Optimal. Leaf size=121 \[ -\frac {\left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}{64 c^4 d}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac {(b d+2 c d x)^{13/2}}{832 c^4 d^7} \]
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Rubi [A]
time = 0.03, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {697}
\begin {gather*} -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac {\left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}{64 c^4 d}+\frac {(b d+2 c d x)^{13/2}}{832 c^4 d^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 \sqrt {b d+2 c d x}}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^{3/2}}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{7/2}}{64 c^3 d^4}+\frac {(b d+2 c d x)^{11/2}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}{64 c^4 d}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac {(b d+2 c d x)^{13/2}}{832 c^4 d^7}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 133, normalized size = 1.10 \begin {gather*} \frac {\sqrt {d (b+2 c x)} \left (-195 b^6+2340 a b^4 c-9360 a^2 b^2 c^2+12480 a^3 c^3+117 b^4 (b+2 c x)^2-936 a b^2 c (b+2 c x)^2+1872 a^2 c^2 (b+2 c x)^2-65 b^2 (b+2 c x)^4+260 a c (b+2 c x)^4+15 (b+2 c x)^6\right )}{12480 c^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 147, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {\frac {\left (2 c d x +b d \right )^{\frac {13}{2}}}{13}+\frac {\left (12 a c \,d^{2}-3 b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {9}{2}}}{9}+\frac {\left (\left (4 a c \,d^{2}-b^{2} d^{2}\right ) \left (8 a c \,d^{2}-2 b^{2} d^{2}\right )+\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{3} \sqrt {2 c d x +b d}}{64 d^{7} c^{4}}\) | \(147\) |
default | \(\frac {\frac {\left (2 c d x +b d \right )^{\frac {13}{2}}}{13}+\frac {\left (12 a c \,d^{2}-3 b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {9}{2}}}{9}+\frac {\left (\left (4 a c \,d^{2}-b^{2} d^{2}\right ) \left (8 a c \,d^{2}-2 b^{2} d^{2}\right )+\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{3} \sqrt {2 c d x +b d}}{64 d^{7} c^{4}}\) | \(147\) |
trager | \(\frac {\left (15 c^{6} x^{6}+45 b \,c^{5} x^{5}+65 a \,c^{5} x^{4}+40 b^{2} c^{4} x^{4}+130 a b \,c^{4} x^{3}+5 b^{3} x^{3} c^{3}+117 a^{2} c^{4} x^{2}+39 a \,b^{2} c^{3} x^{2}-3 b^{4} c^{2} x^{2}+117 a^{2} b \,c^{3} x -26 a \,b^{3} c^{2} x +2 b^{5} c x +195 a^{3} c^{3}-117 a^{2} b^{2} c^{2}+26 a \,b^{4} c -2 b^{6}\right ) \sqrt {2 c d x +b d}}{195 d \,c^{4}}\) | \(171\) |
gosper | \(\frac {\left (2 c x +b \right ) \left (15 c^{6} x^{6}+45 b \,c^{5} x^{5}+65 a \,c^{5} x^{4}+40 b^{2} c^{4} x^{4}+130 a b \,c^{4} x^{3}+5 b^{3} x^{3} c^{3}+117 a^{2} c^{4} x^{2}+39 a \,b^{2} c^{3} x^{2}-3 b^{4} c^{2} x^{2}+117 a^{2} b \,c^{3} x -26 a \,b^{3} c^{2} x +2 b^{5} c x +195 a^{3} c^{3}-117 a^{2} b^{2} c^{2}+26 a \,b^{4} c -2 b^{6}\right )}{195 c^{4} \sqrt {2 c d x +b d}}\) | \(174\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 778 vs.
\(2 (105) = 210\).
