Optimal. Leaf size=460 \[ \frac {7 \sqrt {d x}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.22, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1126, 296,
335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {7 \sqrt {d x}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 \left (a+b x^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 \left (a+b x^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 296
Rule 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{8 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {7 \sqrt {d x}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{32 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {7 \sqrt {d x}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {7 \sqrt {d x}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d-\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{32 a^{5/2} d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d+\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{32 a^{5/2} d^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {7 \sqrt {d x}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{64 a^{5/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{64 a^{5/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (21 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{64 \sqrt {2} a^{11/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (21 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{64 \sqrt {2} a^{11/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {7 \sqrt {d x}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (21 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{11/4} b^{5/4} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {7 \sqrt {d x}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{b} \sqrt {d} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 181, normalized size = 0.39 \begin {gather*} \frac {\sqrt {x} \left (4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (11 a+7 b x^2\right )-21 \sqrt {2} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+21 \sqrt {2} \left (a+b x^2\right )^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{64 a^{11/4} \sqrt [4]{b} \sqrt {d x} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(637\) vs.
\(2(294)=588\).
time = 0.04, size = 638, normalized size = 1.39
method | result | size |
default | \(\frac {\left (21 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right ) b^{2} x^{4}+42 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) b^{2} x^{4}+42 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) b^{2} x^{4}+42 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right ) a b \,x^{2}+84 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) a b \,x^{2}+84 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) a b \,x^{2}+21 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right ) a^{2}+42 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) a^{2}+42 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) a^{2}+56 \sqrt {d x}\, a b \,x^{2}+88 \sqrt {d x}\, a^{2}\right ) \left (b \,x^{2}+a \right )}{128 d \,a^{3} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}\) | \(638\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 298, normalized size = 0.65 \begin {gather*} \frac {84 \, {\left (a^{2} b^{2} d x^{4} + 2 \, a^{3} b d x^{2} + a^{4} d\right )} \left (-\frac {1}{a^{11} b d^{2}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{6} d^{2} \sqrt {-\frac {1}{a^{11} b d^{2}}} + d x} a^{8} b d \left (-\frac {1}{a^{11} b d^{2}}\right )^{\frac {3}{4}} - \sqrt {d x} a^{8} b d \left (-\frac {1}{a^{11} b d^{2}}\right )^{\frac {3}{4}}\right ) + 21 \, {\left (a^{2} b^{2} d x^{4} + 2 \, a^{3} b d x^{2} + a^{4} d\right )} \left (-\frac {1}{a^{11} b d^{2}}\right )^{\frac {1}{4}} \log \left (a^{3} d \left (-\frac {1}{a^{11} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 21 \, {\left (a^{2} b^{2} d x^{4} + 2 \, a^{3} b d x^{2} + a^{4} d\right )} \left (-\frac {1}{a^{11} b d^{2}}\right )^{\frac {1}{4}} \log \left (-a^{3} d \left (-\frac {1}{a^{11} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, {\left (7 \, b x^{2} + 11 \, a\right )} \sqrt {d x}}{64 \, {\left (a^{2} b^{2} d x^{4} + 2 \, a^{3} b d x^{2} + a^{4} d\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {d x} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.82, size = 339, normalized size = 0.74 \begin {gather*} \frac {7 \, \sqrt {d x} b d^{3} x^{2} + 11 \, \sqrt {d x} a d^{3}}{16 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {21 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} b d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {21 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} b d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {21 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{128 \, a^{3} b d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {21 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{128 \, a^{3} b d \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {d\,x}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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