Optimal. Leaf size=158 \[ -\frac {(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 (2 c d-b e) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {742, 736, 632,
212} \begin {gather*} -\frac {6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac {(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 (d+e x) (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 736
Rule 742
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(3 (2 c d-b e)) \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac {(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 (2 c d-b e) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 (2 c d-b e) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac {(b+2 c x) (d+e x)^3}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 (2 c d-b e) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 308, normalized size = 1.95 \begin {gather*} \frac {1}{2} \left (\frac {3 b^3 c d e^2-b^4 e^3+b^2 c e \left (-9 c d^2+5 a e^2+6 c d e x\right )+4 c^2 \left (-4 a^2 e^3+3 c^2 d^3 x+3 a c d e^2 x\right )+6 b c^2 \left (c d^2 (d-3 e x)+a e^2 (d-e x)\right )}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {-b^3 e^3 x+b^2 e^2 (-a e+3 c d x)+2 c \left (a^2 e^3+c^2 d^3 x-3 a c d e (d+e x)\right )+b c \left (c d^2 (d-3 e x)+3 a e^2 (d+e x)\right )}{c^2 \left (-b^2+4 a c\right ) (a+x (b+c x))^2}+\frac {12 (2 c d-b e) \left (c d^2+e (-b d+a e)\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}\right ) \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. \(484\) vs.
\(2(150)=300\).
time = 0.79, size = 485, normalized size = 3.07
method | result | size |
default | \(\frac {-\frac {3 c \left (a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) x^{3}}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}-\frac {\left (16 a^{2} c^{2} e^{3}+a \,b^{2} c \,e^{3}-18 a b \,c^{2} d \,e^{2}+b^{4} e^{3}-9 b^{3} c d \,e^{2}+27 b^{2} c^{2} d^{2} e -18 c^{3} b \,d^{3}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) c}-\frac {\left (5 a^{2} b c \,e^{3}+6 a^{2} c^{2} d \,e^{2}+a \,b^{3} e^{3}-15 a \,b^{2} c d \,e^{2}+15 a b \,c^{2} d^{2} e -10 a \,c^{3} d^{3}+3 b^{3} c \,d^{2} e -2 b^{2} c^{2} d^{3}\right ) x}{c \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {8 a^{3} c \,e^{3}+a^{2} b^{2} e^{3}-18 a^{2} b c d \,e^{2}+24 a^{2} c^{2} d^{2} e +3 a \,b^{2} c \,d^{2} e -10 c^{2} d^{3} a b +b^{3} c \,d^{3}}{2 c \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {6 \left (a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) \sqrt {4 a c -b^{2}}}\) | \(485\) |
risch | \(\frac {-\frac {3 c \left (a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) x^{3}}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}-\frac {\left (16 a^{2} c^{2} e^{3}+a \,b^{2} c \,e^{3}-18 a b \,c^{2} d \,e^{2}+b^{4} e^{3}-9 b^{3} c d \,e^{2}+27 b^{2} c^{2} d^{2} e -18 c^{3} b \,d^{3}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) c}-\frac {\left (5 a^{2} b c \,e^{3}+6 a^{2} c^{2} d \,e^{2}+a \,b^{3} e^{3}-15 a \,b^{2} c d \,e^{2}+15 a b \,c^{2} d^{2} e -10 a \,c^{3} d^{3}+3 b^{3} c \,d^{2} e -2 b^{2} c^{2} d^{3}\right ) x}{c \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {8 a^{3} c \,e^{3}+a^{2} b^{2} e^{3}-18 a^{2} b c d \,e^{2}+24 a^{2} c^{2} d^{2} e +3 a \,b^{2} c \,d^{2} e -10 c^{2} d^{3} a b +b^{3} c \,d^{3}}{2 c \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {3 \ln \left (\left (32 a^{2} c^{3}-16 b^{2} c^{2} a +2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) a b \,e^{3}}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {6 \ln \left (\left (32 a^{2} c^{3}-16 b^{2} c^{2} a +2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) a d \,e^{2} c}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 \ln \left (\left (32 a^{2} c^{3}-16 b^{2} c^{2} a +2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) b^{2} d \,e^{2}}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}-\frac {9 \ln \left (\left (32 a^{2} c^{3}-16 b^{2} c^{2} a +2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) d^{2} e b c}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {6 \ln \left (\left (32 a^{2} c^{3}-16 b^{2} c^{2} a +2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) c^{2} d^{3}}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 \ln \left (\left (-32 a^{2} c^{3}+16 b^{2} c^{2} a -2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) a b \,e^{3}}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}-\frac {6 \ln \left (\left (-32 a^{2} c^{3}+16 b^{2} c^{2} a -2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) a d \,e^{2} c}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}-\frac {3 \ln \left (\left (-32 a^{2} c^{3}+16 b^{2} c^{2} a -2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) b^{2} d \,e^{2}}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {9 \ln \left (\left (-32 a^{2} c^{3}+16 b^{2} c^{2} a -2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) d^{2} e b c}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}-\frac {6 \ln \left (\left (-32 a^{2} c^{3}+16 b^{2} c^{2} a -2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) c^{2} d^{3}}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\) | \(1170\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1024 vs.
\(2 (156) = 312\).
time = 3.35, size = 2068, normalized size = 13.09 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1180 vs.
