Integrand size = 22, antiderivative size = 257 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}+\frac {a^{3/4} (A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {a^{3/4} (A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}} \]
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Time = 0.16 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {470, 327, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {a^{3/4} (A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (A b-a B) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {a^{3/4} (A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {2 x^{3/2} (A b-a B)}{3 b^2}+\frac {2 B x^{7/2}}{7 b} \]
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Rule 210
Rule 303
Rule 327
Rule 335
Rule 470
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {2 B x^{7/2}}{7 b}-\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {7 a B}{2}\right )\right ) \int \frac {x^{5/2}}{a+b x^2} \, dx}{7 b} \\ & = \frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {(a (A b-a B)) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{b^2} \\ & = \frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {(2 a (A b-a B)) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^2} \\ & = \frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}+\frac {(a (A b-a B)) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{5/2}}-\frac {(a (A b-a B)) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{5/2}} \\ & = \frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {(a (A b-a B)) \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 )}{2 b^3}-\frac {(a (A b-a B)) \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 )}{2 b^3}-\frac {\left (a^{3/4} (A b-a B)\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 )}{2 \sqrt {2} b^{11/4}}-\frac {\left (a^{3/4} (A b-a B)\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 )}{2 \sqrt {2} b^{11/4}} \\ & = \frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {a^{3/4} (A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {a^{3/4} (A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}-\frac {\left (a^{3/4} (A b-a B)\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 )}{\sqrt {2} b^{11/4}}+\frac {\left (a^{3/4} (A b-a B)\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 )}{\sqrt {2} b^{11/4}} \\ & = \frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{7/2}}{7 b}+\frac {a^{3/4} (A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (A b-a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}}+\frac {a^{3/4} (A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{11/4}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.59 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {2 x^{3/2} \left (7 A b-7 a B+3 b B x^2\right )}{21 b^2}-\frac {a^{3/4} (-A b+a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} b^{11/4}}-\frac {a^{3/4} (-A b+a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} b^{11/4}} \]
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Time = 2.78 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.54
method | result | size |
risch | \(\frac {2 x^{\frac {3}{2}} \left (3 b B \,x^{2}+7 A b -7 B a \right )}{21 b^{2}}-\frac {a \left (A b -B a \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(140\) |
derivativedivides | \(\frac {\frac {2 b B \,x^{\frac {7}{2}}}{7}+\frac {2 \left (A b -B a \right ) x^{\frac {3}{2}}}{3}}{b^{2}}-\frac {a \left (A b -B a \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(142\) |
default | \(\frac {\frac {2 b B \,x^{\frac {7}{2}}}{7}+\frac {2 \left (A b -B a \right ) x^{\frac {3}{2}}}{3}}{b^{2}}-\frac {a \left (A b -B a \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(142\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 749, normalized size of antiderivative = 2.91 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{a+b x^2} \, dx=-\frac {21 \, b^{2} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {1}{4}} \log \left (b^{8} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{5} - 3 \, A B^{2} a^{4} b + 3 \, A^{2} B a^{3} b^{2} - A^{3} a^{2} b^{3}\right )} \sqrt {x}\right ) - 21 i \, b^{2} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {1}{4}} \log \left (i \, b^{8} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{5} - 3 \, A B^{2} a^{4} b + 3 \, A^{2} B a^{3} b^{2} - A^{3} a^{2} b^{3}\right )} \sqrt {x}\right ) + 21 i \, b^{2} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-i \, b^{8} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{5} - 3 \, A B^{2} a^{4} b + 3 \, A^{2} B a^{3} b^{2} - A^{3} a^{2} b^{3}\right )} \sqrt {x}\right ) - 21 \, b^{2} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-b^{8} \left (-\frac {B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac {3}{4}} - {\left (B^{3} a^{5} - 3 \, A B^{2} a^{4} b + 3 \, A^{2} B a^{3} b^{2} - A^{3} a^{2} b^{3}\right )} \sqrt {x}\right ) - 4 \, {\left (3 \, B b x^{3} - 7 \, {\left (B a - A b\right )} x\right )} \sqrt {x}}{42 \, b^{2}} \]
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Time = 16.09 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.35 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{a+b x^2} \, dx=\begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {7}{2}}}{7}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {11}{2}}}{11}}{a} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {7}{2}}}{7}}{b} & \text {for}\: a = 0 \\- \frac {2 A a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {2 A x^{\frac {3}{2}}}{3 b} + \frac {A \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {A \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {A \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} + \frac {2 B a^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{3} \sqrt [4]{- \frac {a}{b}}} - \frac {2 B a x^{\frac {3}{2}}}{3 b^{2}} - \frac {B a \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {B a \left (- \frac {a}{b}\right )^{\frac {3}{4}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {B a \left (- \frac {a}{b}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} + \frac {2 B x^{\frac {7}{2}}}{7 b} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.83 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {{\left (B a^{2} - A a b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, b^{2}} + \frac {2 \, {\left (3 \, B b x^{\frac {7}{2}} - 7 \, {\left (B a - A b\right )} x^{\frac {3}{2}}\right )}}{21 \, b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.03 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, b^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, b^{5}} + \frac {2 \, {\left (3 \, B b^{6} x^{\frac {7}{2}} - 7 \, B a b^{5} x^{\frac {3}{2}} + 7 \, A b^{6} x^{\frac {3}{2}}\right )}}{21 \, b^{7}} \]
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Time = 5.01 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.36 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{a+b x^2} \, dx=x^{3/2}\,\left (\frac {2\,A}{3\,b}-\frac {2\,B\,a}{3\,b^2}\right )+\frac {2\,B\,x^{7/2}}{7\,b}+\frac {{\left (-a\right )}^{3/4}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b-B\,a\right )}{b^{11/4}}+\frac {{\left (-a\right )}^{3/4}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{b^{11/4}} \]
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