Integrand size = 17, antiderivative size = 40 \[ \int \sqrt {1-2 x} (3+5 x)^2 \, dx=-\frac {121}{12} (1-2 x)^{3/2}+\frac {11}{2} (1-2 x)^{5/2}-\frac {25}{28} (1-2 x)^{7/2} \]
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Time = 0.01 (sec) , antiderivative size = 40, 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} \[ \int \sqrt {1-2 x} (3+5 x)^2 \, dx=-\frac {25}{28} (1-2 x)^{7/2}+\frac {11}{2} (1-2 x)^{5/2}-\frac {121}{12} (1-2 x)^{3/2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {121}{4} \sqrt {1-2 x}-\frac {55}{2} (1-2 x)^{3/2}+\frac {25}{4} (1-2 x)^{5/2}\right ) \, dx \\ & = -\frac {121}{12} (1-2 x)^{3/2}+\frac {11}{2} (1-2 x)^{5/2}-\frac {25}{28} (1-2 x)^{7/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int \sqrt {1-2 x} (3+5 x)^2 \, dx=-\frac {1}{21} (1-2 x)^{3/2} \left (115+156 x+75 x^2\right ) \]
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Time = 0.93 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (75 x^{2}+156 x +115\right )}{21}\) | \(20\) |
trager | \(\left (\frac {50}{7} x^{3}+\frac {79}{7} x^{2}+\frac {74}{21} x -\frac {115}{21}\right ) \sqrt {1-2 x}\) | \(24\) |
pseudoelliptic | \(\frac {\sqrt {1-2 x}\, \left (150 x^{3}+237 x^{2}+74 x -115\right )}{21}\) | \(25\) |
derivativedivides | \(-\frac {121 \left (1-2 x \right )^{\frac {3}{2}}}{12}+\frac {11 \left (1-2 x \right )^{\frac {5}{2}}}{2}-\frac {25 \left (1-2 x \right )^{\frac {7}{2}}}{28}\) | \(29\) |
default | \(-\frac {121 \left (1-2 x \right )^{\frac {3}{2}}}{12}+\frac {11 \left (1-2 x \right )^{\frac {5}{2}}}{2}-\frac {25 \left (1-2 x \right )^{\frac {7}{2}}}{28}\) | \(29\) |
risch | \(-\frac {\left (150 x^{3}+237 x^{2}+74 x -115\right ) \left (-1+2 x \right )}{21 \sqrt {1-2 x}}\) | \(30\) |
meijerg | \(\frac {3 \sqrt {\pi }-\frac {3 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{2}}{\sqrt {\pi }}-\frac {15 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{4 \sqrt {\pi }}+\frac {\frac {10 \sqrt {\pi }}{21}-\frac {5 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{84}}{\sqrt {\pi }}\) | \(91\) |
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Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.60 \[ \int \sqrt {1-2 x} (3+5 x)^2 \, dx=\frac {1}{21} \, {\left (150 \, x^{3} + 237 \, x^{2} + 74 \, x - 115\right )} \sqrt {-2 \, x + 1} \]
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Result contains complex when optimal does not.
Time = 0.88 (sec) , antiderivative size = 185, normalized size of antiderivative = 4.62 \[ \int \sqrt {1-2 x} (3+5 x)^2 \, dx=\begin {cases} \frac {10 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{7} - \frac {11 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{35} - \frac {242 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{525} - \frac {2662 \sqrt {5} i \sqrt {10 x - 5}}{2625} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {10 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{7} - \frac {11 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{35} - \frac {242 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{525} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x}}{2625} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (3+5 x)^2 \, dx=-\frac {25}{28} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {11}{2} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {121}{12} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \sqrt {1-2 x} (3+5 x)^2 \, dx=\frac {25}{28} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {11}{2} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {121}{12} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
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Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int \sqrt {1-2 x} (3+5 x)^2 \, dx=-\frac {{\left (1-2\,x\right )}^{3/2}\,\left (924\,x+75\,{\left (2\,x-1\right )}^2+385\right )}{84} \]
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