Integrand size = 17, antiderivative size = 53 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x}} \, dx=-\frac {1331}{8} \sqrt {1-2 x}+\frac {605}{8} (1-2 x)^{3/2}-\frac {165}{8} (1-2 x)^{5/2}+\frac {125}{56} (1-2 x)^{7/2} \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x}} \, dx=\frac {125}{56} (1-2 x)^{7/2}-\frac {165}{8} (1-2 x)^{5/2}+\frac {605}{8} (1-2 x)^{3/2}-\frac {1331}{8} \sqrt {1-2 x} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1331}{8 \sqrt {1-2 x}}-\frac {1815}{8} \sqrt {1-2 x}+\frac {825}{8} (1-2 x)^{3/2}-\frac {125}{8} (1-2 x)^{5/2}\right ) \, dx \\ & = -\frac {1331}{8} \sqrt {1-2 x}+\frac {605}{8} (1-2 x)^{3/2}-\frac {165}{8} (1-2 x)^{5/2}+\frac {125}{56} (1-2 x)^{7/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.53 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x}} \, dx=-\frac {1}{7} \sqrt {1-2 x} \left (764+575 x+390 x^2+125 x^3\right ) \]
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Time = 1.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45
method | result | size |
trager | \(\left (-\frac {125}{7} x^{3}-\frac {390}{7} x^{2}-\frac {575}{7} x -\frac {764}{7}\right ) \sqrt {1-2 x}\) | \(24\) |
gosper | \(-\frac {\sqrt {1-2 x}\, \left (125 x^{3}+390 x^{2}+575 x +764\right )}{7}\) | \(25\) |
pseudoelliptic | \(-\frac {\sqrt {1-2 x}\, \left (125 x^{3}+390 x^{2}+575 x +764\right )}{7}\) | \(25\) |
risch | \(\frac {\left (-1+2 x \right ) \left (125 x^{3}+390 x^{2}+575 x +764\right )}{7 \sqrt {1-2 x}}\) | \(30\) |
derivativedivides | \(\frac {605 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {165 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {125 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {1331 \sqrt {1-2 x}}{8}\) | \(38\) |
default | \(\frac {605 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {165 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {125 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {1331 \sqrt {1-2 x}}{8}\) | \(38\) |
meijerg | \(-\frac {27 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{2 \sqrt {\pi }}+\frac {45 \sqrt {\pi }-\frac {45 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{8}}{\sqrt {\pi }}-\frac {225 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{8 \sqrt {\pi }}+\frac {\frac {50 \sqrt {\pi }}{7}-\frac {25 \sqrt {\pi }\, \left (320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{448}}{\sqrt {\pi }}\) | \(124\) |
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Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x}} \, dx=-\frac {1}{7} \, {\left (125 \, x^{3} + 390 \, x^{2} + 575 \, x + 764\right )} \sqrt {-2 \, x + 1} \]
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Result contains complex when optimal does not.
Time = 1.13 (sec) , antiderivative size = 189, normalized size of antiderivative = 3.57 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x}} \, dx=\begin {cases} - \frac {25 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{7} - \frac {33 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{7} - \frac {242 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{35} - \frac {2662 \sqrt {5} i \sqrt {10 x - 5}}{175} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\- \frac {25 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{7} - \frac {33 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{7} - \frac {242 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{35} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x}}{175} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x}} \, dx=\frac {125}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {165}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {605}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1331}{8} \, \sqrt {-2 \, x + 1} \]
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x}} \, dx=-\frac {125}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {165}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {605}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1331}{8} \, \sqrt {-2 \, x + 1} \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x}} \, dx=\frac {605\,{\left (1-2\,x\right )}^{3/2}}{8}-\frac {1331\,\sqrt {1-2\,x}}{8}-\frac {165\,{\left (1-2\,x\right )}^{5/2}}{8}+\frac {125\,{\left (1-2\,x\right )}^{7/2}}{56} \]
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