Integrand size = 8, antiderivative size = 56 \[ \int x \text {arcsinh}\left (\sqrt {x}\right ) \, dx=\frac {3}{16} \sqrt {x} \sqrt {1+x}-\frac {1}{8} x^{3/2} \sqrt {1+x}-\frac {3 \text {arcsinh}\left (\sqrt {x}\right )}{16}+\frac {1}{2} x^2 \text {arcsinh}\left (\sqrt {x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5875, 12, 52, 56, 221} \[ \int x \text {arcsinh}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} x^2 \text {arcsinh}\left (\sqrt {x}\right )-\frac {3 \text {arcsinh}\left (\sqrt {x}\right )}{16}-\frac {1}{8} \sqrt {x+1} x^{3/2}+\frac {3}{16} \sqrt {x+1} \sqrt {x} \]
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Rule 12
Rule 52
Rule 56
Rule 221
Rule 5875
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {arcsinh}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {x^{3/2}}{2 \sqrt {1+x}} \, dx \\ & = \frac {1}{2} x^2 \text {arcsinh}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {x^{3/2}}{\sqrt {1+x}} \, dx \\ & = -\frac {1}{8} x^{3/2} \sqrt {1+x}+\frac {1}{2} x^2 \text {arcsinh}\left (\sqrt {x}\right )+\frac {3}{16} \int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx \\ & = \frac {3}{16} \sqrt {x} \sqrt {1+x}-\frac {1}{8} x^{3/2} \sqrt {1+x}+\frac {1}{2} x^2 \text {arcsinh}\left (\sqrt {x}\right )-\frac {3}{32} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx \\ & = \frac {3}{16} \sqrt {x} \sqrt {1+x}-\frac {1}{8} x^{3/2} \sqrt {1+x}+\frac {1}{2} x^2 \text {arcsinh}\left (\sqrt {x}\right )-\frac {3}{16} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {3}{16} \sqrt {x} \sqrt {1+x}-\frac {1}{8} x^{3/2} \sqrt {1+x}-\frac {3 \text {arcsinh}\left (\sqrt {x}\right )}{16}+\frac {1}{2} x^2 \text {arcsinh}\left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.66 \[ \int x \text {arcsinh}\left (\sqrt {x}\right ) \, dx=\frac {1}{16} \left ((3-2 x) \sqrt {x} \sqrt {1+x}+\left (-3+8 x^2\right ) \text {arcsinh}\left (\sqrt {x}\right )\right ) \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(-\frac {3 \,\operatorname {arcsinh}\left (\sqrt {x}\right )}{16}+\frac {x^{2} \operatorname {arcsinh}\left (\sqrt {x}\right )}{2}-\frac {x^{\frac {3}{2}} \sqrt {1+x}}{8}+\frac {3 \sqrt {x}\, \sqrt {1+x}}{16}\) | \(37\) |
default | \(-\frac {3 \,\operatorname {arcsinh}\left (\sqrt {x}\right )}{16}+\frac {x^{2} \operatorname {arcsinh}\left (\sqrt {x}\right )}{2}-\frac {x^{\frac {3}{2}} \sqrt {1+x}}{8}+\frac {3 \sqrt {x}\, \sqrt {1+x}}{16}\) | \(37\) |
parts | \(\frac {x^{2} \operatorname {arcsinh}\left (\sqrt {x}\right )}{2}-\frac {x^{\frac {3}{2}} \sqrt {1+x}}{8}+\frac {3 \sqrt {x}\, \sqrt {1+x}}{16}-\frac {3 \sqrt {x \left (1+x \right )}\, \ln \left (\frac {1}{2}+x +\sqrt {x^{2}+x}\right )}{32 \sqrt {x}\, \sqrt {1+x}}\) | \(59\) |
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Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.62 \[ \int x \text {arcsinh}\left (\sqrt {x}\right ) \, dx=-\frac {1}{16} \, {\left (2 \, x - 3\right )} \sqrt {x + 1} \sqrt {x} + \frac {1}{16} \, {\left (8 \, x^{2} - 3\right )} \log \left (\sqrt {x + 1} + \sqrt {x}\right ) \]
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\[ \int x \text {arcsinh}\left (\sqrt {x}\right ) \, dx=\int x \operatorname {asinh}{\left (\sqrt {x} \right )}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.64 \[ \int x \text {arcsinh}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {arsinh}\left (\sqrt {x}\right ) - \frac {1}{8} \, \sqrt {x + 1} x^{\frac {3}{2}} + \frac {3}{16} \, \sqrt {x + 1} \sqrt {x} - \frac {3}{16} \, \operatorname {arsinh}\left (\sqrt {x}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int x \text {arcsinh}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{16} \, \sqrt {x^{2} + x} {\left (2 \, x - 3\right )} + \frac {3}{32} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \]
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Timed out. \[ \int x \text {arcsinh}\left (\sqrt {x}\right ) \, dx=\int x\,\mathrm {asinh}\left (\sqrt {x}\right ) \,d x \]
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