Integrand size = 86, antiderivative size = 238 \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\frac {3 \left (2-2 x-2 k x+5 b x^2+2 k x^2\right ) \left (x-x^2-k x^2+k x^3\right )^{2/3}}{10 x^4}+\frac {\left (-\sqrt {3} a-\sqrt {3} b^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {\left (a+b^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}+\frac {\left (-a-b^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{b}} \]
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\[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = \frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \left (\frac {8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )}{(b-k)^4 \sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x}}+\frac {(1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right )}{(b-k)^3 \sqrt [3]{1-x} x^{10/3} \sqrt [3]{1-k x}}+\frac {a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \left (1+4 k+k^2\right )}{(b-k)^2 \sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x}}-\frac {(1+k) \left (a+k^2\right )}{(b-k) \sqrt [3]{1-x} x^{4/3} \sqrt [3]{1-k x}}-\frac {\left (a+b^2\right ) \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right ) x}{(b-k)^4 \sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (1+(-1-k) x+(-b+k) x^2\right )}\right ) \, dx}{\sqrt [3]{(1-x) x (1-k x)}} \\ & = -\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {\left (a+b^2\right ) \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right ) x}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (1+(-1-k) x+(-b+k) x^2\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left ((1+k) \left (a+k^2\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{4/3} \sqrt [3]{1-k x}} \, dx}{(b-k) \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \left (1+4 k+k^2\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x}} \, dx}{(b-k)^2 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left ((1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{10/3} \sqrt [3]{1-k x}} \, dx}{(b-k)^3 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x}} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}} \\ & = -\frac {3 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{10 (b-k)^4 x^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{7 (b-k)^3 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 \left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \left (1+4 k+k^2\right )\right ) (1-x) (1-k x)}{4 (b-k)^2 x \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 (1+k) \left (a+k^2\right ) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{4/3} (1-k x) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},\frac {1-x}{1-k x}\right )}{2 (1-k) (b-k) x \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \left (\frac {-\frac {\left (a+b^2\right ) \left (1+7 b+13 b^2+4 b^3-k+2 b k+14 b^2 k+2 b k^2+13 b^2 k^2+2 b k^3+7 b k^4-k^5+k^6\right )}{\sqrt {1+4 b-2 k+k^2}}-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right )}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )}+\frac {\frac {\left (a+b^2\right ) \left (1+7 b+13 b^2+4 b^3-k+2 b k+14 b^2 k+2 b k^2+13 b^2 k^2+2 b k^3+7 b k^4-k^5+k^6\right )}{\sqrt {1+4 b-2 k+k^2}}-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right )}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (-1-k+\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )}\right ) \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left ((1+k) \left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \left (1+4 k+k^2\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{4/3} \sqrt [3]{1-k x}} \, dx}{2 (b-k)^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (3 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {-\frac {5}{3} (1+k)+k x}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x}} \, dx}{7 (b-k)^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (3 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {-\frac {8}{3} (1+k)+2 k x}{\sqrt [3]{1-x} x^{10/3} \sqrt [3]{1-k x}} \, dx}{10 (b-k)^4 \sqrt [3]{(1-x) x (1-k x)}} \\ & = -\frac {3 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{10 (b-k)^4 x^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{7 (b-k)^3 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {12 (1+k) \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{35 (b-k)^4 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 \left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \left (1+4 k+k^2\right )\right ) (1-x) (1-k x)}{4 (b-k)^2 x \sqrt [3]{(1-x) x (1-k x)}}-\frac {15 (1+k)^2 \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{28 (b-k)^3 x \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 (1+k) \left (a+k^2\right ) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{4/3} (1-k x) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},\frac {1-x}{1-k x}\right )}{2 (1-k) (b-k) x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (a \left (1+2 b+k^2\right )+k \left (2 b \left (1+3 k+k^2\right )-k \left (1+4 k+k^2\right )\right )\right ) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{4/3} (1-k x) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},\frac {1-x}{1-k x}\right )}{4 (1-k) (b-k)^2 x \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (9 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {2 \left (5+4 k+5 k^2\right )}{9 \sqrt [3]{1-x} x^{4/3} \sqrt [3]{1-k x}} \, dx}{28 (b-k)^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (\left (\frac {\left (a+b^2\right ) \left (1+7 b+13 b^2+4 b^3-k+2 b k+14 b^2 k+2 b k^2+13 b^2 k^2+2 b k^3+7 b k^4-k^5+k^6\right )}{\sqrt {1+4 b-2 k+k^2}}-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (-1-k+\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (9 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {\frac {2}{9} \left (20+19 k+20 k^2\right )-\frac {8}{3} k (1+k) x}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x}} \, dx}{70 (b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (a+b^2\right ) \left (1+5 b^2+5 b^2 k+k^5+5 b \left (1+k^2\right )+5 b k \left (1+k^2\right )+\frac {1+4 b^3-k-k^5+k^6+b^2 \left (13+14 k+13 k^2\right )+b \left (7+2 k+2 k^2+2 k^3+7 k^4\right )}{\sqrt {4 b+(-1+k)^2}}\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 10.48 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.81 \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\frac {3 ((-1+x) x (-1+k x))^{2/3} \left (2-2 (1+k) x+(5 b+2 k) x^2\right )}{10 x^4}-\frac {\sqrt {3} \left (a+b^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{(-1+x) x (-1+k x)}}\right )}{\sqrt [3]{b}}+\frac {\left (a+b^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{(-1+x) x (-1+k x)}\right )}{\sqrt [3]{b}}-\frac {\left (a+b^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{(-1+x) x (-1+k x)}+((-1+x) x (-1+k x))^{2/3}\right )}{2 \sqrt [3]{b}} \]
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Time = 1.18 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {-3 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}} b^{\frac {4}{3}} x^{2}-\frac {6 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}} \left (-1+x \right ) \left (k x -1\right ) b^{\frac {1}{3}}}{5}+x^{4} \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}}{x}\right )\right ) \left (b^{2}+a \right )}{2 b^{\frac {1}{3}} x^{4}}\) | \(170\) |
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Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\int { -\frac {{\left ({\left (k^{2} + a\right )} x^{4} - 2 \, {\left (k^{2} + k\right )} x^{3} + {\left (k^{2} + 4 \, k + 1\right )} x^{2} - 2 \, {\left (k + 1\right )} x + 1\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left ({\left (b - k\right )} x^{2} + {\left (k + 1\right )} x - 1\right )} x^{4}} \,d x } \]
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\[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\int { -\frac {{\left ({\left (k^{2} + a\right )} x^{4} - 2 \, {\left (k^{2} + k\right )} x^{3} + {\left (k^{2} + 4 \, k + 1\right )} x^{2} - 2 \, {\left (k + 1\right )} x + 1\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left ({\left (b - k\right )} x^{2} + {\left (k + 1\right )} x - 1\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\int -\frac {\left (x\,\left (k+1\right )-2\right )\,\left (x^2\,\left (k^2+4\,k+1\right )-2\,x\,\left (k+1\right )+x^4\,\left (k^2+a\right )-2\,x^3\,\left (k^2+k\right )+1\right )}{x^4\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (\left (b-k\right )\,x^2+\left (k+1\right )\,x-1\right )} \,d x \]
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