Integrand size = 22, antiderivative size = 727 \[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac {9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)-\frac {81 \sqrt {(3-2 x)^2} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{7 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )}+\frac {81 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right )|-7-4 \sqrt {3}\right )}{14 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}}-\frac {27 \sqrt [6]{2} 3^{3/4} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2} \left (1+2^{2/3} \sqrt [3]{2-3 x+x^2}\right ) \sqrt {\frac {1-2^{2/3} \sqrt [3]{2-3 x+x^2}+2 \sqrt [3]{2} \left (2-3 x+x^2\right )^{2/3}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}{1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}}\right ),-7-4 \sqrt {3}\right )}{7 (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {1+2^{2/3} \sqrt [3]{2-3 x+x^2}}{\left (1+\sqrt {3}+2^{2/3} \sqrt [3]{2-3 x+x^2}\right )^2}}} \]
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Time = 0.37 (sec) , antiderivative size = 727, 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 = {102, 152, 63, 637, 309, 224, 1891} \[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=-\frac {27 \sqrt [6]{2} 3^{3/4} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt {3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{7 (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}}}+\frac {81 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt {3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{14 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}}}+\frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2-\frac {81 \sqrt {(3-2 x)^2} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{7 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )}+\frac {9}{70} (1-x)^{2/3} (2-x)^{2/3} (8 x+23) \]
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Rule 63
Rule 102
Rule 152
Rule 224
Rule 309
Rule 637
Rule 1891
Rubi steps \begin{align*} \text {integral}& = \frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac {3}{10} \int \frac {x (-4+8 x)}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx \\ & = \frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac {9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)+\frac {27}{7} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx \\ & = \frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac {9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)+\frac {\left (27 \sqrt [3]{2-3 x+x^2}\right ) \int \frac {1}{\sqrt [3]{2-3 x+x^2}} \, dx}{7 \sqrt [3]{1-x} \sqrt [3]{2-x}} \\ & = \frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac {9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)+\frac {\left (81 \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 )}{7 \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)} \\ & = \frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac {9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)+\frac {\left (81 \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 )}{7\ 2^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}-\frac {\left (81 \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 )}{7\ 2^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)} \\ & = \frac {3}{10} (1-x)^{2/3} (2-x)^{2/3} x^2+\frac {9}{70} (1-x)^{2/3} (2-x)^{2/3} (23+8 x)-\frac {81 \sqrt {(3-2 x)^2} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{7 \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 {81 \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 )}{14 \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 {27 \sqrt [6]{2} 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 (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 = 10.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.10 \[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\frac {3}{10} (1-x)^{2/3} \left ((2-x)^{2/3} x^2-12 \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {2}{3},\frac {5}{3},-1+x\right )+42 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {2}{3},\frac {5}{3},-1+x\right )-36 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-1+x\right )\right ) \]
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\[\int \frac {x^{3}}{\left (1-x \right )^{\frac {1}{3}} \left (2-x \right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x^{3}}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int \frac {x^{3}}{\sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \]
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\[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x^{3}}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int { \frac {x^{3}}{{\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx=\int \frac {x^3}{{\left (1-x\right )}^{1/3}\,{\left (2-x\right )}^{1/3}} \,d x \]
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