Integrand size = 18, antiderivative size = 18 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x^4} \, dx=-\frac {4 \left (x^3+x^4\right )^{9/4}}{9 x^9} \]
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Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2077, 2041, 2039} \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x^4} \, dx=-\frac {4 \left (x^4+x^3\right )^{5/4}}{9 x^6}-\frac {4 \left (x^4+x^3\right )^{5/4}}{9 x^5} \]
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Rule 2039
Rule 2041
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt [4]{x^3+x^4}}{x^4}+\frac {\sqrt [4]{x^3+x^4}}{x^3}\right ) \, dx \\ & = \int \frac {\sqrt [4]{x^3+x^4}}{x^4} \, dx+\int \frac {\sqrt [4]{x^3+x^4}}{x^3} \, dx \\ & = -\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}-\frac {4 \left (x^3+x^4\right )^{5/4}}{5 x^5}-\frac {4}{9} \int \frac {\sqrt [4]{x^3+x^4}}{x^3} \, dx \\ & = -\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x^4} \, dx=-\frac {4 \left (x^3 (1+x)\right )^{9/4}}{9 x^9} \]
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Time = 0.79 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11
method | result | size |
gosper | \(-\frac {4 \left (1+x \right )^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(20\) |
pseudoelliptic | \(-\frac {4 \left (1+x \right )^{2} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(20\) |
trager | \(-\frac {4 \left (x^{2}+2 x +1\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(23\) |
meijerg | \(-\frac {4 \left (-\frac {4}{5} x^{2}+\frac {1}{5} x +1\right ) \left (1+x \right )^{\frac {1}{4}}}{9 x^{\frac {9}{4}}}-\frac {4 \left (1+x \right )^{\frac {5}{4}}}{5 x^{\frac {5}{4}}}\) | \(32\) |
risch | \(-\frac {4 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (x^{3}+3 x^{2}+3 x +1\right )}{9 \left (1+x \right ) x^{3}}\) | \(33\) |
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Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x^4} \, dx=-\frac {4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 2 \, x + 1\right )}}{9 \, x^{3}} \]
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\[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x^4} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )} \left (x + 1\right )}{x^{4}}\, dx \]
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\[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x^4} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{x^{4}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.50 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x^4} \, dx=-\frac {4}{9} \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} \]
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Time = 5.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.39 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^4}}{x^4} \, dx=-\frac {8\,x\,{\left (x^4+x^3\right )}^{1/4}+4\,{\left (x^4+x^3\right )}^{1/4}+4\,x^2\,{\left (x^4+x^3\right )}^{1/4}}{9\,x^3} \]
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