Optimal. Leaf size=448 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{7 e^7 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5 (B d-A e)}{e^7 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-5 a B e-A b e+6 b B d)}{11 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{9 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)} \]
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Rubi [A] time = 0.22, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} -\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-5 a B e-A b e+6 b B d)}{11 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{9 e^7 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{7 e^7 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5 (B d-A e)}{e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{\sqrt {d+e x}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 \sqrt {d+e x}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) \sqrt {d+e x}}{e^6}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e) (d+e x)^{3/2}}{e^6}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{5/2}}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{7/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{9/2}}{e^6}+\frac {b^{10} B (d+e x)^{11/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^5 (B d-A e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {2 (b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {2 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}+\frac {10 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 239, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \sqrt {d+e x} \left (-819 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+5005 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-12870 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+9009 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-3003 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+9009 (b d-a e)^5 (B d-A e)+693 b^5 B (d+e x)^6\right )}{9009 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 53.12, size = 812, normalized size = 1.81 \begin {gather*} \frac {2 \sqrt {d+e x} \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (9009 b^5 B d^6-9009 A b^5 e d^5-45045 a b^4 B e d^5-18018 b^5 B (d+e x) d^5+45045 a A b^4 e^2 d^4+90090 a^2 b^3 B e^2 d^4+27027 b^5 B (d+e x)^2 d^4+15015 A b^5 e (d+e x) d^4+75075 a b^4 B e (d+e x) d^4-90090 a^2 A b^3 e^3 d^3-90090 a^3 b^2 B e^3 d^3-25740 b^5 B (d+e x)^3 d^3-18018 A b^5 e (d+e x)^2 d^3-90090 a b^4 B e (d+e x)^2 d^3-60060 a A b^4 e^2 (d+e x) d^3-120120 a^2 b^3 B e^2 (d+e x) d^3+90090 a^3 A b^2 e^4 d^2+45045 a^4 b B e^4 d^2+15015 b^5 B (d+e x)^4 d^2+12870 A b^5 e (d+e x)^3 d^2+64350 a b^4 B e (d+e x)^3 d^2+54054 a A b^4 e^2 (d+e x)^2 d^2+108108 a^2 b^3 B e^2 (d+e x)^2 d^2+90090 a^2 A b^3 e^3 (d+e x) d^2+90090 a^3 b^2 B e^3 (d+e x) d^2-45045 a^4 A b e^5 d-9009 a^5 B e^5 d-4914 b^5 B (d+e x)^5 d-5005 A b^5 e (d+e x)^4 d-25025 a b^4 B e (d+e x)^4 d-25740 a A b^4 e^2 (d+e x)^3 d-51480 a^2 b^3 B e^2 (d+e x)^3 d-54054 a^2 A b^3 e^3 (d+e x)^2 d-54054 a^3 b^2 B e^3 (d+e x)^2 d-60060 a^3 A b^2 e^4 (d+e x) d-30030 a^4 b B e^4 (d+e x) d+9009 a^5 A e^6+693 b^5 B (d+e x)^6+819 A b^5 e (d+e x)^5+4095 a b^4 B e (d+e x)^5+5005 a A b^4 e^2 (d+e x)^4+10010 a^2 b^3 B e^2 (d+e x)^4+12870 a^2 A b^3 e^3 (d+e x)^3+12870 a^3 b^2 B e^3 (d+e x)^3+18018 a^3 A b^2 e^4 (d+e x)^2+9009 a^4 b B e^4 (d+e x)^2+15015 a^4 A b e^5 (d+e x)+3003 a^5 B e^5 (d+e x)\right )}{9009 e^6 (a e+b x e)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 560, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (693 \, B b^{5} e^{6} x^{6} + 3072 \, B