Integrand size = 23, antiderivative size = 349 \[ \int \frac {1}{x^2 (1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=\frac {22}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}-\frac {55 \left (1+x^3\right )}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {55 \left (1+x^3\right )}{27 \sqrt {1+x} \left (1+\sqrt {3}+x\right ) \sqrt {1-x+x^2}}-\frac {55 \sqrt {2-\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{18\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}}+\frac {55 \sqrt {2} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {929, 296, 331, 309, 224, 1891} \[ \int \frac {1}{x^2 (1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=\frac {55 \sqrt {2} \sqrt {x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^2-x+1}}-\frac {55 \sqrt {2-\sqrt {3}} \sqrt {x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{18\ 3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^2-x+1}}+\frac {22}{27 x \sqrt {x+1} \sqrt {x^2-x+1}}-\frac {55 \left (x^3+1\right )}{27 x \sqrt {x+1} \sqrt {x^2-x+1}}+\frac {55 \left (x^3+1\right )}{27 \sqrt {x+1} \left (x+\sqrt {3}+1\right ) \sqrt {x^2-x+1}}+\frac {2}{9 x \sqrt {x+1} \sqrt {x^2-x+1} \left (x^3+1\right )} \]
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Rule 224
Rule 296
Rule 309
Rule 331
Rule 929
Rule 1891
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^3} \int \frac {1}{x^2 \left (1+x^3\right )^{5/2}} \, dx}{\sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}+\frac {\left (11 \sqrt {1+x^3}\right ) \int \frac {1}{x^2 \left (1+x^3\right )^{3/2}} \, dx}{9 \sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {22}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}+\frac {\left (55 \sqrt {1+x^3}\right ) \int \frac {1}{x^2 \sqrt {1+x^3}} \, dx}{27 \sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {22}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}-\frac {55 \left (1+x^3\right )}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {\left (55 \sqrt {1+x^3}\right ) \int \frac {x}{\sqrt {1+x^3}} \, dx}{54 \sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {22}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}-\frac {55 \left (1+x^3\right )}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {\left (55 \sqrt {1+x^3}\right ) \int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx}{54 \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {\left (55 \left (-1+\sqrt {3}\right ) \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{54 \sqrt {1+x} \sqrt {1-x+x^2}} \\ & = \frac {22}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {2}{9 x \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}-\frac {55 \left (1+x^3\right )}{27 x \sqrt {1+x} \sqrt {1-x+x^2}}+\frac {55 \left (1+x^3\right )}{27 \sqrt {1+x} \left (1+\sqrt {3}+x\right ) \sqrt {1-x+x^2}}-\frac {55 \sqrt {2-\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{18\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}}+\frac {55 \sqrt {2} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.54 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^2 (1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=-\frac {27+88 x^3+55 x^6}{27 x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}+\frac {55 (1+x)^{3/2} \left (\frac {12 \sqrt {-\frac {i}{3 i+\sqrt {3}}} \left (1-x+x^2\right )}{(1+x)^2}+\frac {3 \sqrt {2} \left (1-i \sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}-\frac {6 i}{1+x}}{3 i+\sqrt {3}}} \sqrt {\frac {-3 i+\sqrt {3}+\frac {6 i}{1+x}}{-3 i+\sqrt {3}}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right )|\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {1+x}}+\frac {i \sqrt {2} \left (3 i+\sqrt {3}\right ) \sqrt {\frac {3 i+\sqrt {3}-\frac {6 i}{1+x}}{3 i+\sqrt {3}}} \sqrt {\frac {-3 i+\sqrt {3}+\frac {6 i}{1+x}}{-3 i+\sqrt {3}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {1+x}}\right )}{324 \sqrt {-\frac {i}{3 i+\sqrt {3}}} \sqrt {1-x+x^2}} \]
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Time = 0.71 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.69
method | result | size |
elliptic | \(\frac {\sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (-\frac {\sqrt {x^{3}+1}}{x}-\frac {2 x^{2}}{9 \left (x^{3}+1\right )^{\frac {3}{2}}}-\frac {28 x^{2}}{27 \sqrt {x^{3}+1}}+\frac {55 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) E\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{27 \sqrt {x^{3}+1}}\right )}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(240\) |
default | \(\frac {55 i \sqrt {3}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x^{4} \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}+165 F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x^{4} \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}-330 E\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x^{4} \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}+55 i \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) \sqrt {3}\, x -110 x^{6}+165 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x -330 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, E\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x -176 x^{3}-54}{54 x \left (x^{2}-x +1\right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}\) | \(695\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.18 \[ \int \frac {1}{x^2 (1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=-\frac {{\left (55 \, x^{6} + 88 \, x^{3} + 27\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} + 55 \, {\left (x^{7} + 2 \, x^{4} + x\right )} {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right )}{27 \, {\left (x^{7} + 2 \, x^{4} + x\right )}} \]
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\[ \int \frac {1}{x^2 (1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (x + 1\right )^{\frac {5}{2}} \left (x^{2} - x + 1\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{x^2 (1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {5}{2}} {\left (x + 1\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 (1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {5}{2}} {\left (x + 1\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 (1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=\int \frac {1}{x^2\,{\left (x+1\right )}^{5/2}\,{\left (x^2-x+1\right )}^{5/2}} \,d x \]
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