Integrand size = 90, antiderivative size = 115 \[ \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(-1+x) x \sqrt [4]{(1-x) x (1-k x)} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx=\frac {4 \left (x-x^2-k x^2+k x^3\right )^{3/4}}{(-1+x) x}+2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x+(-1-k) x^2+k x^3}}{-1+k x}\right )-2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{x+(-1-k) x^2+k x^3}}{-1+k x}\right ) \]
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\[ \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(-1+x) x \sqrt [4]{(1-x) x (1-k x)} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx=\int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(-1+x) x \sqrt [4]{(1-x) x (1-k x)} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{\sqrt [4]{1-x} (-1+x) x^{5/4} \sqrt [4]{1-k x} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = -\frac {\left (\sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(1-x)^{5/4} x^{5/4} \sqrt [4]{1-k x} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = -\frac {\left (\sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \int \frac {(1-k x)^{3/4} \left (-1-2 (-1+k) x+k x^2\right ) \left (-1+2 k x-k^2 x^2\right )}{(1-x)^{5/4} x^{5/4} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = \frac {\left (\sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \int \frac {(1-k x)^{3/4} (-1+k x)^2 \left (-1-2 (-1+k) x+k x^2\right )}{(1-x)^{5/4} x^{5/4} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = \frac {\left (\sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \int \frac {(1-k x)^{11/4} \left (-1-2 (-1+k) x+k x^2\right )}{(1-x)^{5/4} x^{5/4} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = \frac {\left (4 \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \frac {\left (1-k x^4\right )^{11/4} \left (-1-2 (-1+k) x^4+k x^8\right )}{x^2 \left (1-x^4\right )^{5/4} \left (-1+(d+3 k) x^4-\left (d+3 k^2\right ) x^8+k^3 x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = \frac {\left (4 \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \frac {\left (1-k x^4\right )^{11/4} \left (-1-2 (-1+k) x^4+k x^8\right )}{x^2 \left (1-x^4\right )^{5/4} \left (\left (-1+k x^4\right )^3+d \left (x^4-x^8\right )\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = \frac {\left (4 \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \left (\frac {\left (1-k x^4\right )^{11/4}}{x^2 \left (1-x^4\right )^{5/4}}+\frac {x^2 \left (1-k x^4\right )^{11/4} \left (-2+d+5 k-\left (d+k+3 k^2\right ) x^4+k^3 x^8\right )}{\left (1-x^4\right )^{5/4} \left (1-d \left (1+\frac {3 k}{d}\right ) x^4+d \left (1+\frac {3 k^2}{d}\right ) x^8-k^3 x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = \frac {\left (4 \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \frac {\left (1-k x^4\right )^{11/4}}{x^2 \left (1-x^4\right )^{5/4}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}}+\frac {\left (4 \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \frac {x^2 \left (1-k x^4\right )^{11/4} \left (-2+d+5 k-\left (d+k+3 k^2\right ) x^4+k^3 x^8\right )}{\left (1-x^4\right )^{5/4} \left (1-d \left (1+\frac {3 k}{d}\right ) x^4+d \left (1+\frac {3 k^2}{d}\right ) x^8-k^3 x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = -\frac {4 \sqrt [4]{1-x} \sqrt [4]{1-k x} \operatorname {AppellF1}\left (-\frac {1}{4},\frac {5}{4},-\frac {11}{4},\frac {3}{4},x,k x\right )}{\sqrt [4]{(1-x) x (1-k x)}}+\frac {\left (4 \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \left (\frac {d \left (1+\frac {-2+5 k}{d}\right ) x^2 \left (1-k x^4\right )^{11/4}}{\left (1-x^4\right )^{5/4} \left (1-d \left (1+\frac {3 k}{d}\right ) x^4+d \left (1+\frac {3 k^2}{d}\right ) x^8-k^3 x^{12}\right )}+\frac {\left (-d-k-3 k^2\right ) x^6 \left (1-k x^4\right )^{11/4}}{\left (1-x^4\right )^{5/4} \left (1-d \left (1+\frac {3 k}{d}\right ) x^4+d \left (1+\frac {3 k^2}{d}\right ) x^8-k^3 x^{12}\right )}+\frac {k^3 x^{10} \left (1-k x^4\right )^{11/4}}{\left (1-x^4\right )^{5/4} \left (1-d \left (1+\frac {3 k}{d}\right ) x^4+d \left (1+\frac {3 k^2}{d}\right ) x^8-k^3 x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = -\frac {4 \sqrt [4]{1-x} \sqrt [4]{1-k x} \operatorname {AppellF1}\left (-\frac {1}{4},\frac {5}{4},-\frac {11}{4},\frac {3}{4},x,k x\right )}{\sqrt [4]{(1-x) x (1-k x)}}+\frac {\left (4 k^3 \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \frac {x^{10} \left (1-k x^4\right )^{11/4}}{\left (1-x^4\right )^{5/4} \left (1-d \left (1+\frac {3 k}{d}\right ) x^4+d \left (1+\frac {3 k^2}{d}\right ) x^8-k^3 x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}}+\frac {\left (4 (-2+d+5 k) \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \frac {x^2 \left (1-k x^4\right )^{11/4}}{\left (1-x^4\right )^{5/4} \left (1-d \left (1+\frac {3 k}{d}\right ) x^4+d \left (1+\frac {3 k^2}{d}\right ) x^8-k^3 x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}}+\frac {\left (4 \left (-d-k-3 k^2\right ) \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \frac {x^6 \left (1-k x^4\right )^{11/4}}{\left (1-x^4\right )^{5/4} \left (1-d \left (1+\frac {3 k}{d}\right ) x^4+d \left (1+\frac {3 k^2}{d}\right ) x^8-k^3 x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}} \\ & = -\frac {4 \sqrt [4]{1-x} \sqrt [4]{1-k x} \operatorname {AppellF1}\left (-\frac {1}{4},\frac {5}{4},-\frac {11}{4},\frac {3}{4},x,k x\right )}{\sqrt [4]{(1-x) x (1-k x)}}+\frac {\left (4 k^3 \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \frac {x^{10} \left (1-k x^4\right )^{11/4}}{\left (1-x^4\right )^{5/4} \left (-\left (-1+k x^4\right )^3-d \left (x^4-x^8\right )\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}}+\frac {\left (4 (-2+d+5 k) \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \frac {x^2 \left (1-k x^4\right )^{11/4}}{\left (1-x^4\right )^{5/4} \left (-\left (-1+k x^4\right )^3-d \left (x^4-x^8\right )\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}}+\frac {\left (4 \left (-d-k-3 k^2\right ) \sqrt [4]{1-x} \sqrt [4]{x} \sqrt [4]{1-k x}\right ) \text {Subst}\left (\int \frac {x^6 \left (1-k x^4\right )^{11/4}}{\left (1-x^4\right )^{5/4} \left (-\left (-1+k x^4\right )^3-d \left (x^4-x^8\right )\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{(1-x) x (1-k x)}} \\ \end{align*}
\[ \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(-1+x) x \sqrt [4]{(1-x) x (1-k x)} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx=\int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(-1+x) x \sqrt [4]{(1-x) x (1-k x)} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx \]
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\[\int \frac {\left (-1-2 \left (-1+k \right ) x +k \,x^{2}\right ) \left (k^{3} x^{3}-3 k^{2} x^{2}+3 k x -1\right )}{\left (-1+x \right ) x \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{4}} \left (-1+\left (d +3 k \right ) x -\left (3 k^{2}+d \right ) x^{2}+k^{3} x^{3}\right )}d x\]
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Timed out. \[ \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(-1+x) x \sqrt [4]{(1-x) x (1-k x)} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(-1+x) x \sqrt [4]{(1-x) x (1-k x)} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(-1+x) x \sqrt [4]{(1-x) x (1-k x)} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx=\int { \frac {{\left (k^{3} x^{3} - 3 \, k^{2} x^{2} + 3 \, k x - 1\right )} {\left (k x^{2} - 2 \, {\left (k - 1\right )} x - 1\right )}}{{\left (k^{3} x^{3} - {\left (3 \, k^{2} + d\right )} x^{2} + {\left (d + 3 \, k\right )} x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{4}} {\left (x - 1\right )} x} \,d x } \]
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\[ \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(-1+x) x \sqrt [4]{(1-x) x (1-k x)} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx=\int { \frac {{\left (k^{3} x^{3} - 3 \, k^{2} x^{2} + 3 \, k x - 1\right )} {\left (k x^{2} - 2 \, {\left (k - 1\right )} x - 1\right )}}{{\left (k^{3} x^{3} - {\left (3 \, k^{2} + d\right )} x^{2} + {\left (d + 3 \, k\right )} x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{4}} {\left (x - 1\right )} x} \,d x } \]
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Timed out. \[ \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (-1+3 k x-3 k^2 x^2+k^3 x^3\right )}{(-1+x) x \sqrt [4]{(1-x) x (1-k x)} \left (-1+(d+3 k) x-\left (d+3 k^2\right ) x^2+k^3 x^3\right )} \, dx=\int \frac {\left (2\,x\,\left (k-1\right )-k\,x^2+1\right )\,\left (-k^3\,x^3+3\,k^2\,x^2-3\,k\,x+1\right )}{x\,\left (x-1\right )\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/4}\,\left (k^3\,x^3-x^2\,\left (3\,k^2+d\right )+x\,\left (d+3\,k\right )-1\right )} \,d x \]
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