Integrand size = 77, antiderivative size = 127 \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{3/4}}{(b-x) (-a+x)}\right )}{d^{3/4}} \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.72 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.23, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6820, 6851, 6860, 142, 141} \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=-\frac {4 \left (1-\sqrt {-4 a d+4 b d+1}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )} \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\frac {a-x}{a-b},-\frac {2 d (a-x)}{-2 a d+2 b d-\sqrt {-4 a d+4 b d+1}+1}\right )}{\left (-\sqrt {-4 a d+4 b d+1}-2 a d+2 b d+1\right ) \sqrt {-\frac {b-x}{a-b}}}-\frac {4 \left (\sqrt {-4 a d+4 b d+1}+1\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )} \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\frac {a-x}{a-b},-\frac {2 d (a-x)}{-2 a d+2 b d+\sqrt {-4 a d+4 b d+1}+1}\right )}{\left (\sqrt {-4 a d+4 b d+1}-2 a d+2 b d+1\right ) \sqrt {-\frac {b-x}{a-b}}} \]
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Rule 141
Rule 142
Rule 6820
Rule 6851
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \frac {(2 a-b-x) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}}{(a-x) \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx \\ & = \frac {\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \int \frac {(2 a-b-x) \sqrt {b-x}}{(a-x)^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx}{\sqrt [4]{a-x} \sqrt {b-x}} \\ & = \frac {\sqrt [4]{-\left ((a-x) (b-x)^2\right )} \int \left (\frac {\left (-1-\sqrt {1-4 a d+4 b d}\right ) \sqrt {b-x}}{(a-x)^{3/4} \left (-1-2 b d-\sqrt {1-4 a d+4 b d}+2 d x\right )}+\frac {\left (-1+\sqrt {1-4 a d+4 b d}\right ) \sqrt {b-x}}{(a-x)^{3/4} \left (-1-2 b d+\sqrt {1-4 a d+4 b d}+2 d x\right )}\right ) \, dx}{\sqrt [4]{a-x} \sqrt {b-x}} \\ & = \frac {\left (\left (-1-\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}\right ) \int \frac {\sqrt {b-x}}{(a-x)^{3/4} \left (-1-2 b d-\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{\sqrt [4]{a-x} \sqrt {b-x}}+\frac {\left (\left (-1+\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}\right ) \int \frac {\sqrt {b-x}}{(a-x)^{3/4} \left (-1-2 b d+\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{\sqrt [4]{a-x} \sqrt {b-x}} \\ & = \frac {\left (\left (-1-\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}\right ) \int \frac {\sqrt {-\frac {b}{a-b}+\frac {x}{a-b}}}{(a-x)^{3/4} \left (-1-2 b d-\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{\sqrt [4]{a-x} \sqrt {-\frac {b-x}{a-b}}}+\frac {\left (\left (-1+\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )}\right ) \int \frac {\sqrt {-\frac {b}{a-b}+\frac {x}{a-b}}}{(a-x)^{3/4} \left (-1-2 b d+\sqrt {1-4 a d+4 b d}+2 d x\right )} \, dx}{\sqrt [4]{a-x} \sqrt {-\frac {b-x}{a-b}}} \\ & = -\frac {4 \left (1-\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )} \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\frac {a-x}{a-b},-\frac {2 d (a-x)}{1-2 a d+2 b d-\sqrt {1-4 a d+4 b d}}\right )}{\left (1-2 a d+2 b d-\sqrt {1-4 a d+4 b d}\right ) \sqrt {-\frac {b-x}{a-b}}}-\frac {4 \left (1+\sqrt {1-4 a d+4 b d}\right ) \sqrt [4]{-\left ((a-x) (b-x)^2\right )} \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\frac {a-x}{a-b},-\frac {2 d (a-x)}{1-2 a d+2 b d+\sqrt {1-4 a d+4 b d}}\right )}{\left (1-2 a d+2 b d+\sqrt {1-4 a d+4 b d}\right ) \sqrt {-\frac {b-x}{a-b}}} \\ \end{align*}
Time = 2.53 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.22 \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=\frac {\sqrt {2} \sqrt [4]{(b-x)^2 (-a+x)} \left (\arctan \left (\frac {-b \sqrt {d}+\sqrt {a-x}+\sqrt {d} x}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{a-x} \sqrt {b-x}}{\sqrt {a-x}+\sqrt {d} (b-x)}\right )\right )}{d^{3/4} \sqrt [4]{a-x} \sqrt {b-x}} \]
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\[\int \frac {-\left (2 a -b \right ) b^{2}+\left (4 a -b \right ) b x -\left (2 a +b \right ) x^{2}+x^{3}}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {3}{4}} \left (a +b^{2} d -\left (2 b d +1\right ) x +d \,x^{2}\right )}d x\]
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Timed out. \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=\int { -\frac {{\left (2 \, a - b\right )} b^{2} - {\left (4 \, a - b\right )} b x + {\left (2 \, a + b\right )} x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {3}{4}} {\left (b^{2} d + d x^{2} - {\left (2 \, b d + 1\right )} x + a\right )}} \,d x } \]
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\[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=\int { -\frac {{\left (2 \, a - b\right )} b^{2} - {\left (4 \, a - b\right )} b x + {\left (2 \, a + b\right )} x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {3}{4}} {\left (b^{2} d + d x^{2} - {\left (2 \, b d + 1\right )} x + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {-\left ((2 a-b) b^2\right )+(4 a-b) b x-(2 a+b) x^2+x^3}{\left ((-a+x) (-b+x)^2\right )^{3/4} \left (a+b^2 d-(1+2 b d) x+d x^2\right )} \, dx=\int -\frac {b^2\,\left (2\,a-b\right )+x^2\,\left (2\,a+b\right )-x^3-b\,x\,\left (4\,a-b\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{3/4}\,\left (a-x\,\left (2\,b\,d+1\right )+b^2\,d+d\,x^2\right )} \,d x \]
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