Integrand size = 52, antiderivative size = 541 \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} (a-b)^{2/3} \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{-2 a (-a+b)^{2/3}+2 (-a+b)^{2/3} x+(a-b)^{2/3} \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{2 \sqrt [3]{a-b} (-a+b)^{2/3} d^{2/3}}-\frac {\log \left (a (-a+b)^{2/3}-(-a+b)^{2/3} x+(a-b)^{2/3} \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{2 \sqrt [3]{a-b} (-a+b)^{2/3} d^{2/3}}+\frac {\log \left (a^3 \sqrt [3]{-a+b}-a^2 b \sqrt [3]{-a+b}-2 a^2 \sqrt [3]{-a+b} x+2 a b \sqrt [3]{-a+b} x+a \sqrt [3]{-a+b} x^2-b \sqrt [3]{-a+b} x^2+\left (a (a-b)^{2/3} (-a+b)^{2/3} \sqrt [3]{d}-(a-b)^{2/3} (-a+b)^{2/3} \sqrt [3]{d} x\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+\left (-a \sqrt [3]{a-b} d^{2/3}+\sqrt [3]{a-b} b d^{2/3}\right ) \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{4 \sqrt [3]{a-b} (-a+b)^{2/3} d^{2/3}} \]
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Time = 0.52 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6851, 925, 132, 61, 12, 93} \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-b}}\right )}{2 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \arctan \left (\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{2 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{x-a} (x-b)^{2/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{x-a} (x-b)^{2/3} \log \left (2 (1-d) x-2 \left (\sqrt {d}+1\right ) \left (a-b \sqrt {d}\right )\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \log \left (-\frac {\sqrt [3]{x-a}}{\sqrt [6]{d}}-\sqrt [3]{x-b}\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \log \left (\frac {\sqrt [3]{x-a}}{\sqrt [6]{d}}-\sqrt [3]{x-b}\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}} \]
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Rule 12
Rule 61
Rule 93
Rule 132
Rule 925
Rule 6851
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}} \\ & = \frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \left (\frac {(-1+d) \sqrt [3]{-b+x}}{(a-b) \sqrt {d} \sqrt [3]{-a+x} \left (2 a-2 (a-b) \sqrt {d}-2 b d-2 (1-d) x\right )}+\frac {(-1+d) \sqrt [3]{-b+x}}{(a-b) \sqrt {d} \sqrt [3]{-a+x} \left (-2 a-2 (a-b) \sqrt {d}+2 b d+2 (1-d) x\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}} \\ & = -\frac {\left ((1-d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (2 a-2 (a-b) \sqrt {d}-2 b d-2 (1-d) x\right )} \, dx}{(a-b) \sqrt {d} \sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left ((1-d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (-2 a-2 (a-b) \sqrt {d}+2 b d+2 (1-d) x\right )} \, dx}{(a-b) \sqrt {d} \sqrt [3]{(-a+x) (-b+x)^2}} \\ & = -\frac {\left ((1-d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {a-b}{\left (1+\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \left (2 a-2 (a-b) \sqrt {d}-2 b d-2 (1-d) x\right )} \, dx}{(a-b) \sqrt {d} \sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left ((1-d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {a-b}{\left (1-\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-2 a-2 (a-b) \sqrt {d}+2 b d+2 (1-d) x\right )} \, dx}{(a-b) \sqrt {d} \sqrt [3]{(-a+x) (-b+x)^2}} \\ & = -\frac {\left ((1-d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (-2 a-2 (a-b) \sqrt {d}+2 b d+2 (1-d) x\right )} \, dx}{\left (1-\sqrt {d}\right ) \sqrt {d} \sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left ((1-d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (2 a-2 (a-b) \sqrt {d}-2 b d-2 (1-d) x\right )} \, dx}{\left (1+\sqrt {d}\right ) \sqrt {d} \sqrt [3]{(-a+x) (-b+x)^2}} \\ & = -\frac {\sqrt {3} \sqrt [3]{-a+x} (-b+x)^{2/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{-b+x}}\right )}{2 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {3} \sqrt [3]{-a+x} (-b+x)^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{-b+x}}\right )}{2 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (-2 \left (1+\sqrt {d}\right ) \left (a-b \sqrt {d}\right )+2 (1-d) x\right )}{4 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (-\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d}}-\sqrt [3]{-b+x}\right )}{4 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d}}-\sqrt [3]{-b+x}\right )}{4 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.47 \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (-2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}}{\sqrt {3}}\right )+\log \left (1+\frac {\sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 \log \left (-1+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+\log \left (1+\frac {\sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )\right )}{4 (a-b) d^{2/3} \sqrt [3]{(b-x)^2 (-a+x)}} \]
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\[\int \frac {-b +x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-a^{2}+b^{2} d +2 \left (-b d +a \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
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Time = 0.26 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.60 \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left ({\left (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, {\left (b^{2} d^{2} - 2 \, b d^{2} x + d^{2} x^{2}\right )}}\right ) - 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d}{b^{2} - 2 \, b x + x^{2}}\right ) + {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (a d - d x\right )} - {\left (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right )}{4 \, {\left (a - b\right )} d^{2}} \]
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Timed out. \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}} \,d x } \]
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\[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\int { -\frac {b - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}} \,d x } \]
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Timed out. \[ \int \frac {-b+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx=\int -\frac {b-x}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (b^2\,d+2\,x\,\left (a-b\,d\right )-a^2+x^2\,\left (d-1\right )\right )} \,d x \]
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