Integrand size = 85, antiderivative size = 296 \[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{d}-\sqrt {3} \sqrt [3]{d} x}{a \sqrt [3]{d}-\sqrt [3]{d} x-2 \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right )}{d^{2/3}}-\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right )}{d^{2/3}}+\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+\left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \]
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\[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left ((-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3}\right ) \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{(-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{2/3}} \\ & = \frac {\left ((-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x} (a b+a c-2 b c+(-2 a+b+c) x)}{(-b+x)^{2/3} (-c+x)^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{2/3}} \\ & = \frac {\left ((-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3}\right ) \int \left (\frac {\left (-2 a+b+c-\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}\right ) \sqrt [3]{-a+x}}{(-b+x)^{2/3} (-c+x)^{2/3} \left (b+c-2 a d-\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )}+\frac {\left (-2 a+b+c+\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}\right ) \sqrt [3]{-a+x}}{(-b+x)^{2/3} (-c+x)^{2/3} \left (b+c-2 a d+\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )}\right ) \, dx}{((-a+x) (-b+x) (-c+x))^{2/3}} \\ & = \frac {\left (\left (-2 a+b+c-\sqrt {b^2+c^2+4 a^2 d-4 a c d-2 b (c+2 a d-2 c d)}\right ) (-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{(-b+x)^{2/3} (-c+x)^{2/3} \left (b+c-2 a d-\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{2/3}}+\frac {\left (\left (-2 a+b+c+\sqrt {b^2+c^2+4 a^2 d-4 a c d-2 b (c+2 a d-2 c d)}\right ) (-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{(-b+x)^{2/3} (-c+x)^{2/3} \left (b+c-2 a d+\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{2/3}} \\ \end{align*}
\[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx \]
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\[\int \frac {-a \left (a b +a c -2 b c \right )+2 \left (a^{2}-b c \right ) x +\left (-2 a +b +c \right ) x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {2}{3}} \left (-b c +a^{2} d +\left (-2 a d +b +c \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
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Timed out. \[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {{\left (2 \, a - b - c\right )} x^{2} + {\left (a b + a c - 2 \, b c\right )} a - 2 \, {\left (a^{2} - b c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {2}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b c - {\left (2 \, a d - b - c\right )} x\right )}} \,d x } \]
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\[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {{\left (2 \, a - b - c\right )} x^{2} + {\left (a b + a c - 2 \, b c\right )} a - 2 \, {\left (a^{2} - b c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {2}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b c - {\left (2 \, a d - b - c\right )} x\right )}} \,d x } \]
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Timed out. \[ \int \frac {-a (a b+a c-2 b c)+2 \left (a^2-b c\right ) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx=-\int \frac {2\,x\,\left (b\,c-a^2\right )-x^2\,\left (b-2\,a+c\right )+a\,\left (a\,b+a\,c-2\,b\,c\right )}{{\left (-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )\right )}^{2/3}\,\left (x\,\left (b+c-2\,a\,d\right )-b\,c+a^2\,d+x^2\,\left (d-1\right )\right )} \,d x \]
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