Integrand size = 22, antiderivative size = 720 \[ \int \frac {x^2}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\frac {45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac {3}{7} (1-x)^{2/3} (2-x)^{2/3} x-\frac {99 \sqrt {(3-2 x)^2} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{14 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )}+\frac {99 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt {3}\right )}{28 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}-\frac {33\ 3^{3/4} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right ),-7-4 \sqrt {3}\right )}{7\ 2^{5/6} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}} \]
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Time = 0.36 (sec) , antiderivative size = 720, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {92, 81, 63, 637, 309, 224, 1891} \[ \int \frac {x^2}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=-\frac {33\ 3^{3/4} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt {3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{7\ 2^{5/6} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}}}+\frac {99 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt {3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{28 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}}}-\frac {99 \sqrt {(3-2 x)^2} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{14 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )}+\frac {3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac {45}{28} (1-x)^{2/3} (2-x)^{2/3} \]
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Rule 63
Rule 81
Rule 92
Rule 224
Rule 309
Rule 637
Rule 1891
Rubi steps \begin{align*} \text {integral}& = \frac {3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac {3}{7} \int \frac {-2+5 x}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx \\ & = \frac {45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac {3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac {33}{14} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx \\ & = \frac {45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac {3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac {\left (33 \sqrt [3]{2-3 x+x^2}\right ) \int \frac {1}{\sqrt [3]{2-3 x+x^2}} \, dx}{14 \sqrt [3]{1-x} \sqrt [3]{2-x}} \\ & = \frac {45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac {3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac {\left (99 \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{14 \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)} \\ & = \frac {45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac {3}{7} (1-x)^{2/3} (2-x)^{2/3} x+\frac {\left (99 \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \text {Subst}\left (\int \frac {1-\sqrt {3}+2^{2/3} x}{\sqrt {1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{14\ 2^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}-\frac {\left (99 \left (1-\sqrt {3}\right ) \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{14\ 2^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)} \\ & = \frac {45}{28} (1-x)^{2/3} (2-x)^{2/3}+\frac {3}{7} (1-x)^{2/3} (2-x)^{2/3} x-\frac {99 \sqrt {(3-2 x)^2} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{14 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )}+\frac {99 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt {3}\right )}{28 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}-\frac {33\ 3^{3/4} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt {3}\right )}{7\ 2^{5/6} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.06 \[ \int \frac {x^2}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\frac {3}{28} (1-x)^{2/3} \left ((2-x)^{2/3} (15+4 x)-33 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-1+x\right )\right ) \]
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\[\int \frac {x^{2}}{\left (1-x \right )^{\frac {1}{3}} \left (2-x \right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {x^2}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x^{2}}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int \frac {x^{2}}{\sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \]
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\[ \int \frac {x^2}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x^{2}}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x^{2}}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int \frac {x^2}{{\left (1-x\right )}^{1/3}\,{\left (2-x\right )}^{1/3}} \,d x \]
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