time = 0.29, size = 778, normalized size = 6.43 \begin {gather*} \frac {960960 \, \sqrt {2 \, c d x + b d} a^{3} - 48048 \, a^{2} {\left (\frac {10 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} b}{c d} - \frac {15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}}{c d^{2}}\right )} + 572 \, a {\left (\frac {84 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} b^{2}}{c^{2} d^{2}} - \frac {36 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b}{c^{2} d^{3}} + \frac {315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{c^{2} d^{4}}\right )} - \frac {3432 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b^{3}}{c^{3} d^{3}} + \frac {572 \, {\left (315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}\right )} b^{2}}{c^{3} d^{4}} - \frac {130 \, {\left (693 \, \sqrt {2 \, c d x + b d} b^{5} d^{5} - 1155 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{4} d^{4} + 1386 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{3} d^{3} - 990 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{2} d^{2} + 385 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b d - 63 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}}\right )} b}{c^{3} d^{5}} + \frac {5 \, {\left (3003 \, \sqrt {2 \, c d x + b d} b^{6} d^{6} - 6006 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{5} d^{5} + 9009 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{4} d^{4} - 8580 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{3} d^{3} + 5005 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b^{2} d^{2} - 1638 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} b d + 231 \, {\left (2 \, c d x + b d\right )}^{\frac {13}{2}}\right )}}{c^{3} d^{6}}}{960960 \, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.66, size = 165, normalized size = 1.36 \begin {gather*} \frac {{\left (15 \, c^{6} x^{6} + 45 \, b c^{5} x^{5} - 2 \, b^{6} + 26 \, a b^{4} c - 117 \, a^{2} b^{2} c^{2} + 195 \, a^{3} c^{3} + 5 \, {\left (8 \, b^{2} c^{4} + 13 \, a c^{5}\right )} x^{4} + 5 \, {\left (b^{3} c^{3} + 26 \, a b c^{4}\right )} x^{3} - 3 \, {\left (b^{4} c^{2} - 13 \, a b^{2} c^{3} - 39 \, a^{2} c^{4}\right )} x^{2} + {\left (2 \, b^{5} c - 26 \, a b^{3} c^{2} + 117 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{195 \, c^{4} d} \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. 1363 vs.
\(2 (116) = 232\).
time = 65.78, size = 1363, normalized size = 11.26 \begin {gather*} \begin {cases} \frac {- \frac {a^{3} b}{\sqrt {b d + 2 c d x}} - \frac {a^{3} \left (- \frac {b d}{\sqrt {b d + 2 c d x}} - \sqrt {b d + 2 c d x}\right )}{d} - \frac {3 a^{2} b^{2} \left (- \frac {b d}{\sqrt {b d + 2 c d x}} - \sqrt {b d + 2 c d x}\right )}{2 c d} - \frac {9 a^{2} b \left (\frac {b^{2} d^{2}}{\sqrt {b d + 2 c d x}} + 2 b d \sqrt {b d + 2 c d x} - \frac {\left (b d + 2 c d x\right )^{\frac {3}{2}}}{3}\right )}{4 c d^{2}} - \frac {3 a^{2} \left (- \frac {b^{3} d^{3}}{\sqrt {b d + 2 c d x}} - 3 b^{2} d^{2} \sqrt {b d + 2 c d x} + b d \left (b d + 2 c d x\right )^{\frac {3}{2}} - \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}}}{5}\right )}{4 c d^{3}} - \frac {3 a b^{3} \left (\frac {b^{2} d^{2}}{\sqrt {b d + 2 c d x}} + 2 b d \sqrt {b d + 2 c d x} - \frac {\left (b d + 2 c d x\right )^{\frac {3}{2}}}{3}\right )}{4 c^{2} d^{2}} - \frac {3 a b^{2} \left (- \frac {b^{3} d^{3}}{\sqrt {b d + 2 c d x}} - 3 b^{2} d^{2} \sqrt {b d + 2 c d x} + b d \left (b d + 2 c d x\right )^{\frac {3}{2}} - \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}}}{5}\right )}{2 c^{2} d^{3}} - \frac {15 a b \left (\frac {b^{4} d^{4}}{\sqrt {b d + 2 c d x}} + 4 b^{3} d^{3} \sqrt {b d + 2 c d x} - 2 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {3}{2}} + \frac {4 b d \left (b d + 2 c d x\right )^{\frac {5}{2}}}{5} - \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}}}{7}\right )}{16 c^{2} d^{4}} - \frac {3 a \left (- \frac {b^{5} d^{5}}{\sqrt {b d + 2 c d x}} - 5 b^{4} d^{4} \sqrt {b d + 2 c d x} + \frac {10 b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} - 2 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {5}{2}} + \frac {5 b d \left (b d + 2 c d x\right )^{\frac {7}{2}}}{7} - \frac {\left (b d + 2 c d x\right )^{\frac {9}{2}}}{9}\right )}{16 c^{2} d^{5}} - \frac {b^{4} \left (- \frac {b^{3} d^{3}}{\sqrt {b d + 2 c d x}} - 3 b^{2} d^{2} \sqrt {b d + 2 c d