Optimal. Leaf size=109 \[ \frac {\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {\left (26+7 (-1+x)^2\right ) (-1+x)}{432 \sqrt {3-2 (-1+x)^2-(-1+x)^4}}+\frac {7 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{144 \sqrt {3}}-\frac {11 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{144 \sqrt {3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1120, 1106,
1192, 1194, 538, 435, 430} \begin {gather*} -\frac {11 F\left (\text {ArcSin}(1-x)\left |-\frac {1}{3}\right .\right )}{144 \sqrt {3}}+\frac {7 E\left (\text {ArcSin}(1-x)\left |-\frac {1}{3}\right .\right )}{144 \sqrt {3}}+\frac {\left (7 (x-1)^2+26\right ) (x-1)}{432 \sqrt {-(x-1)^4-2 (x-1)^2+3}}+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 538
Rule 1106
Rule 1120
Rule 1192
Rule 1194
Rubi steps
\begin {align*} \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx &=\text {Subst}\left (\int \frac {1}{\left (3-2 x^2-x^4\right )^{5/2}} \, dx,x,-1+x\right )\\ &=\frac {\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}-\frac {1}{144} \text {Subst}\left (\int \frac {-38-6 x^2}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx,x,-1+x\right )\\ &=-\frac {\left (26+7 (1-x)^2\right ) (1-x)}{432 \sqrt {3-2 (1-x)^2-(1-x)^4}}+\frac {\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {192-112 x^2}{\sqrt {3-2 x^2-x^4}} \, dx,x,-1+x\right )}{6912}\\ &=-\frac {\left (26+7 (1-x)^2\right ) (1-x)}{432 \sqrt {3-2 (1-x)^2-(1-x)^4}}+\frac {\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {192-112 x^2}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )}{3456}\\ &=-\frac {\left (26+7 (1-x)^2\right ) (1-x)}{432 \sqrt {3-2 (1-x)^2-(1-x)^4}}+\frac {\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}-\frac {7}{432} \text {Subst}\left (\int \frac {\sqrt {6+2 x^2}}{\sqrt {2-2 x^2}} \, dx,x,-1+x\right )+\frac {11}{72} \text {Subst}\left (\int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx,x,-1+x\right )\\ &=-\frac {\left (26+7 (1-x)^2\right ) (1-x)}{432 \sqrt {3-2 (1-x)^2-(1-x)^4}}+\frac {\left (5+(-1+x)^2\right ) (-1+x)}{72 \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {7 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{144 \sqrt {3}}-\frac {11 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{144 \sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 20.75, size = 298, normalized size = 2.73 \begin {gather*} \frac {\frac {7 i \sqrt {2} (-2+x) x^2 \sqrt {\frac {4-2 x+x^2}{x^2}} 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 )}{\sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}}}+\frac {36-232 x+274 x^2-226 x^3+115 x^4-37 x^5+7 x^6-19 i \sqrt {2} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} x^3 \sqrt {\frac {4-2 x+x^2}{x^2}} \left (-8+8 x-4 x^2+x^3\right ) 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 )}{-8+8 x-4 x^2+x^3}}{432 x \sqrt {-x \left (-8+8 x-4 x^2+x^3\right )}} \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. 1038 vs. \(2 (93 ) = 186\).
time = 0.50, size = 1039, normalized size = 9.53 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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 195, normalized size = 1.79 \begin {gather*} -\frac {43 \, \sqrt {2} {\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\right )} {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right ) - 84 \, \sqrt {2} {\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\right )} {\rm weierstrassZeta}\left (-\frac {2}{3}, \frac {7}{54}, {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right )\right ) + 6 \, {\left (7 \, x^{6} - 37 \, x^{5} + 115 \, x^{4} - 226 \, x^{3} + 274 \, x^{2} - 232 \, x + 36\right )} \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{2592 \, {\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\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 \frac {1}{\left (- x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {5}{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 \frac {1}{{\left (-x^4+4\,x^3-8\,x^2+8\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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