Optimal. Leaf size=671 \[ -\frac{\sqrt [6]{2} 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{(3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}-\frac{3 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{\sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}+\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}} \]
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Rubi [A] time = 0.286413, antiderivative size = 671, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {61, 623, 303, 218, 1877} \[ -\frac{3 \sqrt{(3-2 x)^2} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{\sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )}-\frac{\sqrt [6]{2} 3^{3/4} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{(3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt{3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [3]{2} (3-2 x) \sqrt{(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt{\frac{2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt{3}+1\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 61
Rule 623
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx &=\frac{\sqrt [3]{2-3 x+x^2} \int \frac{1}{\sqrt [3]{2-3 x+x^2}} \, dx}{\sqrt [3]{1-x} \sqrt [3]{2-x}}\\ &=\frac{\left (3 \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{\sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=\frac{\left (3 \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \operatorname{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 )}{2^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}+\frac{\left (3 \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+4 x^3}} \, dx,x,\sqrt [3]{2-3 x+x^2}\right )}{\sqrt [6]{2} \sqrt{2+\sqrt{3}} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=-\frac{3 \sqrt{(3-2 x)^2} \sqrt{(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{\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{3 \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 )}{2 \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{\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 )}{(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*}
Mathematica [C] time = 0.003955, size = 26, normalized size = 0.04 \[ -\frac{3}{2} (1-x)^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x-1\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{2-x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x + 2\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-x + 2\right )}^{\frac{2}{3}}{\left (-x + 1\right )}^{\frac{2}{3}}}{x^{2} - 3 \, x + 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.83808, size = 41, normalized size = 0.06 \begin{align*} - \frac{\left (-1\right )^{\frac{2}{3}} \left (x - 1\right )^{\frac{2}{3}} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\left (x - 1\right ) e^{2 i \pi }} \right )}}{\Gamma \left (\frac{5}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x + 2\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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