Integrand size = 16, antiderivative size = 90 \[ \int \left (1+\frac {2 x}{1+x^2}\right )^{3/2} \, dx=-\left ((1-x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}\right )-\frac {x \left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {3 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}} \text {arcsinh}(x)}{1+x} \]
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Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6855, 984, 753, 531, 396, 221} \[ \int \left (1+\frac {2 x}{1+x^2}\right )^{3/2} \, dx=\frac {3 \sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1} \text {arcsinh}(x)}{x+1}-\left ((1-x) \sqrt {\frac {2 x}{x^2+1}+1} (x+1)\right )-\frac {x \left (x^2+1\right ) \sqrt {\frac {2 x}{x^2+1}+1}}{x+1} \]
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Rule 221
Rule 396
Rule 531
Rule 753
Rule 984
Rule 6855
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {\left (1+2 x+x^2\right )^{3/2}}{\left (1+x^2\right )^{3/2}} \, dx}{\sqrt {1+2 x+x^2}} \\ & = \frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {(2+2 x)^3}{\left (1+x^2\right )^{3/2}} \, dx}{4 (2+2 x)} \\ & = -\left ((1-x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}\right )+\frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {(8-8 x) (2+2 x)}{\sqrt {1+x^2}} \, dx}{4 (2+2 x)} \\ & = -\left ((1-x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}\right )+\frac {\left (\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {16-16 x^2}{\sqrt {1+x^2}} \, dx}{4 (2+2 x)} \\ & = -\left ((1-x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}\right )-\frac {x \left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {\left (6 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}\right ) \int \frac {1}{\sqrt {1+x^2}} \, dx}{2+2 x} \\ & = -\left ((1-x) (1+x) \sqrt {1+\frac {2 x}{1+x^2}}\right )-\frac {x \left (1+x^2\right ) \sqrt {1+\frac {2 x}{1+x^2}}}{1+x}+\frac {3 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}} \sinh ^{-1}(x)}{1+x} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62 \[ \int \left (1+\frac {2 x}{1+x^2}\right )^{3/2} \, dx=\frac {\sqrt {\frac {(1+x)^2}{1+x^2}} \left (-1-2 x+x^2-3 \sqrt {1+x^2} \log \left (-x+\sqrt {1+x^2}\right )\right )}{1+x} \]
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Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.54
method | result | size |
default | \(\frac {\left (\frac {x^{2}+2 x +1}{x^{2}+1}\right )^{\frac {3}{2}} \left (x^{2}+1\right ) \left (3 \,\operatorname {arcsinh}\left (x \right ) \sqrt {x^{2}+1}+x^{2}-2 x -1\right )}{\left (x +1\right )^{3}}\) | \(49\) |
risch | \(\frac {\left (x^{2}-2 x -1\right ) \sqrt {\frac {\left (x +1\right )^{2}}{x^{2}+1}}}{x +1}+\frac {3 \,\operatorname {arcsinh}\left (x \right ) \sqrt {x^{2}+1}\, \sqrt {\frac {\left (x +1\right )^{2}}{x^{2}+1}}}{x +1}\) | \(62\) |
trager | \(\frac {\left (x^{2}-2 x -1\right ) \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}}{x +1}+3 \ln \left (\frac {\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, x^{2}+x^{2}+\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}+x}{x +1}\right )\) | \(97\) |
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Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \left (1+\frac {2 x}{1+x^2}\right )^{3/2} \, dx=-\frac {3 \, {\left (x + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} + 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + x}{x + 1}\right ) - {\left (x^{2} - 2 \, x - 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + 2 \, x + 2}{x + 1} \]
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\[ \int \left (1+\frac {2 x}{1+x^2}\right )^{3/2} \, dx=\int \left (\frac {2 x}{x^{2} + 1} + 1\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (1+\frac {2 x}{1+x^2}\right )^{3/2} \, dx=\int { {\left (\frac {2 \, x}{x^{2} + 1} + 1\right )}^{\frac {3}{2}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \left (1+\frac {2 x}{1+x^2}\right )^{3/2} \, dx=-{\left (\sqrt {2} - 3 \, \log \left (\sqrt {2} + 1\right )\right )} \mathrm {sgn}\left (x + 1\right ) - 3 \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \mathrm {sgn}\left (x + 1\right ) + \frac {{\left (x \mathrm {sgn}\left (x + 1\right ) - 2 \, \mathrm {sgn}\left (x + 1\right )\right )} x - \mathrm {sgn}\left (x + 1\right )}{\sqrt {x^{2} + 1}} \]
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Timed out. \[ \int \left (1+\frac {2 x}{1+x^2}\right )^{3/2} \, dx=\int {\left (\frac {2\,x}{x^2+1}+1\right )}^{3/2} \,d x \]
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