Integrand size = 24, antiderivative size = 376 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (11 b c+a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}} \]
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Time = 0.31 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 479, 584, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (a d+11 b c)}{4 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (a d+11 b c)}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {(b c-a d)^2 (a d+11 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (a d+11 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (6 a^2 d^2-21 a b c d+11 b^2 c^2\right )}{6 a^3 b x^{3/2}}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{7/2} \left (a+b x^2\right )} \]
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Rule 210
Rule 217
Rule 477
Rule 479
Rule 584
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^8 \left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (11 b c-7 a d)-d (3 b c+a d) x^4\right )}{x^8 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a b} \\ & = \frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {c^2 (-11 b c+7 a d)}{a x^8}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{a^2 x^4}-\frac {(-b c+a d)^2 (11 b c+a d)}{a^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b} \\ & = -\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^3 b} \\ & = -\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{7/2} b}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{7/2} b} \\ & = -\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^{7/2} b^{3/2}}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^{7/2} b^{3/2}}-\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\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,x,\sqrt {x}\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}-\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\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,x,\sqrt {x}\right )}{8 \sqrt {2} a^{15/4} b^{5/4}} \\ & = -\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}} \\ & = -\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.63 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx=\frac {-\frac {4 a^{3/4} \sqrt [4]{b} \left (-77 b^3 c^3 x^4+21 a^3 d^3 x^4+a b^2 c^2 x^2 \left (-44 c+147 d x^2\right )+3 a^2 b c \left (4 c^2+28 c d x^2-21 d^2 x^4\right )\right )}{x^{7/2} \left (a+b x^2\right )}-21 \sqrt {2} (b c-a d)^2 (11 b c+a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+21 \sqrt {2} (b c-a d)^2 (11 b c+a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{168 a^{15/4} b^{5/4}} \]
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Time = 2.78 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.53
method | result | size |
risch | \(-\frac {2 c^{2} \left (21 a d \,x^{2}-14 c b \,x^{2}+3 a c \right )}{21 a^{3} x^{\frac {7}{2}}}+\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (-\frac {\left (a d -b c \right ) \sqrt {x}}{4 b \left (b \,x^{2}+a \right )}+\frac {\left (a d +11 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b a}\right )}{a^{3}}\) | \(201\) |
derivativedivides | \(-\frac {2 c^{3}}{7 a^{2} x^{\frac {7}{2}}}-\frac {2 c^{2} \left (3 a d -2 b c \right )}{3 a^{3} x^{\frac {3}{2}}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {x}}{2 b \left (b \,x^{2}+a \right )}+\frac {\left (a^{3} d^{3}+9 a^{2} b c \,d^{2}-21 a \,b^{2} c^{2} d +11 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b a}}{a^{3}}\) | \(236\) |
default | \(-\frac {2 c^{3}}{7 a^{2} x^{\frac {7}{2}}}-\frac {2 c^{2} \left (3 a d -2 b c \right )}{3 a^{3} x^{\frac {3}{2}}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {x}}{2 b \left (b \,x^{2}+a \right )}+\frac {\left (a^{3} d^{3}+9 a^{2} b c \,d^{2}-21 a \,b^{2} c^{2} d +11 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b a}}{a^{3}}\) | \(236\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 1761, normalized size of antiderivative = 4.68 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.34 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.13 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx=-\frac {12 \, a^{2} b c^{3} - 7 \, {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3}\right )} x^{4} - 4 \, {\left (11 \, a b^{2} c^{3} - 21 \, a^{2} b c^{2} d\right )} x^{2}}{42 \, {\left (a^{3} b^{2} x^{\frac {11}{2}} + a^{4} b x^{\frac {7}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, a^{3} b} \]
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Time = 0.32 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.35 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{4} b^{2}} + \frac {b^{3} c^{3} \sqrt {x} - 3 \, a b^{2} c^{2} d \sqrt {x} + 3 \, a^{2} b c d^{2} \sqrt {x} - a^{3} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a^{3} b} + \frac {2 \, {\left (14 \, b c^{3} x^{2} - 21 \, a c^{2} d x^{2} - 3 \, a c^{3}\right )}}{21 \, a^{3} x^{\frac {7}{2}}} \]
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Time = 5.94 (sec) , antiderivative size = 1746, normalized size of antiderivative = 4.64 \[ \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
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