Integrand size = 13, antiderivative size = 28 \[ \int \frac {x}{\sqrt {x^3+x^4}} \, dx=\log (x)-\log \left (-x-2 x^2+2 \sqrt {x^3+x^4}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2054, 212} \[ \int \frac {x}{\sqrt {x^3+x^4}} \, dx=2 \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+x^3}}\right ) \]
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Rule 212
Rule 2054
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x^3+x^4}}\right ) \\ & = 2 \text {arctanh}\left (\frac {x^2}{\sqrt {x^3+x^4}}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {x}{\sqrt {x^3+x^4}} \, dx=-\frac {2 x^{3/2} \sqrt {1+x} \log \left (-\sqrt {x}+\sqrt {1+x}\right )}{\sqrt {x^3 (1+x)}} \]
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Time = 0.82 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.25
method | result | size |
meijerg | \(2 \,\operatorname {arcsinh}\left (\sqrt {x}\right )\) | \(7\) |
pseudoelliptic | \(\ln \left (\frac {2 x^{2}+2 \sqrt {x^{3} \left (1+x \right )}+x}{x}\right )\) | \(24\) |
trager | \(-\ln \left (\frac {-x -2 x^{2}+2 \sqrt {x^{4}+x^{3}}}{x}\right )\) | \(28\) |
default | \(\frac {x \sqrt {\left (1+x \right ) x}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{\sqrt {x^{4}+x^{3}}}\) | \(30\) |
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {x}{\sqrt {x^3+x^4}} \, dx=-\log \left (-\frac {2 \, x^{2} + x - 2 \, \sqrt {x^{4} + x^{3}}}{x}\right ) \]
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\[ \int \frac {x}{\sqrt {x^3+x^4}} \, dx=\int \frac {x}{\sqrt {x^{3} \left (x + 1\right )}}\, dx \]
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\[ \int \frac {x}{\sqrt {x^3+x^4}} \, dx=\int { \frac {x}{\sqrt {x^{4} + x^{3}}} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {x}{\sqrt {x^3+x^4}} \, dx=-\frac {\log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right )}{\mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x}{\sqrt {x^3+x^4}} \, dx=\int \frac {x}{\sqrt {x^4+x^3}} \,d x \]
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