Optimal. Leaf size=687 \[ -\frac {25 d^{4/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{27648 \sqrt {3} c^{23/6}}+\frac {25 d^{4/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{82944 c^{23/6}}-\frac {25 d^{4/3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{82944 c^{23/6}}-\frac {5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{432 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {5 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {5 d^{4/3} \sqrt {c+d x^3}}{864 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {5 d \sqrt {c+d x^3}}{864 c^4 x}-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )} \]
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Rubi [A] time = 0.93, antiderivative size = 687, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {472, 583, 584, 303, 218, 1877, 486, 444, 63, 206, 2138, 2145, 205} \[ -\frac {5 d^{4/3} \sqrt {c+d x^3}}{864 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {25 d^{4/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{27648 \sqrt {3} c^{23/6}}+\frac {25 d^{4/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{82944 c^{23/6}}-\frac {25 d^{4/3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{82944 c^{23/6}}-\frac {5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{432 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {5 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {5 d \sqrt {c+d x^3}}{864 c^4 x}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 206
Rule 218
Rule 303
Rule 444
Rule 472
Rule 486
Rule 583
Rule 584
Rule 1877
Rule 2138
Rule 2145
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}+\frac {\int \frac {31 c d+\frac {11 d^2 x^3}{2}}{x^5 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{216 c^2 d}\\ &=-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac {\int \frac {320 c^2 d^2-\frac {155}{2} c d^3 x^3}{x^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{6912 c^4 d}\\ &=-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4}+\frac {5 d \sqrt {c+d x^3}}{864 c^4 x}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}+\frac {\int \frac {x \left (-980 c^3 d^3+160 c^2 d^4 x^3\right )}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{55296 c^6 d}\\ &=-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4}+\frac {5 d \sqrt {c+d x^3}}{864 c^4 x}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}+\frac {\int \left (-\frac {160 c^2 d^3 x}{\sqrt {c+d x^3}}+\frac {300 c^3 d^3 x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}}\right ) \, dx}{55296 c^6 d}\\ &=-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4}+\frac {5 d \sqrt {c+d x^3}}{864 c^4 x}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac {\left (5 d^2\right ) \int \frac {x}{\sqrt {c+d x^3}} \, dx}{1728 c^4}+\frac {\left (25 d^2\right ) \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{4608 c^3}\\ &=-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4}+\frac {5 d \sqrt {c+d x^3}}{864 c^4 x}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac {(25 d) \int \frac {2 \sqrt [3]{c} d^{2/3}-2 d x-\frac {d^{4/3} x^2}{\sqrt [3]{c}}}{\left (4+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+\frac {d^{2/3} x^2}{c^{2/3}}\right ) \sqrt {c+d x^3}} \, dx}{55296 c^4}-\frac {\left (5 d^{5/3}\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt {c+d x^3}} \, dx}{1728 c^4}+\frac {\left (25 d^{5/3}\right ) \int \frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (2-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt {c+d x^3}} \, dx}{55296 c^{11/3}}-\frac {\left (5 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} d^{5/3}\right ) \int \frac {1}{\sqrt {c+d x^3}} \, dx}{864 c^{11/3}}-\frac {\left (25 d^{7/3}\right ) \int \frac {x^2}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx}{18432 c^{10/3}}\\ &=-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4}+\frac {5 d \sqrt {c+d x^3}}{864 c^4 x}-\frac {5 d^{4/3} \sqrt {c+d x^3}}{864 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}+\frac {5 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{432 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {\left (25 d^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{9-c x^2} \, dx,x,\frac {\left (1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\sqrt {c+d x^3}}\right )}{27648 c^{10/3}}-\frac {\left (25 d^{7/3}\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{55296 c^{10/3}}+\frac {\left (25 d^{10/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {2 d^2}{c}-6 d^2 x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {c+d x^3}}\right )}{13824 c^{13/3}}\\ &=-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4}+\frac {5 d \sqrt {c+d x^3}}{864 c^4 x}-\frac {5 d^{4/3} \sqrt {c+d x^3}}{864 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac {25 d^{4/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{27648 \sqrt {3} c^{23/6}}+\frac {25 d^{4/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{82944 c^{23/6}}+\frac {5 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{432 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {\left (25 d^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{27648 c^{10/3}}\\ &=-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4}+\frac {5 d \sqrt {c+d x^3}}{864 c^4 x}-\frac {5 d^{4/3} \sqrt {c+d x^3}}{864 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac {25 d^{4/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{27648 \sqrt {3} c^{23/6}}+\frac {25 d^{4/3} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{82944 c^{23/6}}-\frac {25 d^{4/3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{82944 c^{23/6}}+\frac {5 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{432 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 196, normalized size = 0.29 \[ \frac {245 c d^2 x^6 \left (d x^3-8 c\right ) \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-16 \left (d^3 x^9 \left (d x^3-8 c\right ) \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {5}{3};\frac {1}{2},1;\frac {8}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+2 c \left (216 c^3-135 c^2 d x^3-311 c d^2 x^6+40 d^3 x^9\right )\right )}{221184 c^5 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 6.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x^{3} + c}}{d^{3} x^{14} - 15 \, c d^{2} x^{11} + 48 \, c^{2} d x^{8} + 64 \, c^{3} x^{5}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}^{2} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.24, size = 2240, normalized size = 3.26 \[ \text {Expression too large to display} \]
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 \[ \int \frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}^{2} x^{5}}\,{d x} \]
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 \[ \int \frac {1}{x^5\,\sqrt {d\,x^3+c}\,{\left (8\,c-d\,x^3\right )}^2} \,d x \]
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 \[ \int \frac {1}{x^{5} \left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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