Optimal. Leaf size=703 \[ \frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{192 c^2 e^4}+\frac {\sqrt {b x+c x^2} \left (3 \left (4 A c e \left (-3 b^4 e^4-10 b^3 c d e^3+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-5 b^5 e^5-12 b^4 c d e^4-40 b^3 c^2 d^2 e^3+704 b^2 c^3 d^3 e^2-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{1536 c^3 e^6}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e \left (-3 b^5 e^5-10 b^4 c d e^4-80 b^3 c^2 d^2 e^3+480 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{512 c^{7/2} e^7}-\frac {d^{5/2} (B d-A e) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e^7}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2} \]
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Rubi [A] time = 1.12, antiderivative size = 703, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {814, 843, 620, 206, 724} \begin {gather*} \frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (12 b^2 c d e^2+5 b^3 e^3-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{192 c^2 e^4}+\frac {\sqrt {b x+c x^2} \left (3 \left (4 A c e \left (176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{1536 c^3 e^6}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e \left (480 b^2 c^3 d^3 e^2-80 b^3 c^2 d^2 e^3-10 b^4 c d e^4-3 b^5 e^5-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (1920 b^2 c^4 d^4 e^2-320 b^3 c^3 d^3 e^3-40 b^4 c^2 d^2 e^4-12 b^5 c d e^5-5 b^6 e^6-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{512 c^{7/2} e^7}-\frac {d^{5/2} (B d-A e) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e^7}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 724
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{d+e x} \, dx &=-\frac {(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}-\frac {\int \frac {\left (-\frac {1}{2} b d (12 B c d-5 b B e-12 A c e)+\frac {1}{2} \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx}{12 c e^2}\\ &=\frac {\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac {(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}+\frac {\int \frac {\left (-\frac {3}{4} b d \left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )\right )-\frac {1}{4} \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt {b x+c x^2}}{d+e x} \, dx}{96 c^2 e^4}\\ &=\frac {\left (3 \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )-2 c e \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt {b x+c x^2}}{1536 c^3 e^6}+\frac {\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac {(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}-\frac {\int \frac {\frac {3}{8} b d \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )+\frac {3}{8} \left (4 A c e \left (256 c^5 d^5-640 b c^4 d^4 e+480 b^2 c^3 d^3 e^2-80 b^3 c^2 d^2 e^3-10 b^4 c d e^4-3 b^5 e^5\right )-B \left (1024 c^6 d^6-2560 b c^5 d^5 e+1920 b^2 c^4 d^4 e^2-320 b^3 c^3 d^3 e^3-40 b^4 c^2 d^2 e^4-12 b^5 c d e^5-5 b^6 e^6\right )\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{384 c^3 e^6}\\ &=\frac {\left (3 \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )-2 c e \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt {b x+c x^2}}{1536 c^3 e^6}+\frac {\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac {(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}-\frac {\left (d^3 (B d-A e) (c d-b e)^3\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{e^7}-\frac {\left (4 A c e \left (256 c^5 d^5-640 b c^4 d^4 e+480 b^2 c^3 d^3 e^2-80 b^3 c^2 d^2 e^3-10 b^4 c d e^4-3 b^5 e^5\right )-B \left (1024 c^6 d^6-2560 b c^5 d^5 e+1920 b^2 c^4 d^4 e^2-320 b^3 c^3 d^3 e^3-40 b^4 c^2 d^2 e^4-12 b^5 c d e^5-5 b^6 e^6\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{1024 c^3 e^7}\\ &=\frac {\left (3 \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )-2 c e \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt {b x+c x^2}}{1536 c^3 e^6}+\frac {\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac {(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}+\frac {\left (2 d^3 (B d-A e) (c d-b e)^3\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{e^7}-\frac {\left (4 A c e \left (256 c^5 d^5-640 b c^4 d^4 e+480 b^2 c^3 d^3 e^2-80 b^3 c^2 d^2 e^3-10 b^4 c d e^4-3 b^5 e^5\right )-B \left (1024 c^6 