Integrand size = 20, antiderivative size = 14 \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4 x}{\sqrt [4]{x^2 (1+x)}} \]
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Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2081, 75} \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4 x}{\sqrt [4]{x^3+x^2}} \]
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Rule 75
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{1+x}\right ) \int \frac {2+x}{\sqrt {x} (1+x)^{5/4}} \, dx}{\sqrt [4]{x^2+x^3}} \\ & = \frac {4 x}{\sqrt [4]{x^2+x^3}} \\ \end{align*}
Time = 1.99 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4 x}{\sqrt [4]{x^2 (1+x)}} \]
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Time = 1.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(\frac {4 x}{\left (x^{3}+x^{2}\right )^{\frac {1}{4}}}\) | \(13\) |
risch | \(\frac {4 x}{\left (x^{2} \left (1+x \right )\right )^{\frac {1}{4}}}\) | \(13\) |
trager | \(\frac {4 \left (x^{3}+x^{2}\right )^{\frac {3}{4}}}{\left (1+x \right ) x}\) | \(20\) |
meijerg | \(4 \sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -x \right )+\frac {2 x^{\frac {3}{2}} \operatorname {hypergeom}\left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], -x \right )}{3}\) | \(30\) |
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none
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4 \, {\left (x^{3} + x^{2}\right )}^{\frac {3}{4}}}{x^{2} + x} \]
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\[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\int \frac {x + 2}{\sqrt [4]{x^{2} \left (x + 1\right )} \left (x + 1\right )}\, dx \]
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\[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\int { \frac {x + 2}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{4}} {\left (x + 1\right )}} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4}{{\left (\frac {1}{x} + \frac {1}{x^{2}}\right )}^{\frac {1}{4}}} \]
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Time = 4.91 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {2+x}{(1+x) \sqrt [4]{x^2+x^3}} \, dx=\frac {4\,{\left (x^3+x^2\right )}^{3/4}}{x\,\left (x+1\right )} \]
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