Integrand size = 18, antiderivative size = 18 \[ \int \frac {2+x}{x^2 \sqrt [4]{x^2+x^3}} \, dx=-\frac {4 \left (x^2+x^3\right )^{3/4}}{3 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1604} \[ \int \frac {2+x}{x^2 \sqrt [4]{x^2+x^3}} \, dx=-\frac {4 \left (x^3+x^2\right )^{3/4}}{3 x^3} \]
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Rule 1604
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \left (x^2+x^3\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{x^2 \sqrt [4]{x^2+x^3}} \, dx=-\frac {4 \left (x^2 (1+x)\right )^{3/4}}{3 x^3} \]
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Time = 0.81 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
trager | \(-\frac {4 \left (x^{3}+x^{2}\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(15\) |
gosper | \(-\frac {4 \left (1+x \right )}{3 x \left (x^{3}+x^{2}\right )^{\frac {1}{4}}}\) | \(18\) |
risch | \(-\frac {4 \left (1+x \right )}{3 x \left (x^{2} \left (1+x \right )\right )^{\frac {1}{4}}}\) | \(18\) |
meijerg | \(-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {3}{2}, \frac {1}{4}\right ], \left [-\frac {1}{2}\right ], -x \right )}{3 x^{\frac {3}{2}}}-\frac {2 \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {1}{2}\right ], -x \right )}{\sqrt {x}}\) | \(30\) |
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Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {2+x}{x^2 \sqrt [4]{x^2+x^3}} \, dx=-\frac {4 \, {\left (x^{3} + x^{2}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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\[ \int \frac {2+x}{x^2 \sqrt [4]{x^2+x^3}} \, dx=\int \frac {x + 2}{x^{2} \sqrt [4]{x^{2} \left (x + 1\right )}}\, dx \]
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\[ \int \frac {2+x}{x^2 \sqrt [4]{x^2+x^3}} \, dx=\int { \frac {x + 2}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{4}} x^{2}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {2+x}{x^2 \sqrt [4]{x^2+x^3}} \, dx=-\frac {4}{3} \, {\left (\frac {1}{x} + \frac {1}{x^{2}}\right )}^{\frac {3}{4}} \]
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Time = 4.94 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {2+x}{x^2 \sqrt [4]{x^2+x^3}} \, dx=-\frac {4\,{\left (x^3+x^2\right )}^{3/4}}{3\,x^3} \]
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