Optimal. Leaf size=53 \[ -\frac {1331}{40} (1-2 x)^{5/2}+\frac {1815}{56} (1-2 x)^{7/2}-\frac {275}{24} (1-2 x)^{9/2}+\frac {125}{88} (1-2 x)^{11/2} \]
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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 = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45}
\begin {gather*} \frac {125}{88} (1-2 x)^{11/2}-\frac {275}{24} (1-2 x)^{9/2}+\frac {1815}{56} (1-2 x)^{7/2}-\frac {1331}{40} (1-2 x)^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (1-2 x)^{3/2} (3+5 x)^3 \, dx &=\int \left (\frac {1331}{8} (1-2 x)^{3/2}-\frac {1815}{8} (1-2 x)^{5/2}+\frac {825}{8} (1-2 x)^{7/2}-\frac {125}{8} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {1331}{40} (1-2 x)^{5/2}+\frac {1815}{56} (1-2 x)^{7/2}-\frac {275}{24} (1-2 x)^{9/2}+\frac {125}{88} (1-2 x)^{11/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {(1-2 x)^{5/2} \left (12592+31775 x+33250 x^2+13125 x^3\right )}{1155} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 38, normalized size = 0.72
method | result | size |
gosper | \(-\frac {\left (13125 x^{3}+33250 x^{2}+31775 x +12592\right ) \left (1-2 x \right )^{\frac {5}{2}}}{1155}\) | \(25\) |
trager | \(\left (-\frac {500}{11} x^{5}-\frac {2300}{33} x^{4}-\frac {1445}{231} x^{3}+\frac {14494}{385} x^{2}+\frac {18593}{1155} x -\frac {12592}{1155}\right ) \sqrt {1-2 x}\) | \(34\) |
derivativedivides | \(-\frac {1331 \left (1-2 x \right )^{\frac {5}{2}}}{40}+\frac {1815 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {275 \left (1-2 x \right )^{\frac {9}{2}}}{24}+\frac {125 \left (1-2 x \right )^{\frac {11}{2}}}{88}\) | \(38\) |
default | \(-\frac {1331 \left (1-2 x \right )^{\frac {5}{2}}}{40}+\frac {1815 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {275 \left (1-2 x \right )^{\frac {9}{2}}}{24}+\frac {125 \left (1-2 x \right )^{\frac {11}{2}}}{88}\) | \(38\) |
risch | \(\frac {\left (52500 x^{5}+80500 x^{4}+7225 x^{3}-43482 x^{2}-18593 x +12592\right ) \left (-1+2 x \right )}{1155 \sqrt {1-2 x}}\) | \(40\) |
meijerg | \(-\frac {81 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{8 \sqrt {\pi }}+\frac {\frac {27 \sqrt {\pi }}{7}-\frac {27 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{56}}{\sqrt {\pi }}-\frac {675 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{32 \sqrt {\pi }}+\frac {\frac {50 \sqrt {\pi }}{231}-\frac {25 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{14784}}{\sqrt {\pi }}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 37, normalized size = 0.70 \begin {gather*} \frac {125}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {275}{24} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1815}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1331}{40} \, {\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 = 1.40, size = 34, normalized size = 0.64 \begin {gather*} -\frac {1}{1155} \, {\left (52500 \, x^{5} + 80500 \, x^{4} + 7225 \, x^{3} - 43482 \, x^{2} - 18593 \, x + 12592\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.02, size = 284, normalized size = 5.36 \begin {gather*} \begin {cases} - \frac {100 \sqrt {5} i \left (x + \frac {3}{5}\right )^{5} \sqrt {10 x - 5}}{11} + \frac {40 \sqrt {5} i \left (x + \frac {3}{5}\right )^{4} \sqrt {10 x - 5}}{3} - \frac {11 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{21} - \frac {121 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{175} - \frac {2662 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{2625} - \frac {29282 \sqrt {5} i \sqrt {10 x - 5}}{13125} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\- \frac {100 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{5}}{11} + \frac {40 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{4}}{3} - \frac {11 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{21} - \frac {121 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{175} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{2625} - \frac {29282 \sqrt {5} \sqrt {5 - 10 x}}{13125} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.99, size = 65, normalized size = 1.23 \begin {gather*} -\frac {125}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {275}{24} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1815}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1331}{40} \, {\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 = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {1815\,{\left (1-2\,x\right )}^{7/2}}{56}-\frac {1331\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {275\,{\left (1-2\,x\right )}^{9/2}}{24}+\frac {125\,{\left (1-2\,x\right )}^{11/2}}{88} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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