x} + b d \left (b d + 2 c d x\right )^{\frac {3}{2}} - \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}}}{5}\right )}{8 c^{3} d^{3}} - \frac {5 b^{3} \left (\frac {b^{4} d^{4}}{\sqrt {b d + 2 c d x}} + 4 b^{3} d^{3} \sqrt {b d + 2 c d x} - 2 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {3}{2}} + \frac {4 b d \left (b d + 2 c d x\right )^{\frac {5}{2}}}{5} - \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}}}{7}\right )}{16 c^{3} d^{4}} - \frac {9 b^{2} \left (- \frac {b^{5} d^{5}}{\sqrt {b d + 2 c d x}} - 5 b^{4} d^{4} \sqrt {b d + 2 c d x} + \frac {10 b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} - 2 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {5}{2}} + \frac {5 b d \left (b d + 2 c d x\right )^{\frac {7}{2}}}{7} - \frac {\left (b d + 2 c d x\right )^{\frac {9}{2}}}{9}\right )}{32 c^{3} d^{5}} - \frac {7 b \left (\frac {b^{6} d^{6}}{\sqrt {b d + 2 c d x}} + 6 b^{5} d^{5} \sqrt {b d + 2 c d x} - 5 b^{4} d^{4} \left (b d + 2 c d x\right )^{\frac {3}{2}} + 4 b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac {5}{2}} - \frac {15 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {7}{2}}}{7} + \frac {2 b d \left (b d + 2 c d x\right )^{\frac {9}{2}}}{3} - \frac {\left (b d + 2 c d x\right )^{\frac {11}{2}}}{11}\right )}{64 c^{3} d^{6}} - \frac {- \frac {b^{7} d^{7}}{\sqrt {b d + 2 c d x}} - 7 b^{6} d^{6} \sqrt {b d + 2 c d x} + 7 b^{5} d^{5} \left (b d + 2 c d x\right )^{\frac {3}{2}} - 7 b^{4} d^{4} \left (b d + 2 c d x\right )^{\frac {5}{2}} + 5 b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac {7}{2}} - \frac {7 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {9}{2}}}{3} + \frac {7 b d \left (b d + 2 c d x\right )^{\frac {11}{2}}}{11} - \frac {\left (b d + 2 c d x\right )^{\frac {13}{2}}}{13}}{64 c^{3} d^{7}}}{c} & \text {for}\: c \neq 0 \\\frac {\begin {cases} a^{3} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{4}}{4 b} & \text {otherwise} \end {cases}}{\sqrt {b d}} & \text {otherwise} \end {cases} \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. 778 vs.
\(2 (105) = 210\).
time = 1.42, size = 778, normalized size = 6.43 \begin {gather*} \frac {960960 \, \sqrt {2 \, c d x + b d} a^{3} - \frac {480480 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} a^{2} b}{c d} + \frac {48048 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} a b^{2}}{c^{2} d^{2}} + \frac {48048 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} a^{2}}{c d^{2}} - \frac {3432 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b^{3}}{c^{3} d^{3}} - \frac {20592 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} a b}{c^{2} d^{3}} + \frac {572 \, {\left (315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}\right )} b^{2}}{c^{3} d^{4}} + \frac {572 \, {\left (315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}\right )} a}{c^{2} d^{4}} - \frac {130 \, {\left (693 \, \sqrt {2 \, c d x + b d} b^{5} d^{5} - 1155 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{4} d^{4} + 1386 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{3} d^{3} - 990 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{2} d^{2} + 385 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b d - 63 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}}\right )} b}{c^{3} d^{5}} + \frac {5 \, {\left (3003 \, \sqrt {2 \, c d x + b d} b^{6} d^{6} - 6006 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{5} d^{5} + 9009 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{4} d^{4} - 8580 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{3} d^{3} + 5005 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b^{2} d^{2} - 1638 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} b d + 231 \, {\left (2 \, c d x + b d\right )}^{\frac {13}{2}}\right )}}{c^{3} d^{6}}}{960960 \, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 111, normalized size = 0.92 \begin {gather*} \frac {{\left (b\,d+2\,c\,d\,x\right )}^{13/2}}{832\,c^4\,d^7}+\frac {{\left (b\,d+2\,c\,d\,x\right )}^{9/2}\,\left (4\,a\,c-b^2\right )}{192\,c^4\,d^5}+\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (4\,a\,c-b^2\right )}^3}{64\,c^4\,d}+\frac {3\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,{\left (4\,a\,c-b^2\right )}^2}{320\,c^4\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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