Optimal. Leaf size=821 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt {3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}}}-\frac {\sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt {3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2^{5/6} \sqrt [4]{3} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}}}+\frac {\log \left (\frac {(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\sqrt {(3-2 x)^2} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2} \]
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Rubi [A] time = 0.57, antiderivative size = 821, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {124, 151, 157, 61, 623, 303, 218, 1877, 123} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt {3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}}}-\frac {\sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+1\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-3 x+2\right )^{2/3}-2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}-\sqrt {3}+1}{2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2^{5/6} \sqrt [4]{3} (3-2 x) \sqrt {(3-2 x)^2} \sqrt [3]{1-x} \sqrt [3]{2-x} \sqrt {\frac {2^{2/3} \sqrt [3]{x^2-3 x+2}+1}{\left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )^2}}}+\frac {\log \left (\frac {(2-x)^{2/3}}{2^{2/3}}-\sqrt [3]{1-x}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\sqrt {(3-2 x)^2} \sqrt {(2 x-3)^2} \sqrt [3]{x^2-3 x+2}}{2 \sqrt [3]{2} (3-2 x) \sqrt [3]{1-x} \sqrt [3]{2-x} \left (2^{2/3} \sqrt [3]{x^2-3 x+2}+\sqrt {3}+1\right )}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 61
Rule 123
Rule 124
Rule 151
Rule 157
Rule 218
Rule 303
Rule 623
Rule 1877
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^3} \, dx &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}+\frac {1}{24} \int \frac {24-4 x}{\sqrt [3]{1-x} \sqrt [3]{2-x} x^2} \, dx\\ &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {1}{48} \int \frac {-16-8 x}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx\\ &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}+\frac {1}{6} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x}} \, dx+\frac {1}{3} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{2-x} x} \, dx\\ &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}+\frac {\sqrt [3]{2-3 x+x^2} \int \frac {1}{\sqrt [3]{2-3 x+x^2}} \, dx}{6 \sqrt [3]{1-x} \sqrt [3]{2-x}}\\ &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}+\frac {\left (\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 )}{2 \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}+\frac {\left (\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^{2/3} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}+\frac {\left (\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 )}{2 \sqrt [6]{2} \sqrt {2+\sqrt {3}} \sqrt [3]{1-x} \sqrt [3]{2-x} (-3+2 x)}\\ &=-\frac {(1-x)^{2/3} (2-x)^{2/3}}{4 x^2}-\frac {(1-x)^{2/3} (2-x)^{2/3}}{2 x}-\frac {\sqrt {(3-2 x)^2} \sqrt {(-3+2 x)^2} \sqrt [3]{2-3 x+x^2}}{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 {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\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 )}{4 \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 {(-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 )}{2^{5/6} \sqrt [4]{3} (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 {\log \left (-\sqrt [3]{1-x}+\frac {(2-x)^{2/3}}{2^{2/3}}\right )}{4 \sqrt [3]{2}}-\frac {\log (x)}{6 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 84, normalized size = 0.10 \[ -\frac {(1-x)^{2/3} \left (15 x^2 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x-1,1-x\right )+2 (x-1) x^2 F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};x-1,1-x\right )+5 (2-x)^{2/3} (2 x+1)\right )}{20 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 3.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x + 2\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {2}{3}}}{x^{5} - 3 \, x^{4} + 2 \, x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-x +1\right )^{\frac {1}{3}} \left (-x +2\right )^{\frac {1}{3}} x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} {\left (-x + 2\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^3\,{\left (1-x\right )}^{1/3}\,{\left (2-x\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \sqrt [3]{1 - x} \sqrt [3]{2 - x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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