Integrand size = 11, antiderivative size = 32 \[ \int \frac {x^2}{\sqrt {-1+x}} \, dx=2 \sqrt {-1+x}+\frac {4}{3} (-1+x)^{3/2}+\frac {2}{5} (-1+x)^{5/2} \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^2}{\sqrt {-1+x}} \, dx=\frac {2}{5} (x-1)^{5/2}+\frac {4}{3} (x-1)^{3/2}+2 \sqrt {x-1} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {-1+x}}+2 \sqrt {-1+x}+(-1+x)^{3/2}\right ) \, dx \\ & = 2 \sqrt {-1+x}+\frac {4}{3} (-1+x)^{3/2}+\frac {2}{5} (-1+x)^{5/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{\sqrt {-1+x}} \, dx=\frac {2}{15} \sqrt {-1+x} \left (8+4 x+3 x^2\right ) \]
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Time = 0.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.53
method | result | size |
trager | \(\left (\frac {2}{5} x^{2}+\frac {8}{15} x +\frac {16}{15}\right ) \sqrt {-1+x}\) | \(17\) |
gosper | \(\frac {2 \sqrt {-1+x}\, \left (3 x^{2}+4 x +8\right )}{15}\) | \(18\) |
risch | \(\frac {2 \sqrt {-1+x}\, \left (3 x^{2}+4 x +8\right )}{15}\) | \(18\) |
derivativedivides | \(\frac {4 \left (-1+x \right )^{\frac {3}{2}}}{3}+\frac {2 \left (-1+x \right )^{\frac {5}{2}}}{5}+2 \sqrt {-1+x}\) | \(23\) |
default | \(\frac {4 \left (-1+x \right )^{\frac {3}{2}}}{3}+\frac {2 \left (-1+x \right )^{\frac {5}{2}}}{5}+2 \sqrt {-1+x}\) | \(23\) |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (-1+x \right )}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 x^{2}+8 x +16\right ) \sqrt {1-x}}{15}\right )}{\sqrt {\pi }\, \sqrt {\operatorname {signum}\left (-1+x \right )}}\) | \(48\) |
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none
Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.53 \[ \int \frac {x^2}{\sqrt {-1+x}} \, dx=\frac {2}{15} \, {\left (3 \, x^{2} + 4 \, x + 8\right )} \sqrt {x - 1} \]
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Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.38 \[ \int \frac {x^2}{\sqrt {-1+x}} \, dx=\begin {cases} \frac {2 x^{2} \sqrt {x - 1}}{5} + \frac {8 x \sqrt {x - 1}}{15} + \frac {16 \sqrt {x - 1}}{15} & \text {for}\: \left |{x}\right | > 1 \\\frac {2 i x^{2} \sqrt {1 - x}}{5} + \frac {8 i x \sqrt {1 - x}}{15} + \frac {16 i \sqrt {1 - x}}{15} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{\sqrt {-1+x}} \, dx=\frac {2}{5} \, {\left (x - 1\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} + 2 \, \sqrt {x - 1} \]
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none
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{\sqrt {-1+x}} \, dx=\frac {2}{5} \, {\left (x - 1\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} + 2 \, \sqrt {x - 1} \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59 \[ \int \frac {x^2}{\sqrt {-1+x}} \, dx=\frac {2\,\sqrt {x-1}\,\left (10\,x+3\,{\left (x-1\right )}^2+5\right )}{15} \]
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