Integrand size = 20, antiderivative size = 28 \[ \int \frac {1}{x (1+x) \sqrt [4]{x^3+x^4}} \, dx=-\frac {4 (1+4 x) \left (x^3+x^4\right )^{3/4}}{3 x^3 (1+x)} \]
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Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2081, 47, 37} \[ \int \frac {1}{x (1+x) \sqrt [4]{x^3+x^4}} \, dx=\frac {4}{\sqrt [4]{x^4+x^3}}-\frac {16 (x+1)}{3 \sqrt [4]{x^4+x^3}} \]
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Rule 37
Rule 47
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{3/4} \sqrt [4]{1+x}\right ) \int \frac {1}{x^{7/4} (1+x)^{5/4}} \, dx}{\sqrt [4]{x^3+x^4}} \\ & = \frac {4}{\sqrt [4]{x^3+x^4}}+\frac {\left (4 x^{3/4} \sqrt [4]{1+x}\right ) \int \frac {1}{x^{7/4} \sqrt [4]{1+x}} \, dx}{\sqrt [4]{x^3+x^4}} \\ & = \frac {4}{\sqrt [4]{x^3+x^4}}-\frac {16 (1+x)}{3 \sqrt [4]{x^3+x^4}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x (1+x) \sqrt [4]{x^3+x^4}} \, dx=-\frac {4 (1+4 x)}{3 \sqrt [4]{x^3 (1+x)}} \]
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Time = 0.93 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.57
method | result | size |
meijerg | \(-\frac {4 \left (1+4 x \right )}{3 x^{\frac {3}{4}} \left (1+x \right )^{\frac {1}{4}}}\) | \(16\) |
gosper | \(-\frac {4 \left (1+4 x \right )}{3 \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}\) | \(17\) |
risch | \(-\frac {4 \left (1+4 x \right )}{3 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\) | \(17\) |
pseudoelliptic | \(\frac {-\frac {4}{3}-\frac {16 x}{3}}{\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\) | \(17\) |
trager | \(-\frac {4 \left (1+4 x \right ) \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{3 x^{3} \left (1+x \right )}\) | \(25\) |
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Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x (1+x) \sqrt [4]{x^3+x^4}} \, dx=-\frac {4 \, {\left (4 \, x + 1\right )}}{3 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}} \]
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\[ \int \frac {1}{x (1+x) \sqrt [4]{x^3+x^4}} \, dx=\int \frac {1}{x \sqrt [4]{x^{3} \left (x + 1\right )} \left (x + 1\right )}\, dx \]
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\[ \int \frac {1}{x (1+x) \sqrt [4]{x^3+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )} x} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x (1+x) \sqrt [4]{x^3+x^4}} \, dx=-\frac {4}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {3}{4}} - \frac {4}{{\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}} \]
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Time = 5.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x (1+x) \sqrt [4]{x^3+x^4}} \, dx=-\frac {16\,x+4}{3\,{\left (x^4+x^3\right )}^{1/4}} \]
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