\(2 (148) = 296\).
time = 5.43, size = 1180, normalized size = 7.47 \begin {gather*} 3 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {- 192 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) + 144 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) - 36 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) + 3 a b^{2} e^{3} - 6 a b c d e^{2} + 3 b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) - 3 b^{3} d e^{2} + 9 b^{2} c d^{2} e - 6 b c^{2} d^{3}}{6 a b c e^{3} - 12 a c^{2} d e^{2} - 6 b^{2} c d e^{2} + 18 b c^{2} d^{2} e - 12 c^{3} d^{3}} \right )} - 3 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (x + \frac {192 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) - 144 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) + 36 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) + 3 a b^{2} e^{3} - 6 a b c d e^{2} - 3 b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) - 3 b^{3} d e^{2} + 9 b^{2} c d^{2} e - 6 b c^{2} d^{3}}{6 a b c e^{3} - 12 a c^{2} d e^{2} - 6 b^{2} c d e^{2} + 18 b c^{2} d^{2} e - 12 c^{3} d^{3}} \right )} + \frac {- 8 a^{3} c e^{3} - a^{2} b^{2} e^{3} + 18 a^{2} b c d e^{2} - 24 a^{2} c^{2} d^{2} e - 3 a b^{2} c d^{2} e + 10 a b c^{2} d^{3} - b^{3} c d^{3} + x^{3} \left (- 6 a b c^{2} e^{3} + 12 a c^{3} d e^{2} + 6 b^{2} c^{2} d e^{2} - 18 b c^{3} d^{2} e + 12 c^{4} d^{3}\right ) + x^{2} \left (- 16 a^{2} c^{2} e^{3} - a b^{2} c e^{3} + 18 a b c^{2} d e^{2} - b^{4} e^{3} + 9 b^{3} c d e^{2} - 27 b^{2} c^{2} d^{2} e + 18 b c^{3} d^{3}\right ) + x \left (- 10 a^{2} b c e^{3} - 12 a^{2} c^{2} d e^{2} - 2 a b^{3} e^{3} + 30 a b^{2} c d e^{2} - 30 a b c^{2} d^{2} e + 20 a c^{3} d^{3} - 6 b^{3} c d^{2} e + 4 b^{2} c^{2} d^{3}\right )}{32 a^{4} c^{3} - 16 a^{3} b^{2} c^{2} + 2 a^{2} b^{4} c + x^{4} \cdot \left (32 a^{2} c^{5} - 16 a b^{2} c^{4} + 2 b^{4} c^{3}\right ) + x^{3} \cdot \left (64 a^{2} b c^{4} - 32 a b^{3} c^{3} + 4 b^{5} c^{2}\right ) + x^{2} \cdot \left (64 a^{3} c^{4} - 12 a b^{4} c^{2} + 2 b^{6} c\right ) + x \left (64 a^{3} b c^{3} - 32 a^{2} b^{3} c^{2} + 4 a b^{5} c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 446 vs.
\(2 (156) = 312\).
time = 0.84, size = 446, normalized size = 2.82 \begin {gather*} \frac {6 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, c^{4} d^{3} x^{3} - 18 \, b c^{3} d^{2} x^{3} e + 18 \, b c^{3} d^{3} x^{2} + 6 \, b^{2} c^{2} d x^{3} e^{2} + 12 \, a c^{3} d x^{3} e^{2} - 27 \, b^{2} c^{2} d^{2} x^{2} e + 4 \, b^{2} c^{2} d^{3} x + 20 \, a c^{3} d^{3} x - 6 \, a b c^{2} x^{3} e^{3} + 9 \, b^{3} c d x^{2} e^{2} + 18 \, a b c^{2} d x^{2} e^{2} - 6 \, b^{3} c d^{2} x e - 30 \, a b c^{2} d^{2} x e - b^{3} c d^{3} + 10 \, a b c^{2} d^{3} - b^{4} x^{2} e^{3} - a b^{2} c x^{2} e^{3} - 16 \, a^{2} c^{2} x^{2} e^{3} + 30 \, a b^{2} c d x e^{2} - 12 \, a^{2} c^{2} d x e^{2} - 3 \, a b^{2} c d^{2} e - 24 \, a^{2} c^{2} d^{2} e - 2 \, a b^{3} x e^{3} - 10 \, a^{2} b c x e^{3} + 18 \, a^{2} b c d e^{2} - a^{2} b^{2} e^{3} - 8 \, a^{3} c e^{3}}{2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.36, size = 636, normalized size = 4.03 \begin {gather*} \frac {6\,\mathrm {atan}\left (\frac {\left (\frac {3\,\left (b\,e-2\,c\,d\right )\,\left (16\,a^2\,b\,c^2-8\,a\,b^3\,c+b^5\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {6\,c\,x\,\left (b\,e-2\,c\,d\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{3\,b^2\,d\,e^2-9\,b\,c\,d^2\,e-3\,a\,b\,e^3+6\,c^2\,d^3+6\,a\,c\,d\,e^2}\right )\,\left (b\,e-2\,c\,d\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {8\,a^3\,c\,e^3+a^2\,b^2\,e^3-18\,a^2\,b\,c\,d\,e^2+24\,a^2\,c^2\,d^2\,e+3\,a\,b^2\,c\,d^2\,e-10\,a\,b\,c^2\,d^3+b^3\,c\,d^3}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (16\,a^2\,c^2\,e^3+a\,b^2\,c\,e^3-18\,a\,b\,c^2\,d\,e^2+b^4\,e^3-9\,b^3\,c\,d\,e^2+27\,b^2\,c^2\,d^2\,e-18\,b\,c^3\,d^3\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {3\,c\,x^3\,\left (b^2\,d\,e^2-3\,b\,c\,d^2\,e-a\,b\,e^3+2\,c^2\,d^3+2\,a\,c\,d\,e^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {x\,\left (5\,a^2\,b\,c\,e^3+6\,a^2\,c^2\,d\,e^2+a\,b^3\,e^3-15\,a\,b^2\,c\,d\,e^2+15\,a\,b\,c^2\,d^2\,e-10\,a\,c^3\,d^3+3\,b^3\,c\,d^2\,e-2\,b^2\,c^2\,d^3\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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