Integrand size = 25, antiderivative size = 156 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\frac {2}{3} \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )-\frac {1}{3} \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^3+x^5}}{-x^2+\sqrt {x^3+x^5}}\right )-\frac {2}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^3+x^5}}{\sqrt {2}}}{x \sqrt [4]{x^3+x^5}}\right ) \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.50 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.05, number of steps used = 18, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2081, 6857, 973, 477, 524} \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=-\frac {4 \left (1-\sqrt [3]{-1}\right ) \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^2,-\sqrt [3]{-1} x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {4 \left (1+(-1)^{2/3}\right ) \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^2,(-1)^{2/3} x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {8 \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {3}{8},1,-\frac {1}{4},\frac {11}{8},x^2,-x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {4 \left (1+(-1)^{2/3}\right ) x \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {7}{8},-\frac {1}{4},1,\frac {15}{8},-x^2,-\sqrt [3]{-1} x^2\right )}{21 \sqrt [4]{x^2+1}}-\frac {4 \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {7}{8},-\frac {1}{4},1,\frac {15}{8},-x^2,(-1)^{2/3} x^2\right )}{21 \sqrt [4]{x^2+1}}-\frac {8 x \sqrt [4]{x^5+x^3} \operatorname {AppellF1}\left (\frac {7}{8},1,-\frac {1}{4},\frac {15}{8},x^2,-x^2\right )}{21 \sqrt [4]{x^2+1}} \]
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Rule 477
Rule 524
Rule 973
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^3+x^5} \int \frac {(1+x) \sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (-1+x^3\right )} \, dx}{x^{3/4} \sqrt [4]{1+x^2}} \\ & = \frac {\sqrt [4]{x^3+x^5} \int \left (-\frac {2 \sqrt [4]{1+x^2}}{3 (1-x) \sqrt [4]{x}}-\frac {\left (1+(-1)^{2/3}\right ) \sqrt [4]{1+x^2}}{3 \sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt [4]{1+x^2}}{3 \sqrt [4]{x} \left (1-(-1)^{2/3} x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x^2}} \\ & = -\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{(1-x) \sqrt [4]{x}} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-(-1)^{2/3} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}} \\ & = -\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1-x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1+\sqrt [3]{-1} x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1+\sqrt [3]{-1} x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-(-1)^{2/3} x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (\sqrt [3]{-1} \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1-(-1)^{2/3} x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}} \\ & = -\frac {\left (8 \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (8 \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1-x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1+\sqrt [3]{-1} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 (-1)^{2/3} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1+\sqrt [3]{-1} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-(-1)^{2/3} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (4 \sqrt [3]{-1} \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1-(-1)^{2/3} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}} \\ & = -\frac {4 \left (1-\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^2,-\sqrt [3]{-1} x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {4 \left (1+(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^2,(-1)^{2/3} x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {8 \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {3}{8},1,-\frac {1}{4},\frac {11}{8},x^2,-x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {4 (-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {7}{8},-\frac {1}{4},1,\frac {15}{8},-x^2,-\sqrt [3]{-1} x^2\right )}{21 \sqrt [4]{1+x^2}}-\frac {4 \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {7}{8},-\frac {1}{4},1,\frac {15}{8},-x^2,(-1)^{2/3} x^2\right )}{21 \sqrt [4]{1+x^2}}-\frac {8 x \sqrt [4]{x^3+x^5} \operatorname {AppellF1}\left (\frac {7}{8},1,-\frac {1}{4},\frac {15}{8},x^2,-x^2\right )}{21 \sqrt [4]{1+x^2}} \\ \end{align*}
Time = 1.84 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.10 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\frac {\sqrt [4]{2} \sqrt [4]{x^3+x^5} \left (2 \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\sqrt [4]{2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}-\sqrt {1+x^2}}\right )-2 \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}+\sqrt {1+x^2}}\right )\right )}{3 x^{3/4} \sqrt [4]{1+x^2}} \]
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Time = 12.35 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.30
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{3}-\frac {2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}}{3}+\frac {\ln \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{-\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right ) \sqrt {2}}{6}+\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}}{3}+\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}}{3}\) | \(203\) |
trager | \(\text {Expression too large to display}\) | \(737\) |
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Result contains complex when optimal does not.
Time = 3.00 (sec) , antiderivative size = 656, normalized size of antiderivative = 4.21 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\left (\frac {1}{12} i - \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} - \left (i + 1\right ) \, x^{3} + \left (i + 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) - \left (\frac {1}{12} i - \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} + \left (i + 1\right ) \, x^{3} - \left (i + 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) - \left (\frac {1}{12} i + \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} + \left (i - 1\right ) \, x^{3} - \left (i - 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) + \left (\frac {1}{12} i + \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} - \left (i - 1\right ) \, x^{3} + \left (i - 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x + 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x + 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) - \frac {1}{6} i \cdot 2^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (i \, x^{4} + 2 i \, x^{3} + i \, x^{2}\right )} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) + \frac {1}{6} i \cdot 2^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (-i \, x^{4} - 2 i \, x^{3} - i \, x^{2}\right )} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) \]
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\[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x^{2} + 1\right )} \left (x + 1\right )}{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
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\[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\int { \frac {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{{\left (x^{3} - 1\right )} x} \,d x } \]
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\[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\int { \frac {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{{\left (x^{3} - 1\right )} x} \,d x } \]
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Timed out. \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\int \frac {{\left (x^5+x^3\right )}^{1/4}\,\left (x+1\right )}{x\,\left (x^3-1\right )} \,d x \]
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