Integrand size = 28, antiderivative size = 340 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac {e \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}-\frac {3 \left (\sqrt {b} d+\sqrt {a} f\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d+\sqrt {a} f\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}-\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}} \]
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Time = 0.22 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {1837, 1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} f+\sqrt {b} d\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} f+\sqrt {b} d\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {e \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}-\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )} \]
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Rule 210
Rule 211
Rule 281
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1837
Rule 1869
Rule 1890
Rubi steps \begin{align*} \text {integral}& = -\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {\int \frac {d+2 e x+3 f x^2}{\left (a+b x^4\right )^2} \, dx}{8 b} \\ & = -\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}-\frac {\int \frac {-3 d-4 e x-3 f x^2}{a+b x^4} \, dx}{32 a b} \\ & = -\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}-\frac {\int \left (-\frac {4 e x}{a+b x^4}+\frac {-3 d-3 f x^2}{a+b x^4}\right ) \, dx}{32 a b} \\ & = -\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}-\frac {\int \frac {-3 d-3 f x^2}{a+b x^4} \, dx}{32 a b}+\frac {e \int \frac {x}{a+b x^4} \, dx}{8 a b} \\ & = -\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac {e \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{16 a b}+\frac {\left (3 \left (\frac {\sqrt {b} d}{\sqrt {a}}-f\right )\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{64 a b^2}+\frac {\left (3 \left (\frac {\sqrt {b} d}{\sqrt {a}}+f\right )\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{64 a b^2} \\ & = -\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac {e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}+\frac {\left (3 \left (\frac {\sqrt {b} d}{\sqrt {a}}+f\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a b^2}+\frac {\left (3 \left (\frac {\sqrt {b} d}{\sqrt {a}}+f\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a b^2}-\frac {\left (3 \left (\sqrt {b} d-\sqrt {a} f\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{128 \sqrt {2} a^{7/4} b^{7/4}}-\frac {\left (3 \left (\sqrt {b} d-\sqrt {a} f\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{128 \sqrt {2} a^{7/4} b^{7/4}} \\ & = -\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac {e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}-\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\left (3 \left (\sqrt {b} d+\sqrt {a} f\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}-\frac {\left (3 \left (\sqrt {b} d+\sqrt {a} f\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}} \\ & = -\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac {x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac {e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}-\frac {3 \left (\sqrt {b} d+\sqrt {a} f\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d+\sqrt {a} f\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}-\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{7/4} b^{7/4}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {8 b^{3/4} x (d+x (2 e+3 f x))}{a \left (a+b x^4\right )}-\frac {32 b^{3/4} (c+x (d+x (e+f x)))}{\left (a+b x^4\right )^2}-\frac {2 \left (3 \sqrt {2} \sqrt {b} d+8 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt {2} \sqrt {a} f\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {2 \left (3 \sqrt {2} \sqrt {b} d-8 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt {2} \sqrt {a} f\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {3 \sqrt {2} \left (-\sqrt {b} d+\sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}+\frac {3 \sqrt {2} \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}}{256 b^{7/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.63 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.34
method | result | size |
risch | \(\frac {\frac {3 f \,x^{7}}{32 a}+\frac {e \,x^{6}}{16 a}+\frac {d \,x^{5}}{32 a}-\frac {f \,x^{3}}{32 b}-\frac {e \,x^{2}}{16 b}-\frac {3 d x}{32 b}-\frac {c}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (3 f \,\textit {\_R}^{2}+4 e \textit {\_R} +3 d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a \,b^{2}}\) | \(114\) |
default | \(\frac {\frac {3 f \,x^{7}}{32 a}+\frac {e \,x^{6}}{16 a}+\frac {d \,x^{5}}{32 a}-\frac {f \,x^{3}}{32 b}-\frac {e \,x^{2}}{16 b}-\frac {3 d x}{32 b}-\frac {c}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\frac {3 d \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {2 e \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{\sqrt {a b}}+\frac {3 f \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{32 b a}\) | \(304\) |
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Result contains complex when optimal does not.