b^{5} d^{6} + 9009 \, A a^{5} e^{6} - 3328 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 18304 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 41184 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 24024 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 6006 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 63 \, {\left (12 \, B b^{5} d e^{5} - 13 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 35 \, {\left (24 \, B b^{5} d^{2} e^{4} - 26 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 143 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 10 \, {\left (96 \, B b^{5} d^{3} e^{3} - 104 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 572 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 1287 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 3 \, {\left (384 \, B b^{5} d^{4} e^{2} - 416 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 2288 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 5148 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 3003 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - {\left (1536 \, B b^{5} d^{5} e - 1664 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 9152 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 20592 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 12012 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 758, normalized size = 1.69
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 689, normalized size = 1.54 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (693 B \,b^{5} e^{6} x^{6}+819 A \,b^{5} e^{6} x^{5}+4095 B a \,b^{4} e^{6} x^{5}-756 B \,b^{5} d \,e^{5} x^{5}+5005 A a \,b^{4} e^{6} x^{4}-910 A \,b^{5} d \,e^{5} x^{4}+10010 B \,a^{2} b^{3} e^{6} x^{4}-4550 B a \,b^{4} d \,e^{5} x^{4}+840 B \,b^{5} d^{2} e^{4} x^{4}+12870 A \,a^{2} b^{3} e^{6} x^{3}-5720 A a \,b^{4} d \,e^{5} x^{3}+1040 A \,b^{5} d^{2} e^{4} x^{3}+12870 B \,a^{3} b^{2} e^{6} x^{3}-11440 B \,a^{2} b^{3} d \,e^{5} x^{3}+5200 B a \,b^{4} d^{2} e^{4} x^{3}-960 B \,b^{5} d^{3} e^{3} x^{3}+18018 A \,a^{3} b^{2} e^{6} x^{2}-15444 A \,a^{2} b^{3} d \,e^{5} x^{2}+6864 A a \,b^{4} d^{2} e^{4} x^{2}-1248 A \,b^{5} d^{3} e^{3} x^{2}+9009 B \,a^{4} b \,e^{6} x^{2}-15444 B \,a^{3} b^{2} d \,e^{5} x^{2}+13728 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-6240 B a \,b^{4} d^{3} e^{3} x^{2}+1152 B \,b^{5} d^{4} e^{2} x^{2}+15015 A \,a^{4} b \,e^{6} x -24024 A \,a^{3} b^{2} d \,e^{5} x +20592 A \,a^{2} b^{3} d^{2} e^{4} x -9152 A a \,b^{4} d^{3} e^{3} x +1664 A \,b^{5} d^{4} e^{2} x +3003 B \,a^{5} e^{6} x -12012 B \,a^{4} b d \,e^{5} x +20592 B \,a^{3} b^{2} d^{2} e^{4} x -18304 B \,a^{2} b^{3} d^{3} e^{3} x +8320 B a \,b^{4} d^{4} e^{2} x -1536 B \,b^{5} d^{5} e x +9009 A \,a^{5} e^{6}-30030 A \,a^{4} b d \,e^{5}+48048 A \,a^{3} b^{2} d^{2} e^{4}-41184 A \,a^{2} b^{3} d^{3} e^{3}+18304 A a \,b^{4} d^{4} e^{2}-3328 A \,b^{5} d^{5} e -6006 B \,a^{5} d \,e^{5}+24024 B \,a^{4} b \,d^{2} e^{4}-41184 B \,a^{3} b^{2} d^{3} e^{3}+36608 B \,a^{2} b^{3} d^{4} e^{2}-16640 B a \,b^{4} d^{5} e +3072 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{9009 \left (b x +a \right )^{5} e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.74, size = 758, normalized size = 1.69 \begin {gather*} \frac {2 \, {\left (63 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1408 \, a b^{4} d^{5} e - 3168 \, a^{2} b^{3} d^{4} e^{2} + 3696 \, a^{3} b^{2} d^{3} e^{3} - 2310 \, a^{4} b d^{2} e^{4} + 693 \, a^{5} d e^{5} - 7 \, {\left (b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 5 \, {\left (2 \, b^{5} d^{2} e^{4} - 11 \, a b^{4} d e^{5} + 198 