d^6-2560 b c^5 d^5 e+1920 b^2 c^4 d^4 e^2-320 b^3 c^3 d^3 e^3-40 b^4 c^2 d^2 e^4-12 b^5 c d e^5-5 b^6 e^6\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{512 c^3 e^7}\\ &=\frac {\left (3 \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )-2 c e \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt {b x+c x^2}}{1536 c^3 e^6}+\frac {\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac {(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}-\frac {\left (4 A c e \left (256 c^5 d^5-640 b c^4 d^4 e+480 b^2 c^3 d^3 e^2-80 b^3 c^2 d^2 e^3-10 b^4 c d e^4-3 b^5 e^5\right )-B \left (1024 c^6 d^6-2560 b c^5 d^5 e+1920 b^2 c^4 d^4 e^2-320 b^3 c^3 d^3 e^3-40 b^4 c^2 d^2 e^4-12 b^5 c d e^5-5 b^6 e^6\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{7/2} e^7}-\frac {d^{5/2} (B d-A e) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^7}\\ \end {align*}
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Mathematica [A] time = 2.47, size = 650, normalized size = 0.92 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (e \sqrt {x} \left (4 A c e \left (-45 b^4 e^4+30 b^3 c e^3 (e x-5 d)+4 b^2 c^2 e^2 \left (660 d^2-295 d e x+186 e^2 x^2\right )+16 b c^3 e \left (-270 d^3+130 d^2 e x-85 d e^2 x^2+63 e^3 x^3\right )+32 c^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+B \left (75 b^5 e^5+10 b^4 c e^4 (18 d-5 e x)+40 b^3 c^2 e^3 \left (15 d^2-3 d e x+e^2 x^2\right )+16 b^2 c^3 e^2 \left (-660 d^3+295 d^2 e x-186 d e^2 x^2+135 e^3 x^3\right )+64 b c^4 e \left (270 d^4-130 d^3 e x+85 d^2 e^2 x^2-63 d e^3 x^3+50 e^4 x^4\right )-128 c^5 \left (60 d^5-30 d^4 e x+20 d^3 e^2 x^2-15 d^2 e^3 x^3+12 d e^4 x^4-10 e^5 x^5\right )\right )\right )-\frac {15360 c^3 d^{5/2} (B d-A e) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {b+c x}}\right )+\frac {15 \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (4 A c e \left (3 b^5 e^5+10 b^4 c d e^4+80 b^3 c^2 d^2 e^3-480 b^2 c^3 d^3 e^2+640 b c^4 d^4 e-256 c^5 d^5\right )+B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{\sqrt {b} \sqrt {\frac {c x}{b}+1}}\right )}{7680 c^{7/2} e^7 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 11.42, size = 905, normalized size = 1.29 \begin {gather*} \frac {\sqrt {c x^2+b x} \left (-7680 B d^5 c^5+1280 B e^5 x^5 c^5+1536 A e^5 x^4 c^5-1536 B d e^4 x^4 c^5-1920 A d e^4 x^3 c^5+1920 B d^2 e^3 x^3 c^5+2560 A d^2 e^3 x^2 c^5-2560 B d^3 e^2 x^2 c^5+7680 A d^4 e c^5-3840 A d^3 e^2 x c^5+3840 B d^4 e x c^5+3200 b B e^5 x^4 c^4+4032 A b e^5 x^3 c^4-4032 b B d e^4 x^3 c^4-17280 A b d^3 e^2 c^4-5440 A b d e^4 x^2 c^4+5440 b B d^2 e^3 x^2 c^4+17280 b B d^4 e c^4+8320 A b d^2 e^3 x c^4-8320 b B d^3 e^2 x c^4+10560 A b^2 d^2 e^3 c^3+2160 b^2 B e^5 x^3 c^3-10560 b^2 B d^3 e^2 c^3+2976 A b^2 e^5 x^2 c^3-2976 b^2 B d e^4 x^2 c^3-4720 A b^2 d e^4 x c^3+4720 b^2 B d^2 e^3 x c^3-600 A b^3 d e^4 c^2+600 b^3 B d^2 e^3 c^2+40 b^3 B e^5 x^2 c^2+120 A b^3 e^5 x c^2-120 b^3 B d e^4 x c^2-180 A b^4 e^5 c+180 b^4 B d e^4 c-50 b^4 B e^5 x c+75 b^5 B e^5\right )}{7680 c^3 e^6}-\frac {2 \left (B c^2 \sqrt {c d-b e} d^{11/2}-A c^2 e \sqrt {c d-b e} d^{9/2}-2 b B c e \sqrt {c d-b e} d^{9/2}+b^2 B e^2 \sqrt {c d-b e} d^{7/2}+2 A b c e^2 \sqrt {c d-b e} d^{7/2}-A b^2 e^3 \sqrt {c d-b e} d^{5/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {c} d+\sqrt {c} e x-e \sqrt {c x^2+b x}}{\sqrt {d} \sqrt {c d-b e}}\right )}{e^7}+\frac {\left (-1024 B d^6 c^6+1024 A d^5 e c^6-2560 A b d^4 e^2 c^5+2560 b B d^5 e c^5+1920 A b^2 d^3 e^3 c^4-1920 b^2 B d^4 e^2 c^4-320 A b^3 d^2 e^4 c^3+320 b^3 B d^3 e^3 c^3-40 A b^4 d e^5 c^2+40 b^4 B d^2 e^4 c^2-12 A b^5 e^6 c+12 b^5 B d e^5 c+5 b^6 B e^6\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x}\right )}{1024 c^{7/2} e^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [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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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 4097, normalized size = 5.83 \begin {gather*} \text {output too large to display} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{d+e\,x} \,d x \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 {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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