Integrand size = 23, antiderivative size = 16 \[ \int \frac {\left (1+x^2\right ) \left (3+x^2\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4 \left (x+x^3\right )^{7/4}}{7 x^7} \]
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Leaf count is larger than twice the leaf count of optimal. \(33\) vs. \(2(16)=32\).
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06, number of steps used = 14, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2077, 2050, 2036, 371} \[ \int \frac {\left (1+x^2\right ) \left (3+x^2\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4 \left (x^3+x\right )^{3/4}}{7 x^6}-\frac {4 \left (x^3+x\right )^{3/4}}{7 x^4} \]
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Rule 371
Rule 2036
Rule 2050
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{x^6 \sqrt [4]{x+x^3}}+\frac {4}{x^4 \sqrt [4]{x+x^3}}+\frac {1}{x^2 \sqrt [4]{x+x^3}}\right ) \, dx \\ & = 3 \int \frac {1}{x^6 \sqrt [4]{x+x^3}} \, dx+4 \int \frac {1}{x^4 \sqrt [4]{x+x^3}} \, dx+\int \frac {1}{x^2 \sqrt [4]{x+x^3}} \, dx \\ & = -\frac {4 \left (x+x^3\right )^{3/4}}{7 x^6}-\frac {16 \left (x+x^3\right )^{3/4}}{13 x^4}-\frac {4 \left (x+x^3\right )^{3/4}}{5 x^2}+\frac {1}{5} \int \frac {1}{\sqrt [4]{x+x^3}} \, dx-\frac {15}{7} \int \frac {1}{x^4 \sqrt [4]{x+x^3}} \, dx-\frac {28}{13} \int \frac {1}{x^2 \sqrt [4]{x+x^3}} \, dx \\ & = -\frac {4 \left (x+x^3\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^3\right )^{3/4}}{7 x^4}+\frac {12 \left (x+x^3\right )^{3/4}}{13 x^2}-\frac {28}{65} \int \frac {1}{\sqrt [4]{x+x^3}} \, dx+\frac {15}{13} \int \frac {1}{x^2 \sqrt [4]{x+x^3}} \, dx+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{1+x^2}} \, dx}{5 \sqrt [4]{x+x^3}} \\ & = -\frac {4 \left (x+x^3\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^3\right )^{3/4}}{7 x^4}+\frac {4 x \sqrt [4]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{8},\frac {11}{8},-x^2\right )}{15 \sqrt [4]{x+x^3}}+\frac {3}{13} \int \frac {1}{\sqrt [4]{x+x^3}} \, dx-\frac {\left (28 \sqrt [4]{x} \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{1+x^2}} \, dx}{65 \sqrt [4]{x+x^3}} \\ & = -\frac {4 \left (x+x^3\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^3\right )^{3/4}}{7 x^4}-\frac {4 x \sqrt [4]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{8},\frac {11}{8},-x^2\right )}{13 \sqrt [4]{x+x^3}}+\frac {\left (3 \sqrt [4]{x} \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{1+x^2}} \, dx}{13 \sqrt [4]{x+x^3}} \\ & = -\frac {4 \left (x+x^3\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^3\right )^{3/4}}{7 x^4} \\ \end{align*}
Time = 10.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {\left (1+x^2\right ) \left (3+x^2\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4 \left (1+x^2\right ) \left (x+x^3\right )^{3/4}}{7 x^6} \]
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Time = 0.89 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12
method | result | size |
trager | \(-\frac {4 \left (x^{2}+1\right ) \left (x^{3}+x \right )^{\frac {3}{4}}}{7 x^{6}}\) | \(18\) |
gosper | \(-\frac {4 \left (x^{2}+1\right )^{2}}{7 x^{5} \left (x^{3}+x \right )^{\frac {1}{4}}}\) | \(20\) |
risch | \(-\frac {4 \left (x^{4}+2 x^{2}+1\right )}{7 x^{5} {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{4}}}\) | \(25\) |
meijerg | \(-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {21}{8}, \frac {1}{4}\right ], \left [-\frac {13}{8}\right ], -x^{2}\right )}{7 x^{\frac {21}{4}}}-\frac {16 \operatorname {hypergeom}\left (\left [-\frac {13}{8}, \frac {1}{4}\right ], \left [-\frac {5}{8}\right ], -x^{2}\right )}{13 x^{\frac {13}{4}}}-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {5}{8}, \frac {1}{4}\right ], \left [\frac {3}{8}\right ], -x^{2}\right )}{5 x^{\frac {5}{4}}}\) | \(50\) |
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Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {\left (1+x^2\right ) \left (3+x^2\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4 \, {\left (x^{3} + x\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )}}{7 \, x^{6}} \]
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\[ \int \frac {\left (1+x^2\right ) \left (3+x^2\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=\int \frac {\left (x^{2} + 1\right ) \left (x^{2} + 3\right )}{x^{6} \sqrt [4]{x \left (x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {\left (1+x^2\right ) \left (3+x^2\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=\int { \frac {{\left (x^{2} + 3\right )} {\left (x^{2} + 1\right )}}{{\left (x^{3} + x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int \frac {\left (1+x^2\right ) \left (3+x^2\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4}{7} \, {\left (\frac {1}{x} + \frac {1}{x^{3}}\right )}^{\frac {7}{4}} \]
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Time = 5.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\left (1+x^2\right ) \left (3+x^2\right )}{x^6 \sqrt [4]{x+x^3}} \, dx=-\frac {4\,{\left (x^3+x\right )}^{3/4}+4\,x^2\,{\left (x^3+x\right )}^{3/4}}{7\,x^6} \]
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