Integrand size = 31, antiderivative size = 175 \[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\frac {3 \left (-1+2 x+x^3\right )^{2/3}}{4 x^2}-\frac {3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+2 x+x^3}}\right )}{2\ 2^{2/3}}+\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+2 x+x^3}\right )-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+2 x+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+2 x+x^3\right )^{2/3}\right ) \]
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\[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \left (-1+2 x+x^3\right )^{2/3}}{2 x^3}-\frac {\left (-1+2 x+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+2 x+x^3\right )^{2/3}}{x}+\frac {\left (-13+2 x+4 x^2\right ) \left (-1+2 x+x^3\right )^{2/3}}{2 \left (2-4 x+x^3\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {\left (-13+2 x+4 x^2\right ) \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3} \, dx-\frac {3}{2} \int \frac {\left (-1+2 x+x^3\right )^{2/3}}{x^3} \, dx-2 \int \frac {\left (-1+2 x+x^3\right )^{2/3}}{x} \, dx-\int \frac {\left (-1+2 x+x^3\right )^{2/3}}{x^2} \, dx \\ & = \frac {1}{2} \int \left (-\frac {13 \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3}+\frac {2 x \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3}+\frac {4 x^2 \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3}\right ) \, dx-\frac {\left (-1+2 x+x^3\right )^{2/3} \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x^2} \, dx}{\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}-\frac {\left (3 \left (-1+2 x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x^3} \, dx}{2 \left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}-\frac {\left (2 \left (-1+2 x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x} \, dx}{\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}} \\ & = 2 \int \frac {x^2 \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3} \, dx-\frac {13}{2} \int \frac {\left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3} \, dx-\frac {\left (-1+2 x+x^3\right )^{2/3} \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x^2} \, dx}{\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}-\frac {\left (3 \left (-1+2 x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x^3} \, dx}{2 \left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}-\frac {\left (2 \left (-1+2 x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x} \, dx}{\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}+\int \frac {x \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3} \, dx \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00 \[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\frac {3 \left (-1+2 x+x^3\right )^{2/3}}{4 x^2}-\frac {3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+2 x+x^3}}\right )}{2\ 2^{2/3}}+\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+2 x+x^3}\right )-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+2 x+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+2 x+x^3\right )^{2/3}\right ) \]
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Time = 16.96 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(\frac {2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2} \ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 \left (x^{3}+2 x -1\right )^{\frac {1}{3}}}{x}\right )-2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2} \ln \left (\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (x^{3}+2 x -1\right )^{\frac {1}{3}} x +2 \left (x^{3}+2 x -1\right )^{\frac {2}{3}}}{x^{2}}\right )-2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2} \ln \left (2\right )+6 \,3^{\frac {1}{6}} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}+2 x -1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) x^{2}+6 \left (x^{3}+2 x -1\right )^{\frac {2}{3}}}{8 x^{2}}\) | \(169\) |
trager | \(\text {Expression too large to display}\) | \(986\) |
risch | \(\text {Expression too large to display}\) | \(1034\) |
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (133) = 266\).
Time = 10.50 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.44 \[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=-\frac {4 \cdot 9^{\frac {1}{3}} 4^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (4 \cdot 9^{\frac {2}{3}} 4^{\frac {2}{3}} {\left (4 \, x^{7} - 14 \, x^{5} + 7 \, x^{4} - 8 \, x^{3} + 8 \, x^{2} - 2 \, x\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} - 12 \cdot 9^{\frac {1}{3}} {\left (55 \, x^{8} + 100 \, x^{6} - 50 \, x^{5} + 16 \, x^{4} - 16 \, x^{3} + 4 \, x^{2}\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}} {\left (377 \, x^{9} + 1200 \, x^{7} - 600 \, x^{6} + 816 \, x^{5} - 816 \, x^{4} + 268 \, x^{3} - 96 \, x^{2} + 48 \, x - 8\right )}\right )}}{6 \, {\left (487 \, x^{9} + 960 \, x^{7} - 480 \, x^{6} + 48 \, x^{5} - 48 \, x^{4} - 52 \, x^{3} + 96 \, x^{2} - 48 \, x + 8\right )}}\right ) - 2 \cdot 9^{\frac {1}{3}} 4^{\frac {2}{3}} x^{2} \log \left (-\frac {6 \cdot 9^{\frac {2}{3}} 4^{\frac {1}{3}} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {1}{3}} x^{2} - 9^{\frac {1}{3}} 4^{\frac {2}{3}} {\left (x^{3} - 4 \, x + 2\right )} - 36 \, {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} x}{x^{3} - 4 \, x + 2}\right ) + 9^{\frac {1}{3}} 4^{\frac {2}{3}} x^{2} \log \left (\frac {18 \cdot 9^{\frac {1}{3}} 4^{\frac {2}{3}} {\left (4 \, x^{4} + 2 \, x^{2} - x\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} + 9^{\frac {2}{3}} 4^{\frac {1}{3}} {\left (55 \, x^{6} + 100 \, x^{4} - 50 \, x^{3} + 16 \, x^{2} - 16 \, x + 4\right )} + 54 \, {\left (7 \, x^{5} + 8 \, x^{3} - 4 \, x^{2}\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {1}{3}}}{x^{6} - 8 \, x^{4} + 4 \, x^{3} + 16 \, x^{2} - 16 \, x + 4}\right ) - 36 \, {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}}}{48 \, x^{2}} \]
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\[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\int \frac {\left (4 x - 3\right ) \left (x^{3} + 2 x - 1\right )^{\frac {2}{3}}}{x^{3} \left (x^{3} - 4 x + 2\right )}\, dx \]
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\[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} {\left (4 \, x - 3\right )}}{{\left (x^{3} - 4 \, x + 2\right )} x^{3}} \,d x } \]
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\[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} {\left (4 \, x - 3\right )}}{{\left (x^{3} - 4 \, x + 2\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx=\int \frac {\left (4\,x-3\right )\,{\left (x^3+2\,x-1\right )}^{2/3}}{x^3\,\left (x^3-4\,x+2\right )} \,d x \]
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