Optimal. Leaf size=1081 \[ -\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (3 d (B c-A d) f h a^2+b \left (3 A d^2 (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left (3 B c d e g-A \left (-f h c^2+d (f g+e h) c+2 d^2 e g\right )\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {2 b \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {2 d \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}} \]
[Out]
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Rubi [A] time = 3.37, antiderivative size = 1080, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1599, 1602, 12, 170, 419, 176, 424} \[ -\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (A b-a B)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (3 d (B c-A d) f h a^2+b \left (3 A d^2 (f g+e h)-B \left (f h c^2+2 d (f g+e h) c+d^2 e g\right )\right ) a+b^2 \left (A f h c^2+3 B d e g c-A d (f g+e h) c-2 A d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac {2 b \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {2 d \left (3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 170
Rule 176
Rule 419
Rule 424
Rule 1599
Rule 1602
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=-\frac {2 b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac {\int \frac {-3 a^2 A d f h+b^2 (3 B c e g-2 A (d e g+c f g+c e h))-a b (B (d e g+c f g+c e h)-3 A (d f g+d e h+c f h))+(A b-a B) (3 a d f h-b (d f g+d e h+c f h)) x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac {2 b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {\int \frac {b (A b-a B) (b c e g-a (d e g+c f g+c e h)) (3 a d f h-b (d f g+d e h+c f h))+a (a d f h-b (d f g+d e h+c f h)) \left (3 a^2 A d f h-b^2 (3 B c e g-2 A (d e g+c f g+c e h))+a b (B (d e g+c f g+c e h)-3 A (d f g+d e h+c f h))\right )+(a d f h+b (d f g+d e h+c f h)) \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) x+2 b d f h \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2}\\ &=\frac {2 d \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}+\frac {\int -\frac {2 b d f (b e-a f) h (b g-a h) \left (3 a^2 d (B c-A d) f h+b^2 \left (3 B c d e g-2 A d^2 e g+A c^2 f h-A c d (f g+e h)\right )+a b \left (3 A d^2 (f g+e h)-B \left (d^2 e g+c^2 f h+2 c d (f g+e h)\right )\right )\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{6 b d (b c-a d)^2 f (b e-a f)^2 h (b g-a h)^2}+\frac {\left ((d e-c f) (d g-c h) \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right )\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2}\\ &=\frac {2 d \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {\left (3 a^2 d (B c-A d) f h+b^2 \left (3 B c d e g-2 A d^2 e g+A c^2 f h-A c d (f g+e h)\right )+a b \left (3 A d^2 (f g+e h)-B \left (d^2 e g+c^2 f h+2 c d (f g+e h)\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f) (b g-a h)}-\frac {\left (2 (d g-c h) \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {(-b c+a d) x^2}{b e-a f}}}{\sqrt {1-\frac {(d g-c h) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {e+f x}}{\sqrt {c+d x}}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}\\ &=\frac {2 d \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {2 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {\left (2 \left (3 a^2 d (B c-A d) f h+b^2 \left (3 B c d e g-2 A d^2 e g+A c^2 f h-A c d (f g+e h)\right )+a b \left (3 A d^2 (f g+e h)-B \left (d^2 e g+c^2 f h+2 c d (f g+e h)\right )\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {(b c-a d) x^2}{d e-c f}} \sqrt {1-\frac {(b g-a h) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {e+f x}}{\sqrt {a+b x}}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h) (f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}\\ &=\frac {2 d \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b (A b-a B) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 b \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {2 \sqrt {d g-c h} \sqrt {f g-e h} \left (3 a^3 B d f h+b^3 (3 B c e g-2 A (d e g+c f g+c e h))-a b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h))-a^2 b (6 A d f h+B (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (3 a^2 d (B c-A d) f h+b^2 \left (3 B c d e g-2 A d^2 e g+A c^2 f h-A c d (f g+e h)\right )+a b \left (3 A d^2 (f g+e h)-B \left (d^2 e g+c^2 f h+2 c d (f g+e h)\right )\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}\\ \end {align*}
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Mathematica [B] time = 24.06, size = 10667, normalized size = 9.87 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 10.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{b^{3} d f h x^{6} + a^{3} c e g + {\left (b^{3} d f g + {\left (b^{3} d e + {\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} h\right )} x^{5} + {\left ({\left (b^{3} d e + {\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} g + {\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \, {\left (a b^{2} c + a^{2} b d\right )} f\right )} h\right )} x^{4} + {\left ({\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \, {\left (a b^{2} c + a^{2} b d\right )} f\right )} g + {\left (3 \, {\left (a b^{2} c + a^{2} b d\right )} e + {\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} h\right )} x^{3} + {\left ({\left (3 \, {\left (a b^{2} c + a^{2} b d\right )} e + {\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} g + {\left (a^{3} c f + {\left (3 \, a^{2} b c + a^{3} d\right )} e\right )} h\right )} x^{2} + {\left (a^{3} c e h + {\left (a^{3} c f + {\left (3 \, a^{2} b c + a^{3} d\right )} e\right )} g\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.78, size = 71656, normalized size = 66.29 \[ \text {output too large to display} \]
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 \[ \int \frac {B x + A}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]
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 \[ \int \frac {A+B\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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