Optimal. Leaf size=363 \[ -\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (c d^2-3 a e^2\right )\right )}{4 a c \left (a+c x^4\right )}-\frac {e^3 x^3}{c \left (a+c x^4\right )} \]
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Rubi [A] time = 0.41, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {1207, 1858, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}+\frac {x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (c d^2-3 a e^2\right )\right )}{4 a c \left (a+c x^4\right )}-\frac {e^3 x^3}{c \left (a+c x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 1207
Rule 1858
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}-\frac {\int \frac {-c d^3-3 e \left (c d^2+a e^2\right ) x^2-3 c d e^2 x^4}{\left (a+c x^4\right )^2} \, dx}{c}\\ &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {\int \frac {3 c d \left (c d^2+a e^2\right )+3 c e \left (c d^2+a e^2\right ) x^2}{a+c x^4} \, dx}{4 a c^2}\\ &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}+\frac {\left (3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^2}+\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^2}\\ &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {\left (3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} c^{7/4}}-\frac {\left (3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} c^{7/4}}+\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} c^2}+\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} c^2}\\ &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{7/4}}+\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}-\frac {\left (3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}\\ &=-\frac {e^3 x^3}{c \left (a+c x^4\right )}+\frac {x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{7/4}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 371, normalized size = 1.02 \begin {gather*} \frac {-\frac {8 a^{3/4} c^{3/4} \left (a e^2 x \left (3 d+e x^2\right )-c d^2 x \left (d+3 e x^2\right )\right )}{a+c x^4}+3 \sqrt {2} \left (a^{3/2} e^3+\sqrt {a} c d^2 e-a \sqrt {c} d e^2-c^{3/2} d^3\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )+3 \sqrt {2} \left (-a^{3/2} e^3-\sqrt {a} c d^2 e+a \sqrt {c} d e^2+c^{3/2} d^3\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )-6 \sqrt {2} \left (a^{3/2} e^3+\sqrt {a} c d^2 e+a \sqrt {c} d e^2+c^{3/2} d^3\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+6 \sqrt {2} \left (a^{3/2} e^3+\sqrt {a} c d^2 e+a \sqrt {c} d e^2+c^{3/2} d^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{32 a^{7/4} c^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.13, size = 2116, normalized size = 5.83
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 425, normalized size = 1.17 \begin {gather*} \frac {3 \, c d^{2} x^{3} e + c d^{3} x - a x^{3} e^{3} - 3 \, a d x e^{2}}{4 \, {\left (c x^{4} + a\right )} a c} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 624, normalized size = 1.72 \begin {gather*} \frac {3 \sqrt {2}\, d^{2} e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {3 \sqrt {2}\, d^{2} e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {3 \sqrt {2}\, d^{2} e \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{32 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \,e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{16 a c}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \,e^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{16 a c}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \,e^{2} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{32 a c}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{16 a^{2}}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{16 a^{2}}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{32 a^{2}}+\frac {3 \sqrt {2}\, e^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {3 \sqrt {2}\, e^{3} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {3 \sqrt {2}\, e^{3} \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{32 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {-\frac {\left (a \,e^{2}-3 c \,d^{2}\right ) e \,x^{3}}{4 a c}-\frac {\left (3 a \,e^{2}-c \,d^{2}\right ) d x}{4 a c}}{c \,x^{4}+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.36, size = 292, normalized size = 0.80 \begin {gather*} \frac {{\left (3 \, c d^{2} e - a e^{3}\right )} x^{3} + {\left (c d^{3} - 3 \, a d e^{2}\right )} x}{4 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} + \frac {3 \, {\left (c d^{2} + a e^{2}\right )} {\left (\frac {2 \, \sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{32 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.94, size = 2560, normalized size = 7.05
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.37, size = 352, normalized size = 0.97 \begin {gather*} \operatorname {RootSum} {\left (65536 t^{4} a^{7} c^{7} + t^{2} \left (9216 a^{6} c^{4} d e^{5} + 18432 a^{5} c^{5} d^{3} e^{3} + 9216 a^{4} c^{6} d^{5} e\right ) + 81 a^{6} e^{12} + 486 a^{5} c d^{2} e^{10} + 1215 a^{4} c^{2} d^{4} e^{8} + 1620 a^{3} c^{3} d^{6} e^{6} + 1215 a^{2} c^{4} d^{8} e^{4} + 486 a c^{5} d^{10} e^{2} + 81 c^{6} d^{12}, \left (t \mapsto t \log {\left (x + \frac {4096 t^{3} a^{6} c^{5} e + 432 t a^{5} c^{2} d e^{6} + 720 t a^{4} c^{3} d^{3} e^{4} + 144 t a^{3} c^{4} d^{5} e^{2} - 144 t a^{2} c^{5} d^{7}}{27 a^{5} e^{10} + 81 a^{4} c d^{2} e^{8} + 54 a^{3} c^{2} d^{4} e^{6} - 54 a^{2} c^{3} d^{6} e^{4} - 81 a c^{4} d^{8} e^{2} - 27 c^{5} d^{10}} \right )} \right )\right )} + \frac {x^{3} \left (- a e^{3} + 3 c d^{2} e\right ) + x \left (- 3 a d e^{2} + c d^{3}\right )}{4 a^{2} c + 4 a c^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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