Integrand size = 20, antiderivative size = 16 \[ \int \frac {-3+2 x^5}{x^3 \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1604} \[ \int \frac {-3+2 x^5}{x^3 \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (x^6+x\right )^{3/4}}{3 x^3} \]
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Rule 1604
Rubi steps \begin{align*} \text {integral}& = \frac {4 \left (x+x^6\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 10.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-3+2 x^5}{x^3 \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3} \]
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Time = 0.86 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(\frac {4 \left (x^{6}+x \right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
pseudoelliptic | \(\frac {4 \left (x^{6}+x \right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
risch | \(\frac {\frac {4 x^{5}}{3}+\frac {4}{3}}{x^{2} {\left (x \left (x^{5}+1\right )\right )}^{\frac {1}{4}}}\) | \(20\) |
gosper | \(\frac {4 \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right )}{3 x^{2} \left (x^{6}+x \right )^{\frac {1}{4}}}\) | \(32\) |
meijerg | \(\frac {4 \operatorname {hypergeom}\left (\left [-\frac {9}{20}, \frac {1}{4}\right ], \left [\frac {11}{20}\right ], -x^{5}\right )}{3 x^{\frac {9}{4}}}+\frac {8 x^{\frac {11}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {11}{20}\right ], \left [\frac {31}{20}\right ], -x^{5}\right )}{11}\) | \(34\) |
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-3+2 x^5}{x^3 \sqrt [4]{x+x^6}} \, dx=\frac {4 \, {\left (x^{6} + x\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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\[ \int \frac {-3+2 x^5}{x^3 \sqrt [4]{x+x^6}} \, dx=\int \frac {2 x^{5} - 3}{x^{3} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]
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\[ \int \frac {-3+2 x^5}{x^3 \sqrt [4]{x+x^6}} \, dx=\int { \frac {2 \, x^{5} - 3}{{\left (x^{6} + x\right )}^{\frac {1}{4}} x^{3}} \,d x } \]
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\[ \int \frac {-3+2 x^5}{x^3 \sqrt [4]{x+x^6}} \, dx=\int { \frac {2 \, x^{5} - 3}{{\left (x^{6} + x\right )}^{\frac {1}{4}} x^{3}} \,d x } \]
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Time = 5.55 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-3+2 x^5}{x^3 \sqrt [4]{x+x^6}} \, dx=\frac {4\,{\left (x^6+x\right )}^{3/4}}{3\,x^3} \]
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