Integrand size = 36, antiderivative size = 301 \[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} b \sqrt [3]{d}-\sqrt {3} \sqrt [3]{d} x}{b \sqrt [3]{d}-\sqrt [3]{d} x-2 \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{(a-b) d^{2/3}}-\frac {\log \left (b \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{(a-b) d^{2/3}}+\frac {\log \left (b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2+\left (-b \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+\left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{2 (a-b) d^{2/3}} \]
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Time = 0.40 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6851, 93} \[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=-\frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} \arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {(x-a)^{2/3} (x-b)^{4/3} \log (a-b d-(1-d) x)}{2 d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {3 (x-a)^{2/3} (x-b)^{4/3} \log \left (\sqrt [3]{d} \sqrt [3]{x-b}-\sqrt [3]{x-a}\right )}{2 d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}} \]
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Rule 93
Rule 6851
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{(-a+x)^{2/3} \sqrt [3]{-b+x} (a-b d+(-1+d) x)} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}} \\ & = -\frac {\sqrt {3} (-a+x)^{2/3} (-b+x)^{4/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [3]{-a+x}}\right )}{(a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {(-a+x)^{2/3} (-b+x)^{4/3} \log (a-b d-(1-d) x)}{2 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {3 (-a+x)^{2/3} (-b+x)^{4/3} \log \left (-\sqrt [3]{-a+x}+\sqrt [3]{d} \sqrt [3]{-b+x}\right )}{2 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.57 \[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\frac {(b-x)^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-a+x}}{\sqrt [3]{d} \sqrt [3]{b-x}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{d}+\frac {\sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )+\log \left (d^{2/3}-\frac {\sqrt [3]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {(-a+x)^{2/3}}{(b-x)^{2/3}}\right )\right )}{2 (a-b) d^{2/3} \left ((b-x)^2 (-a+x)\right )^{2/3}} \]
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\[\int \frac {-b +x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (a -b d +\left (-1+d \right ) x \right )}d x\]
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none
Time = 0.29 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.96 \[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left ({\left (b d - d x\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, {\left (b d^{2} - d^{2} x\right )}}\right ) + {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} {\left (b - x\right )} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d - {\left (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (d^{2}\right )}^{\frac {2}{3}} {\left (b - x\right )} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b - x}\right )}{2 \, {\left (a - b\right )} d^{2}} \]
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\[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\int \frac {- b + x}{\left (\left (- a + x\right ) \left (- b + x\right )^{2}\right )^{\frac {2}{3}} \left (a - b d + d x - x\right )}\, dx \]
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\[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\int { \frac {b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )}} \,d x } \]
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\[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\int { \frac {b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )}} \,d x } \]
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Timed out. \[ \int \frac {-b+x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx=\int -\frac {b-x}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (a-b\,d+x\,\left (d-1\right )\right )} \,d x \]
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