Integrand size = 23, antiderivative size = 66 \[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x} \, dx=\frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}-\frac {2 \sqrt {1+x} \sqrt {1-x+x^2} \text {arctanh}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x^3}} \]
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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, 52, 65, 213} \[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x} \, dx=\frac {2}{3} \sqrt {x+1} \sqrt {x^2-x+1}-\frac {2 \sqrt {x+1} \sqrt {x^2-x+1} \text {arctanh}\left (\sqrt {x^3+1}\right )}{3 \sqrt {x^3+1}} \]
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Rule 52
Rule 65
Rule 213
Rule 272
Rule 929
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {\sqrt {1+x^3}}{x} \, dx}{\sqrt {1+x^3}} \\ & = \frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^3\right )}{3 \sqrt {1+x^3}} \\ & = \frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}+\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )}{3 \sqrt {1+x^3}} \\ & = \frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}+\frac {\left (2 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )}{3 \sqrt {1+x^3}} \\ & = \frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}-\frac {2 \sqrt {1+x} \sqrt {1-x+x^2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x^3}} \\ \end{align*}
Time = 15.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x} \, dx=\frac {2}{3} \left (\sqrt {1+x} \sqrt {1-x+x^2}-\text {arctanh}\left (\sqrt {1+x} \sqrt {1-x+x^2}\right )\right ) \]
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Time = 0.55 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.65
method | result | size |
default | \(-\frac {2 \sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (-\sqrt {x^{3}+1}+\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )\right )}{3 \sqrt {x^{3}+1}}\) | \(43\) |
elliptic | \(\frac {\sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 \sqrt {x^{3}+1}}{3}-\frac {2 \,\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{3}\right )}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(51\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x} \, dx=\frac {2}{3} \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} - \frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} - 1\right ) \]
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\[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x} \, dx=\int \frac {\sqrt {x + 1} \sqrt {x^{2} - x + 1}}{x}\, dx \]
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\[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x} \, dx=\int { \frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x} \,d x } \]
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\[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x} \, dx=\int { \frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x} \, dx=\int \frac {\sqrt {x+1}\,\sqrt {x^2-x+1}}{x} \,d x \]
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