Time = 8.28 (sec) , antiderivative size = 124542, normalized size of antiderivative = 366.30 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.01 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\frac {3 \, b f x^{7} + 2 \, b e x^{6} + b d x^{5} - a f x^{3} - 2 \, a e x^{2} - 3 \, a d x - 4 \, a c}{32 \, {\left (a b^{3} x^{8} + 2 \, a^{2} b^{2} x^{4} + a^{3} b\right )}} + \frac {\frac {3 \, \sqrt {2} {\left (\sqrt {b} d - \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {3 \, \sqrt {2} {\left (\sqrt {b} d - \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} d + 3 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} f - 8 \, \sqrt {a} \sqrt {b} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} d + 3 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} f + 8 \, \sqrt {a} \sqrt {b} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{256 \, a b} \]
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Time = 0.28 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\frac {3 \, b f x^{7} + 2 \, b e x^{6} + b d x^{5} - a f x^{3} - 2 \, a e x^{2} - 3 \, a d x - 4 \, a c}{32 \, {\left (b x^{4} + a\right )}^{2} a b} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a b} b^{2} e + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a b} b^{2} e + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{2} b^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{2} b^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{2} b^{4}} \]
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Time = 0.40 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.53 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (268435456\,a^7\,b^7\,z^4+589824\,a^4\,b^4\,d\,f\,z^2+524288\,a^4\,b^4\,e^2\,z^2+18432\,a^3\,b^2\,e\,f^2\,z-18432\,a^2\,b^3\,d^2\,e\,z-576\,a\,b\,d\,e^2\,f+162\,a\,b\,d^2\,f^2+256\,a\,b\,e^4+81\,a^2\,f^4+81\,b^2\,d^4,z,k\right )\,\left (\mathrm {root}\left (268435456\,a^7\,b^7\,z^4+589824\,a^4\,b^4\,d\,f\,z^2+524288\,a^4\,b^4\,e^2\,z^2+18432\,a^3\,b^2\,e\,f^2\,z-18432\,a^2\,b^3\,d^2\,e\,z-576\,a\,b\,d\,e^2\,f+162\,a\,b\,d^2\,f^2+256\,a\,b\,e^4+81\,a^2\,f^4+81\,b^2\,d^4,z,k\right )\,\left (\frac {3\,b^2\,d}{2}-2\,b^2\,e\,x\right )+\frac {3\,e\,f}{32\,a}+\frac {x\,\left (144\,a\,b^2\,d^2-144\,a^2\,b\,f^2\right )}{4096\,a^3\,b}\right )-\frac {3\,\left (9\,b\,d^2\,f-16\,b\,d\,e^2+9\,a\,f^3\right )}{32768\,a^3\,b^2}+\frac {x\,\left (8\,e^3-9\,d\,e\,f\right )}{4096\,a^3\,b}\right )\,\mathrm {root}\left (268435456\,a^7\,b^7\,z^4+589824\,a^4\,b^4\,d\,f\,z^2+524288\,a^4\,b^4\,e^2\,z^2+18432\,a^3\,b^2\,e\,f^2\,z-18432\,a^2\,b^3\,d^2\,e\,z-576\,a\,b\,d\,e^2\,f+162\,a\,b\,d^2\,f^2+256\,a\,b\,e^4+81\,a^2\,f^4+81\,b^2\,d^4,z,k\right )\right )-\frac {\frac {c}{8\,b}-\frac {d\,x^5}{32\,a}-\frac {e\,x^6}{16\,a}+\frac {e\,x^2}{16\,b}-\frac {3\,f\,x^7}{32\,a}+\frac {f\,x^3}{32\,b}+\frac {3\,d\,x}{32\,b}}{a^2+2\,a\,b\,x^4+b^2\,x^8} \]
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