}+\frac {8117095}{128 \sqrt {1-2 x}}-\frac {5896905}{128} \sqrt {1-2 x}+\frac {2570085}{128} (1-2 x)^{3/2}-\frac {672003}{128} (1-2 x)^{5/2}+\frac {97605}{128} (1-2 x)^{7/2}-\frac {6075}{128} (1-2 x)^{9/2}\right ) \, dx\\ &=\frac {2033647}{384 (1-2 x)^{3/2}}-\frac {6206585}{128 \sqrt {1-2 x}}-\frac {8117095}{128} \sqrt {1-2 x}+\frac {1965635}{128} (1-2 x)^{3/2}-\frac {514017}{128} (1-2 x)^{5/2}+\frac {672003}{896} (1-2 x)^{7/2}-\frac {10845}{128} (1-2 x)^{9/2}+\frac {6075 (1-2 x)^{11/2}}{1408}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.46 \begin {gather*} -\frac {21852008-65622552 x+32450916 x^2+9702012 x^3+5121279 x^4+2456001 x^5+806085 x^6+127575 x^7}{231 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 74, normalized size = 0.70
method | result | size |
gosper | \(-\frac {127575 x^{7}+806085 x^{6}+2456001 x^{5}+5121279 x^{4}+9702012 x^{3}+32450916 x^{2}-65622552 x +21852008}{231 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(45\) |
trager | \(-\frac {\left (127575 x^{7}+806085 x^{6}+2456001 x^{5}+5121279 x^{4}+9702012 x^{3}+32450916 x^{2}-65622552 x +21852008\right ) \sqrt {1-2 x}}{231 \left (-1+2 x \right )^{2}}\) | \(52\) |
risch | \(\frac {127575 x^{7}+806085 x^{6}+2456001 x^{5}+5121279 x^{4}+9702012 x^{3}+32450916 x^{2}-65622552 x +21852008}{231 \left (-1+2 x \right ) \sqrt {1-2 x}}\) | \(52\) |
derivativedivides | \(\frac {2033647}{384 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {1965635 \left (1-2 x \right )^{\frac {3}{2}}}{128}-\frac {514017 \left (1-2 x \right )^{\frac {5}{2}}}{128}+\frac {672003 \left (1-2 x \right )^{\frac {7}{2}}}{896}-\frac {10845 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {6075 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {6206585}{128 \sqrt {1-2 x}}-\frac {8117095 \sqrt {1-2 x}}{128}\) | \(74\) |
default | \(\frac {2033647}{384 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {1965635 \left (1-2 x \right )^{\frac {3}{2}}}{128}-\frac {514017 \left (1-2 x \right )^{\frac {5}{2}}}{128}+\frac {672003 \left (1-2 x \right )^{\frac {7}{2}}}{896}-\frac {10845 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {6075 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {6206585}{128 \sqrt {1-2 x}}-\frac {8117095 \sqrt {1-2 x}}{128}\) | \(74\) |
meijerg | \(-\frac {192 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {1040 \sqrt {\pi }-\frac {130 \sqrt {\pi }\, \left (-24 x +8\right )}{\left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {7240 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{3 \sqrt {\pi }}+\frac {24880 \sqrt {\pi }-\frac {1555 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{8 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {9615 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{4 \sqrt {\pi }}+\frac {20376 \sqrt {\pi }-\frac {2547 \sqrt {\pi }\, \left (384 x^{5}+384 x^{4}+512 x^{3}+1536 x^{2}-3072 x +1024\right )}{128 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {2295 \left (-\frac {512 \sqrt {\pi }}{21}+\frac {\sqrt {\pi }\, \left (896 x^{6}+768 x^{5}+768 x^{4}+1024 x^{3}+3072 x^{2}-6144 x +2048\right )}{84 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}+\frac {\frac {10800 \sqrt {\pi }}{11}-\frac {675 \sqrt {\pi }\, \left (18432 x^{7}+14336 x^{6}+12288 x^{5}+12288 x^{4}+16384 x^{3}+49152 x^{2}-98304 x +32768\right )}{22528 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}\) | \(324\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 69, normalized size = 0.66 \begin {gather*} \frac {6075}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {10845}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {672003}{896} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {514017}{128} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {1965635}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {8117095}{128} \, \sqrt {-2 \, x + 1} + \frac {26411 \, {\left (705 \, x - 314\right )}}{192 \, {\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 = 1.00, size = 56, normalized size = 0.53 \begin {gather*} -\frac {{\left (127575 \, x^{7} + 806085 \, x^{6} + 2456001 \, x^{5} + 5121279 \, x^{4} + 9702012 \, x^{3} + 32450916 \, x^{2} - 65622552 \, x + 21852008\right )} \sqrt {-2 \, x + 1}}{231 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 22.50, size = 94, normalized size = 0.90 \begin {gather*} \frac {6075 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {10845 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} + \frac {672003 \left (1 - 2 x\right )^{\frac {7}{2}}}{896} - \frac {514017 \left (1 - 2 x\right )^{\frac {5}{2}}}{128} + \frac {1965635 \left (1 - 2 x\right )^{\frac {3}{2}}}{128} - \frac {8117095 \sqrt {1 - 2 x}}{128} - \frac {6206585}{128 \sqrt {1 - 2 x}} + \frac {2033647}{384 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.29, size = 104, normalized size = 0.99 \begin {gather*} -\frac {6075}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {10845}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {672003}{896} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {514017}{128} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {1965635}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {8117095}{128} \, \sqrt {-2 \, x + 1} - \frac {26411 \, {\left (705 \, x - 314\right )}}{192 \, {\left (2 \, x - 1\right )} \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 = 68, normalized size = 0.65 \begin {gather*} \frac {\frac {6206585\,x}{64}-\frac {4146527}{96}}{{\left (1-2\,x\right )}^{3/2}}-\frac {8117095\,\sqrt {1-2\,x}}{128}+\frac {1965635\,{\left (1-2\,x\right )}^{3/2}}{128}-\frac {514017\,{\left (1-2\,x\right )}^{5/2}}{128}+\frac {672003\,{\left (1-2\,x\right )}^{7/2}}{896}-\frac {10845\,{\left (1-2\,x\right )}^{9/2}}{128}+\frac {6075\,{\left (1-2\,x\right )}^{11/2}}{1408} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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