Integrand size = 23, antiderivative size = 66 \[ \int \frac {1}{x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {2}{3 \sqrt {1+x} \sqrt {1-x+x^2}}-\frac {2 \sqrt {1+x^3} \text {arctanh}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x} \sqrt {1-x+x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {929, 272, 53, 65, 213} \[ \int \frac {1}{x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {2}{3 \sqrt {x+1} \sqrt {x^2-x+1}}-\frac {2 \sqrt {x^3+1} \text {arctanh}\left (\sqrt {x^3+1}\right )}{3 \sqrt {x+1} \sqrt {x^2-x+1}} \]
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Rule 53
Rule 65
Rule 213
Rule 272
Rule 929
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^3} \int \frac {1}{x \left (1+x^3\right )^{3/2}} \, dx}{\sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {\sqrt {1+x^3} \text {Subst}\left (\int \frac {1}{x (1+x)^{3/2}} \, dx,x,x^3\right )}{3 \sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {2}{3 \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {\sqrt {1+x^3} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )}{3 \sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {2}{3 \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {\left (2 \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )}{3 \sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {2}{3 \sqrt {1+x} \sqrt {1-x+x^2}}-\frac {2 \sqrt {1+x^3} \tanh ^{-1}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x} \sqrt {1-x+x^2}} \\ \end{align*}
Time = 11.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\frac {2 \left (\sqrt {1+x}-(1+x)^2 \sqrt {\frac {1-x+x^2}{(1+x)^2}} \text {arctanh}\left (\frac {1}{(1+x)^{3/2} \sqrt {\frac {1-x+x^2}{(1+x)^2}}}\right )\right )}{3 (1+x) \sqrt {1-x+x^2}} \]
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Time = 0.67 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.65
method | result | size |
default | \(-\frac {2 \sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right ) \sqrt {x^{3}+1}-1\right )}{3 \left (x^{3}+1\right )}\) | \(43\) |
elliptic | \(\frac {\sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (\frac {2}{3 \sqrt {x^{3}+1}}-\frac {2 \,\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{3}\right )}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(51\) |
risch | \(\frac {2}{3 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}-\frac {2 \,\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right ) \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}}{3 \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(58\) |
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Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=-\frac {{\left (x^{3} + 1\right )} \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} + 1\right ) - {\left (x^{3} + 1\right )} \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} - 1\right ) - 2 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{3 \, {\left (x^{3} + 1\right )}} \]
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\[ \int \frac {1}{x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\int \frac {1}{x \left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}} x} \,d x } \]
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\[ \int \frac {1}{x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}} x} \,d x } \]
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Timed out. \[ \int \frac {1}{x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (x+1\right )}^{3/2}\,{\left (x^2-x+1\right )}^{3/2}} \,d x \]
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