Integrand size = 13, antiderivative size = 27 \[ \int x^3 \sqrt {1+x^2} \, dx=-\frac {1}{3} \left (1+x^2\right )^{3/2}+\frac {1}{5} \left (1+x^2\right )^{5/2} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int x^3 \sqrt {1+x^2} \, dx=\frac {1}{5} \left (x^2+1\right )^{5/2}-\frac {1}{3} \left (x^2+1\right )^{3/2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x \sqrt {1+x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\sqrt {1+x}+(1+x)^{3/2}\right ) \, dx,x,x^2\right ) \\ & = -\frac {1}{3} \left (1+x^2\right )^{3/2}+\frac {1}{5} \left (1+x^2\right )^{5/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int x^3 \sqrt {1+x^2} \, dx=\frac {1}{15} \left (1+x^2\right )^{3/2} \left (-2+3 x^2\right ) \]
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Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(\frac {\left (x^{2}+1\right )^{\frac {3}{2}} \left (3 x^{2}-2\right )}{15}\) | \(17\) |
pseudoelliptic | \(\frac {\left (x^{2}+1\right )^{\frac {3}{2}} \left (3 x^{2}-2\right )}{15}\) | \(17\) |
risch | \(\frac {\left (3 x^{4}+x^{2}-2\right ) \sqrt {x^{2}+1}}{15}\) | \(20\) |
trager | \(\left (\frac {1}{5} x^{4}+\frac {1}{15} x^{2}-\frac {2}{15}\right ) \sqrt {x^{2}+1}\) | \(21\) |
default | \(\frac {x^{2} \left (x^{2}+1\right )^{\frac {3}{2}}}{5}-\frac {2 \left (x^{2}+1\right )^{\frac {3}{2}}}{15}\) | \(23\) |
meijerg | \(-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (x^{2}+1\right )^{\frac {3}{2}} \left (-3 x^{2}+2\right )}{15}}{4 \sqrt {\pi }}\) | \(31\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int x^3 \sqrt {1+x^2} \, dx=\frac {1}{15} \, {\left (3 \, x^{4} + x^{2} - 2\right )} \sqrt {x^{2} + 1} \]
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Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int x^3 \sqrt {1+x^2} \, dx=\frac {x^{4} \sqrt {x^{2} + 1}}{5} + \frac {x^{2} \sqrt {x^{2} + 1}}{15} - \frac {2 \sqrt {x^{2} + 1}}{15} \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int x^3 \sqrt {1+x^2} \, dx=\frac {1}{5} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - \frac {2}{15} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]
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Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int x^3 \sqrt {1+x^2} \, dx=\frac {1}{5} \, {\left (x^{2} + 1\right )}^{\frac {5}{2}} - \frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int x^3 \sqrt {1+x^2} \, dx=\sqrt {x^2+1}\,\left (\frac {x^4}{5}+\frac {x^2}{15}-\frac {2}{15}\right ) \]
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