Optimal. Leaf size=62 \[ \frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac {4 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}+\frac {2 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1106, 1091, 1180, 524, 424, 419} \[ \frac {1}{3} \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac {4 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}+\frac {2 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 1091
Rule 1106
Rule 1180
Rubi steps
\begin {align*} \int \sqrt {(2-x) x \left (4-2 x+x^2\right )} \, dx &=\operatorname {Subst}\left (\int \sqrt {3-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{3} \operatorname {Subst}\left (\int \frac {6-2 x^2}{\sqrt {3-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {2}{3} \operatorname {Subst}\left (\int \frac {6-2 x^2}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)-\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {6+2 x^2}}{\sqrt {2-2 x^2}} \, dx,x,-1+x\right )+8 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )\\ &=\frac {1}{3} \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {2 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}-\frac {4 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.84, size = 256, normalized size = 4.13 \[ \frac {\sqrt {-x \left (x^3-4 x^2+8 x-8\right )} \left (\sqrt {\frac {x^2-2 x+4}{x^2}} \left (x^3-3 x^2+4 x-4\right )+8 i \sqrt {2} \sqrt {-\frac {i (x-2)}{\left (\sqrt {3}-i\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {3}+i-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )+2 \sqrt {2} \left (\sqrt {3}-i\right ) \sqrt {-\frac {i (x-2)}{\left (\sqrt {3}-i\right ) x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {3}+i-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )\right )}{3 (x-2) x \sqrt {\frac {x^2-2 x+4}{x^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}, x\right ) \]
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 \[ \int \sqrt {-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 946, normalized size = 15.26 \[ \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+8 x}\, x}{3}+\frac {8 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \EllipticF \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )}{3 \left (i \sqrt {3}-1\right ) \sqrt {-\left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right ) x}}-\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+8 x}}{3}+\frac {8 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (2 \EllipticF \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )-2 \EllipticPi \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \frac {1+i \sqrt {3}}{i \sqrt {3}-1}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )\right )}{3 \left (i \sqrt {3}-1\right ) \sqrt {-\left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right ) x}}-\frac {2 \left (\left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right ) x +2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (\frac {\left (i \sqrt {3}-1\right ) \EllipticE \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )}{2}+\frac {\left (6+2 i \sqrt {3}\right ) \EllipticF \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )}{2 i \sqrt {3}-2}-\frac {4 \EllipticPi \left (\sqrt {\frac {\left (i \sqrt {3}-1\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \frac {-1-i \sqrt {3}}{1-i \sqrt {3}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (1-i \sqrt {3}\right )}}\right )}{i \sqrt {3}-1}\right )\right )}{3 \sqrt {-\left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right ) x}} \]
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 \[ \int \sqrt {-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {-x\,\left (x-2\right )\,\left (x^2-2\,x+4\right )} \,d x \]
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 \[ \int \sqrt {x \left (2 - x\right ) \left (x^{2} - 2 x + 4\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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