Integrand size = 67, antiderivative size = 89 \[ \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx=\frac {2 \arctan \left (\frac {-\sqrt [4]{d}+\sqrt [4]{d} x}{\sqrt [4]{x+(-1-k) x^2+k x^3}}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {-\sqrt [4]{d}+\sqrt [4]{d} x}{\sqrt [4]{x+(-1-k) x^2+k x^3}}\right )}{d^{3/4}} \]
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\[ \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx=\int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(-1+x)^2 \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx \\ & = \frac {\left ((1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \int \frac {(-1+x)^2 \left (-1+2 (-1+k) x+k x^2\right )}{(1-x)^{3/4} x^{3/4} (1-k x)^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {\left ((1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \int \frac {(1-x)^{5/4} \left (-1+2 (-1+k) x+k x^2\right )}{x^{3/4} (1-k x)^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (1-x^4\right )^{5/4} \left (-1+2 (-1+k) x^4+k x^8\right )}{\left (1-k x^4\right )^{3/4} \left (-d+(1+3 d) x^4-(3 d+k) x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (1-x^4\right )^{5/4} \left (-1+2 (-1+k) x^4+k x^8\right )}{\left (1-k x^4\right )^{3/4} \left (x^4-k x^8+d \left (-1+x^4\right )^3\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {\left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (d-(1+3 d) x^4+3 d \left (1+\frac {k}{3 d}\right ) x^8-d x^{12}\right )}+\frac {2 (1-k) x^4 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (d-(1+3 d) x^4+3 d \left (1+\frac {k}{3 d}\right ) x^8-d x^{12}\right )}+\frac {k x^8 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (-d+(1+3 d) x^4-3 d \left (1+\frac {k}{3 d}\right ) x^8+d x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (d-(1+3 d) x^4+3 d \left (1+\frac {k}{3 d}\right ) x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (8 (1-k) (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^4 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (d-(1+3 d) x^4+3 d \left (1+\frac {k}{3 d}\right ) x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (4 k (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^8 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (-d+(1+3 d) x^4-3 d \left (1+\frac {k}{3 d}\right ) x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {\left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (-x^4+k x^8-d \left (-1+x^4\right )^3\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (8 (1-k) (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^4 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (-x^4+k x^8-d \left (-1+x^4\right )^3\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (4 k (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^8 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (x^4-k x^8+d \left (-1+x^4\right )^3\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}} \\ \end{align*}
\[ \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx=\int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx \]
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\[\int \frac {\left (x^{2}-2 x +1\right ) \left (-1+2 \left (-1+k \right ) x +k \,x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {3}{4}} \left (-d +\left (1+3 d \right ) x -\left (3 d +k \right ) x^{2}+d \,x^{3}\right )}d x\]
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Timed out. \[ \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx=\int { \frac {{\left (k x^{2} + 2 \, {\left (k - 1\right )} x - 1\right )} {\left (x^{2} - 2 \, x + 1\right )}}{{\left (d x^{3} - {\left (3 \, d + k\right )} x^{2} + {\left (3 \, d + 1\right )} x - d\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx=\int { \frac {{\left (k x^{2} + 2 \, {\left (k - 1\right )} x - 1\right )} {\left (x^{2} - 2 \, x + 1\right )}}{{\left (d x^{3} - {\left (3 \, d + k\right )} x^{2} + {\left (3 \, d + 1\right )} x - d\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx=\int -\frac {\left (x^2-2\,x+1\right )\,\left (2\,x\,\left (k-1\right )+k\,x^2-1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{3/4}\,\left (-d\,x^3+\left (3\,d+k\right )\,x^2+\left (-3\,d-1\right )\,x+d\right )} \,d x \]
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