Integrand size = 15, antiderivative size = 27 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {11}{2 \sqrt {1-2 x}}+\frac {5}{2} \sqrt {1-2 x} \]
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Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {5}{2} \sqrt {1-2 x}+\frac {11}{2 \sqrt {1-2 x}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {11}{2 (1-2 x)^{3/2}}-\frac {5}{2 \sqrt {1-2 x}}\right ) \, dx \\ & = \frac {11}{2 \sqrt {1-2 x}}+\frac {5}{2} \sqrt {1-2 x} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {8-5 x}{\sqrt {1-2 x}} \]
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Time = 1.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52
method | result | size |
pseudoelliptic | \(\frac {8-5 x}{\sqrt {1-2 x}}\) | \(14\) |
gosper | \(-\frac {-8+5 x}{\sqrt {1-2 x}}\) | \(15\) |
risch | \(-\frac {-8+5 x}{\sqrt {1-2 x}}\) | \(15\) |
derivativedivides | \(\frac {11}{2 \sqrt {1-2 x}}+\frac {5 \sqrt {1-2 x}}{2}\) | \(20\) |
default | \(\frac {11}{2 \sqrt {1-2 x}}+\frac {5 \sqrt {1-2 x}}{2}\) | \(20\) |
trager | \(\frac {\left (-8+5 x \right ) \sqrt {1-2 x}}{-1+2 x}\) | \(21\) |
meijerg | \(-\frac {3 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-5 \sqrt {\pi }+\frac {5 \sqrt {\pi }\, \left (-8 x +8\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(51\) |
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Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {{\left (5 \, x - 8\right )} \sqrt {-2 \, x + 1}}{2 \, x - 1} \]
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Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {5 x \sqrt {1 - 2 x}}{2 x - 1} - \frac {8 \sqrt {1 - 2 x}}{2 x - 1} \]
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Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {5}{2} \, \sqrt {-2 \, x + 1} + \frac {11}{2 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=\frac {5}{2} \, \sqrt {-2 \, x + 1} + \frac {11}{2 \, \sqrt {-2 \, x + 1}} \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx=-\frac {5\,x-8}{\sqrt {1-2\,x}} \]
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