Integrand size = 11, antiderivative size = 34 \[ \int x^2 \sqrt {1+x} \, dx=\frac {2}{3} (1+x)^{3/2}-\frac {4}{5} (1+x)^{5/2}+\frac {2}{7} (1+x)^{7/2} \]
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Time = 0.00 (sec) , antiderivative size = 34, 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} \[ \int x^2 \sqrt {1+x} \, dx=\frac {2}{7} (x+1)^{7/2}-\frac {4}{5} (x+1)^{5/2}+\frac {2}{3} (x+1)^{3/2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt {1+x}-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx \\ & = \frac {2}{3} (1+x)^{3/2}-\frac {4}{5} (1+x)^{5/2}+\frac {2}{7} (1+x)^{7/2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62 \[ \int x^2 \sqrt {1+x} \, dx=\frac {2}{105} (1+x)^{3/2} \left (8-12 x+15 x^2\right ) \]
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Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(\frac {2 \left (1+x \right )^{\frac {3}{2}} \left (15 x^{2}-12 x +8\right )}{105}\) | \(18\) |
trager | \(\left (\frac {2}{7} x^{3}+\frac {2}{35} x^{2}-\frac {8}{105} x +\frac {16}{105}\right ) \sqrt {1+x}\) | \(22\) |
derivativedivides | \(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-\frac {4 \left (1+x \right )^{\frac {5}{2}}}{5}+\frac {2 \left (1+x \right )^{\frac {7}{2}}}{7}\) | \(23\) |
default | \(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-\frac {4 \left (1+x \right )^{\frac {5}{2}}}{5}+\frac {2 \left (1+x \right )^{\frac {7}{2}}}{7}\) | \(23\) |
risch | \(\frac {2 \left (15 x^{3}+3 x^{2}-4 x +8\right ) \sqrt {1+x}}{105}\) | \(23\) |
meijerg | \(-\frac {\frac {32 \sqrt {\pi }}{105}-\frac {4 \sqrt {\pi }\, \left (1+x \right )^{\frac {3}{2}} \left (15 x^{2}-12 x +8\right )}{105}}{2 \sqrt {\pi }}\) | \(32\) |
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Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int x^2 \sqrt {1+x} \, dx=\frac {2}{105} \, {\left (15 \, x^{3} + 3 \, x^{2} - 4 \, x + 8\right )} \sqrt {x + 1} \]
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Time = 0.78 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41 \[ \int x^2 \sqrt {1+x} \, dx=\frac {2 x^{3} \sqrt {x + 1}}{7} + \frac {2 x^{2} \sqrt {x + 1}}{35} - \frac {8 x \sqrt {x + 1}}{105} + \frac {16 \sqrt {x + 1}}{105} \]
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Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int x^2 \sqrt {1+x} \, dx=\frac {2}{7} \, {\left (x + 1\right )}^{\frac {7}{2}} - \frac {4}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} + \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \]
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int x^2 \sqrt {1+x} \, dx=\frac {2}{7} \, {\left (x + 1\right )}^{\frac {7}{2}} - \frac {4}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} + \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.56 \[ \int x^2 \sqrt {1+x} \, dx=-\frac {2\,{\left (x+1\right )}^{3/2}\,\left (42\,x-15\,{\left (x+1\right )}^2+7\right )}{105} \]
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