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (8 \, b^{5} d^{3} e^{3} - 44 \, a b^{4} d^{2} e^{4} + 99 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} + {\left (32 \, b^{5} d^{4} e^{2} - 176 \, a b^{4} d^{3} e^{3} + 396 \, a^{2} b^{3} d^{2} e^{4} - 462 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} - {\left (128 \, b^{5} d^{5} e - 704 \, a b^{4} d^{4} e^{2} + 1584 \, a^{2} b^{3} d^{3} e^{3} - 1848 \, a^{3} b^{2} d^{2} e^{4} + 1155 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} A}{693 \, \sqrt {e x + d} e^{6}} + \frac {2 \, {\left (693 \, b^{5} e^{7} x^{7} + 3072 \, b^{5} d^{7} - 16640 \, a b^{4} d^{6} e + 36608 \, a^{2} b^{3} d^{5} e^{2} - 41184 \, a^{3} b^{2} d^{4} e^{3} + 24024 \, a^{4} b d^{3} e^{4} - 6006 \, a^{5} d^{2} e^{5} - 63 \, {\left (b^{5} d e^{6} - 65 \, a b^{4} e^{7}\right )} x^{6} + 7 \, {\left (12 \, b^{5} d^{2} e^{5} - 65 \, a b^{4} d e^{6} + 1430 \, a^{2} b^{3} e^{7}\right )} x^{5} - 10 \, {\left (12 \, b^{5} d^{3} e^{4} - 65 \, a b^{4} d^{2} e^{5} + 143 \, a^{2} b^{3} d e^{6} - 1287 \, a^{3} b^{2} e^{7}\right )} x^{4} + {\left (192 \, b^{5} d^{4} e^{3} - 1040 \, a b^{4} d^{3} e^{4} + 2288 \, a^{2} b^{3} d^{2} e^{5} - 2574 \, a^{3} b^{2} d e^{6} + 9009 \, a^{4} b e^{7}\right )} x^{3} - {\left (384 \, b^{5} d^{5} e^{2} - 2080 \, a b^{4} d^{4} e^{3} + 4576 \, a^{2} b^{3} d^{3} e^{4} - 5148 \, a^{3} b^{2} d^{2} e^{5} + 3003 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (1536 \, b^{5} d^{6} e - 8320 \, a b^{4} d^{5} e^{2} + 18304 \, a^{2} b^{3} d^{4} e^{3} - 20592 \, a^{3} b^{2} d^{3} e^{4} + 12012 \, a^{4} b d^{2} e^{5} - 3003 \, a^{5} d e^{6}\right )} x\right )} B}{9009 \, \sqrt {e x + d} e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.26, size = 826, normalized size = 1.84 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,B\,b^4\,x^7}{13}+\frac {-12012\,B\,a^5\,d^2\,e^5+18018\,A\,a^5\,d\,e^6+48048\,B\,a^4\,b\,d^3\,e^4-60060\,A\,a^4\,b\,d^2\,e^5-82368\,B\,a^3\,b^2\,d^4\,e^3+96096\,A\,a^3\,b^2\,d^3\,e^4+73216\,B\,a^2\,b^3\,d^5\,e^2-82368\,A\,a^2\,b^3\,d^4\,e^3-33280\,B\,a\,b^4\,d^6\,e+36608\,A\,a\,b^4\,d^5\,e^2+6144\,B\,b^5\,d^7-6656\,A\,b^5\,d^6\,e}{9009\,b\,e^7}+\frac {x^3\,\left (18018\,B\,a^4\,b\,e^7-5148\,B\,a^3\,b^2\,d\,e^6+36036\,A\,a^3\,b^2\,e^7+4576\,B\,a^2\,b^3\,d^2\,e^5-5148\,A\,a^2\,b^3\,d\,e^6-2080\,B\,a\,b^4\,d^3\,e^4+2288\,A\,a\,b^4\,d^2\,e^5+384\,B\,b^5\,d^4\,e^3-416\,A\,b^5\,d^3\,e^4\right )}{9009\,b\,e^7}+\frac {x^4\,\left (25740\,B\,a^3\,b^2\,e^7-2860\,B\,a^2\,b^3\,d\,e^6+25740\,A\,a^2\,b^3\,e^7+1300\,B\,a\,b^4\,d^2\,e^5-1430\,A\,a\,b^4\,d\,e^6-240\,B\,b^5\,d^3\,e^4+260\,A\,b^5\,d^2\,e^5\right )}{9009\,b\,e^7}+\frac {2\,b^3\,x^6\,\left (13\,A\,b\,e+65\,B\,a\,e-B\,b\,d\right )}{143\,e}+\frac {x\,\left (-6006\,B\,a^5\,d\,e^6+18018\,A\,a^5\,e^7+24024\,B\,a^4\,b\,d^2\,e^5-30030\,A\,a^4\,b\,d\,e^6-41184\,B\,a^3\,b^2\,d^3\,e^4+48048\,A\,a^3\,b^2\,d^2\,e^5+36608\,B\,a^2\,b^3\,d^4\,e^3-41184\,A\,a^2\,b^3\,d^3\,e^4-16640\,B\,a\,b^4\,d^5\,e^2+18304\,A\,a\,b^4\,d^4\,e^3+3072\,B\,b^5\,d^6\,e-3328\,A\,b^5\,d^5\,e^2\right )}{9009\,b\,e^7}+\frac {x^2\,\left (6006\,B\,a^5\,e^7-6006\,B\,a^4\,b\,d\,e^6+30030\,A\,a^4\,b\,e^7+10296\,B\,a^3\,b^2\,d^2\,e^5-12012\,A\,a^3\,b^2\,d\,e^6-9152\,B\,a^2\,b^3\,d^3\,e^4+10296\,A\,a^2\,b^3\,d^2\,e^5+4160\,B\,a\,b^4\,d^4\,e^3-4576\,A\,a\,b^4\,d^3\,e^4-768\,B\,b^5\,d^5\,e^2+832\,A\,b^5\,d^4\,e^3\right )}{9009\,b\,e^7}+\frac {2\,b^2\,x^5\,\left (1430\,B\,a^2\,e^2-65\,B\,a\,b\,d\,e+715\,A\,a\,b\,e^2+12\,B\,b^2\,d^2-13\,A\,b^2\,d\,e\right )}{1287\,e^2}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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