Optimal. Leaf size=102 \[ \frac {2}{35} \left (13-3 (-1+x)^2\right ) \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac {16}{5} \sqrt {3} E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )-\frac {176}{35} \sqrt {3} F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1120, 1105,
1190, 1194, 538, 435, 430} \begin {gather*} -\frac {176}{35} \sqrt {3} F\left (\text {ArcSin}(1-x)\left |-\frac {1}{3}\right .\right )+\frac {16}{5} \sqrt {3} E\left (\text {ArcSin}(1-x)\left |-\frac {1}{3}\right .\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac {2}{35} \left (13-3 (x-1)^2\right ) (x-1) \sqrt {-(x-1)^4-2 (x-1)^2+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 538
Rule 1105
Rule 1120
Rule 1190
Rule 1194
Rubi steps
\begin {align*} \int \left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2} \, dx &=\text {Subst}\left (\int \left (3-2 x^2-x^4\right )^{3/2} \, dx,x,-1+x\right )\\ &=\frac {1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac {3}{7} \text {Subst}\left (\int \left (6-2 x^2\right ) \sqrt {3-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=-\frac {2}{35} \left (13-3 (1-x)^2\right ) \sqrt {3-2 (1-x)^2-(1-x)^4} (1-x)+\frac {1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)-\frac {1}{35} \text {Subst}\left (\int \frac {-192+112 x^2}{\sqrt {3-2 x^2-x^4}} \, dx,x,-1+x\right )\\ &=-\frac {2}{35} \left (13-3 (1-x)^2\right ) \sqrt {3-2 (1-x)^2-(1-x)^4} (1-x)+\frac {1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)-\frac {2}{35} \text {Subst}\left (\int \frac {-192+112 x^2}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )\\ &=-\frac {2}{35} \left (13-3 (1-x)^2\right ) \sqrt {3-2 (1-x)^2-(1-x)^4} (1-x)+\frac {1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)-\frac {16}{5} \text {Subst}\left (\int \frac {\sqrt {6+2 x^2}}{\sqrt {2-2 x^2}} \, dx,x,-1+x\right )+\frac {1056}{35} \text {Subst}\left (\int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )\\ &=-\frac {2}{35} \left (13-3 (1-x)^2\right ) \sqrt {3-2 (1-x)^2-(1-x)^4} (1-x)+\frac {1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac {16}{5} \sqrt {3} E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )-\frac {176}{35} \sqrt {3} F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 22.68, size = 278, normalized size = 2.73 \begin {gather*} \frac {\sqrt {-x \left (-8+8 x-4 x^2+x^3\right )} \left (\sqrt {\frac {4-2 x+x^2}{x^2}} \left (-224+152 x+44 x^2-228 x^3+230 x^4-116 x^5+35 x^6-5 x^7\right )+112 \sqrt {2} \left (-i+\sqrt {3}\right ) \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} E\left (\sin ^{-1}\left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )+304 i \sqrt {2} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )\right )}{35 (-2+x) x \sqrt {\frac {4-2 x+x^2}{x^2}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1049 vs. \(2 (86 ) = 172\).
time = 0.56, size = 1050, normalized size = 10.29 Too large to display
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.10, size = 58, normalized size = 0.57 \begin {gather*} -\frac {{\left (5 \, x^{6} - 30 \, x^{5} + 91 \, x^{4} - 164 \, x^{3} + 130 \, x^{2} - 12 \, x - 132\right )} \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{35 \, {\left (x - 1\right )}} \end {gather*}
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 \begin {gather*} \int \left (x \left (2 - x\right ) \left (x^{2} - 2 x + 4\right )\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (-x\,\left (x-2\right )\,\left (x^2-2\,x+4\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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