Integrand size = 67, antiderivative size = 319 \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{2 b^2-4 b x+2 x^2+\sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{d^{2/3}}+\frac {\log \left (b^2-2 b x+x^2-\sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{d^{2/3}}-\frac {\log \left (b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\left (b^2 \sqrt [3]{d}-2 b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \]
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\[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-b+x} (-4 a+b+3 x)}{\sqrt [3]{-a+x} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}} \\ & = \frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \left (\frac {4 a \left (1-\frac {b}{4 a}\right ) \sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (-b^4-a d+\left (4 b^3+d\right ) x-6 b^2 x^2+4 b x^3-x^4\right )}+\frac {3 x \sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}} \\ & = \frac {\left (3 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {x \sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}+\frac {\left ((4 a-b) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (-b^4-a d+\left (4 b^3+d\right ) x-6 b^2 x^2+4 b x^3-x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}} \\ & = \frac {\left (9 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x \left (a+x^3\right ) \sqrt [3]{a-b+x^3}}{b^4+a d-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left (3 (4 a-b) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{a-b+x^3}}{b^4+a d-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}} \\ & = \frac {\left (9 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x \left (a+x^3\right ) \sqrt [3]{a-b+x^3}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left (3 (4 a-b) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{a-b+x^3}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}} \\ & = \frac {\left (9 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \left (\frac {a x \sqrt [3]{a-b+x^3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+x^{12}}+\frac {x^4 \sqrt [3]{a-b+x^3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+x^{12}}\right ) \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left (3 (4 a-b) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{a-b+x^3}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}} \\ & = \frac {\left (9 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt [3]{a-b+x^3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+x^{12}} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}+\frac {\left (9 a \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{a-b+x^3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+x^{12}} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left (3 (4 a-b) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{a-b+x^3}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 33.72 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.80 \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\frac {(a-b) \left (\frac {b-x}{a-x}\right )^{2/3} (-a+x) \left (4 \text {RootSum}\left [-d+3 d \text {$\#$1}-3 d \text {$\#$1}^2+d \text {$\#$1}^3-a^3 \text {$\#$1}^4+3 a^2 b \text {$\#$1}^4-3 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^4\&,\frac {6 \sqrt [3]{\frac {-b+x}{-a+x}}-2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{\frac {b-x}{a-x}}}{\sqrt [3]{\text {$\#$1}}}}{\sqrt {3}}\right ) \sqrt [3]{\text {$\#$1}}+2 \log \left (-\sqrt [3]{\frac {-b+x}{-a+x}}+\sqrt [3]{\text {$\#$1}}\right ) \sqrt [3]{\text {$\#$1}}-\log \left (\left (\frac {-b+x}{-a+x}\right )^{2/3}+\sqrt [3]{\frac {-b+x}{-a+x}} \sqrt [3]{\text {$\#$1}}+\text {$\#$1}^{2/3}\right ) \sqrt [3]{\text {$\#$1}}}{-3 d+6 d \text {$\#$1}-3 d \text {$\#$1}^2+4 a^3 \text {$\#$1}^3-12 a^2 b \text {$\#$1}^3+12 a b^2 \text {$\#$1}^3-4 b^3 \text {$\#$1}^3}\&\right ]+5 \text {RootSum}\left [-d+3 d \text {$\#$1}-3 d \text {$\#$1}^2+d \text {$\#$1}^3-a^3 \text {$\#$1}^4+3 a^2 b \text {$\#$1}^4-3 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^4\&,\frac {-6 \sqrt [3]{\frac {-b+x}{-a+x}} \text {$\#$1}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{\frac {b-x}{a-x}}}{\sqrt [3]{\text {$\#$1}}}}{\sqrt {3}}\right ) \text {$\#$1}^{4/3}-2 \log \left (-\sqrt [3]{\frac {-b+x}{-a+x}}+\sqrt [3]{\text {$\#$1}}\right ) \text {$\#$1}^{4/3}+\log \left (\left (\frac {-b+x}{-a+x}\right )^{2/3}+\sqrt [3]{\frac {-b+x}{-a+x}} \sqrt [3]{\text {$\#$1}}+\text {$\#$1}^{2/3}\right ) \text {$\#$1}^{4/3}}{-3 d+6 d \text {$\#$1}-3 d \text {$\#$1}^2+4 a^3 \text {$\#$1}^3-12 a^2 b \text {$\#$1}^3+12 a b^2 \text {$\#$1}^3-4 b^3 \text {$\#$1}^3}\&\right ]-\text {RootSum}\left [-d+3 d \text {$\#$1}-3 d \text {$\#$1}^2+d \text {$\#$1}^3-a^3 \text {$\#$1}^4+3 a^2 b \text {$\#$1}^4-3 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^4\&,\frac {-6 \sqrt [3]{\frac {-b+x}{-a+x}} \text {$\#$1}^2+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{\frac {b-x}{a-x}}}{\sqrt [3]{\text {$\#$1}}}}{\sqrt {3}}\right ) \text {$\#$1}^{7/3}-2 \log \left (-\sqrt [3]{\frac {-b+x}{-a+x}}+\sqrt [3]{\text {$\#$1}}\right ) \text {$\#$1}^{7/3}+\log \left (\left (\frac {-b+x}{-a+x}\right )^{2/3}+\sqrt [3]{\frac {-b+x}{-a+x}} \sqrt [3]{\text {$\#$1}}+\text {$\#$1}^{2/3}\right ) \text {$\#$1}^{7/3}}{-3 d+6 d \text {$\#$1}-3 d \text {$\#$1}^2+4 a^3 \text {$\#$1}^3-12 a^2 b \text {$\#$1}^3+12 a b^2 \text {$\#$1}^3-4 b^3 \text {$\#$1}^3}\&\right ]\right )}{2 \sqrt [3]{(b-x)^2 (-a+x)}} \]
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\[\int \frac {\left (-b +x \right ) \left (-4 a +b +3 x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (b^{4}+a d -\left (4 b^{3}+d \right ) x +6 b^{2} x^{2}-4 b \,x^{3}+x^{4}\right )}d x\]
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Time = 0.50 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.06 \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left ({\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d\right )}}{3 \, {\left (b^{2} d - 2 \, b d x + d x^{2}\right )}}\right ) - {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d^{2} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + {\left (b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\right )} {\left (d^{2}\right )}^{\frac {2}{3}}}{b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b^{2} - 2 \, b x + x^{2}}\right )}{2 \, d^{2}} \]
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Timed out. \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\int { \frac {{\left (4 \, a - b - 3 \, x\right )} {\left (b - x\right )}}{{\left (b^{4} + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4} + a d - {\left (4 \, b^{3} + d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\int { \frac {{\left (4 \, a - b - 3 \, x\right )} {\left (b - x\right )}}{{\left (b^{4} + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4} + a d - {\left (4 \, b^{3} + d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\int -\frac {\left (b-x\right )\,\left (b-4\,a+3\,x\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (a\,d-4\,b\,x^3-x\,\left (4\,b^3+d\right )+b^4+x^4+6\,b^2\,x^2\right )} \,d x \]
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