Optimal. Leaf size=109 \[ \frac {\left (7 (x-1)^2+26\right ) (x-1)}{432 \sqrt {-(x-1)^4-2 (x-1)^2+3}}+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}-\frac {11 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{144 \sqrt {3}}+\frac {7 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{144 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 109, 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 = {1106, 1092, 1178, 1180, 524, 424, 419} \[ \frac {\left (7 (x-1)^2+26\right ) (x-1)}{432 \sqrt {-(x-1)^4-2 (x-1)^2+3}}+\frac {\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}-\frac {11 F\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{144 \sqrt {3}}+\frac {7 E\left (\sin ^{-1}(1-x)|-\frac {1}{3}\right )}{144 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 1092
Rule 1106
Rule 1178
Rule 1180
Rubi steps
\begin {align*} \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx &=\operatorname {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} \operatorname {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 {\operatorname {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 {\operatorname {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} \operatorname {Subst}\left (\int \frac {\sqrt {6+2 x^2}}{\sqrt {2-2 x^2}} \, dx,x,-1+x\right )+\frac {11}{72} \operatorname {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] time = 1.09, size = 327, normalized size = 3.00 \[ \frac {(x-2)^3 x^2 \left (x^2-2 x+4\right )^2 \left (-\frac {7 x \left (x^2-2 x+4\right )}{x-2}-19 i \sqrt {2} (x-2) \sqrt {\frac {i x}{\left (\sqrt {3}+i\right ) (x-2)}} \sqrt {\frac {x^2-2 x+4}{(x-2)^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {3}-i-\frac {4 i}{x-2}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{i+\sqrt {3}}\right )+\frac {7 i \sqrt {2} x \sqrt {\frac {x^2-2 x+4}{(x-2)^2}} E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {3}-i-\frac {4 i}{x-2}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{i+\sqrt {3}}\right )}{\sqrt {\frac {i x}{\left (\sqrt {3}+i\right ) (x-2)}}}+\frac {7 x^7-49 x^6+187 x^5-445 x^4+670 x^3-622 x^2+216 x+36}{(x-2)^2 x \left (x^2-2 x+4\right )}\right )}{432 \left (-x \left (x^3-4 x^2+8 x-8\right )\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{x^{12} - 12 \, x^{11} + 72 \, x^{10} - 280 \, x^{9} + 768 \, x^{8} - 1536 \, x^{7} + 2240 \, x^{6} - 2304 \, x^{5} + 1536 \, x^{4} - 512 \, x^{3}}, x\right ) \]
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 \[ \int \frac {1}{\left (-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1039, normalized size = 9.53 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{\left (-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (-x\,\left (x-2\right )\,\left (x^2-2\,x+4\right )\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (x \left (2 - x\right ) \left (x^{2} - 2 x + 4\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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