Optimal. Leaf size=1128 \[ \frac {2 b d \left (9 a^3 C d f h-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 b^2 \left (9 a^3 C d f h-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {2 b \sqrt {d g-c h} \sqrt {f g-e h} \left (9 a^3 C d f h-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (3 a^3 C d^2 f h-b^3 \left (2 B d^2 e g-B c^2 f h-c d (3 C e g-B f g-B e h)\right )-3 a^2 b d (B d f h+C (d f g+d e h-c f h))+a b^2 \left (3 B d^2 (f g+e h)+C \left (d^2 e g-c d f g-c d e h-2 c^2 f h\right )\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]
[Out]
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Rubi [A]
time = 2.79, antiderivative size = 1119, normalized size of antiderivative = 0.99, number of steps
used = 9, number of rules used = 8, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {24, 1613, 1616,
12, 176, 430, 182, 435} \begin {gather*} -\frac {2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} b^2}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} b^2}{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 (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+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 (\text {ArcSin}\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 ) b}{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 (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B f g-B e h) c+2 B 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 (\text {ArcSin}\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 {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 12
Rule 24
Rule 176
Rule 182
Rule 430
Rule 435
Rule 1613
Rule 1616
Rubi steps
\begin {align*} \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\frac {\int \frac {b^2 (b B-a C)+b^3 C x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2}\\ &=-\frac {2 b^2 (b B-2 a C) \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 {b^2 \left (b^2 C (3 b c e g-a (d e g+c f g+c e h))-(b B-a C) \left (3 a^2 d f h+2 b^2 (d e g+c f g+c e h)-3 a b (d f g+d e h+c f h)\right )\right )+b^3 (b B-2 a C) (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^2 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac {2 b^2 (b B-2 a C) \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^2 \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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^2 \left (b^2 (b B-2 a C) (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 (b^2 C (3 b c e g-a (d e g+c f g+c e h))-(b B-a C) \left (3 a^2 d f h+2 b^2 (d e g+c f g+c e h)-3 a b (d f g+d e h+c f h)\right )\right )\right )+b^3 (a d f h+b (d f g+d e h+c f h)) \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) x+2 b^4 d f h \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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^2 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2}\\ &=\frac {2 b d \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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^2 (b B-2 a C) \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^2 \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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^3 d f (b e-a f) h (b g-a h) \left (3 a^3 C d^2 f h-3 a^2 b d (B d f h+C (d f g+d e h-c f h))-b^3 \left (2 B d^2 e g-B c^2 f h-c d (3 C e g-B (f g+e h))\right )+a b^2 \left (3 B d^2 (f g+e h)+C \left (d^2 e g-2 c^2 f h-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^3 d (b c-a d)^2 f (b e-a f)^2 h (b g-a h)^2}+\frac {\left (b (d e-c f) (d g-c h) \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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 b d \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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^2 (b B-2 a C) \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^2 \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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^3 C d^2 f h-3 a^2 b d (B d f h+C (d f g+d e h-c f h))-b^3 \left (2 B d^2 e g-B c^2 f h-c d (3 C e g-B (f g+e h))\right )+a b^2 \left (3 B d^2 (f g+e h)+C \left (d^2 e g-2 c^2 f h-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 b (d g-c h) \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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 ) \text {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 b d \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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^2 (b B-2 a C) \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^2 \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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 b \sqrt {d g-c h} \sqrt {f g-e h} \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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^3 C d^2 f h-3 a^2 b d (B d f h+C (d f g+d e h-c f h))-b^3 \left (2 B d^2 e g-B c^2 f h-c d (3 C e g-B (f g+e h))\right )+a b^2 \left (3 B d^2 (f g+e h)+C \left (d^2 e g-2 c^2 f h-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 ) \text {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 b d \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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^2 (b B-2 a C) \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^2 \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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 b \sqrt {d g-c h} \sqrt {f g-e h} \left (9 a^3 C d f h+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (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^3 C d^2 f h-b^3 \left (2 B d^2 e g-B c^2 f h-c d (3 C e g-B f g-B e h)\right )-3 a^2 b d (B d f h+C (d f g+d e h-c f h))+a b^2 \left (3 B d^2 (f g+e h)+C \left (d^2 e g-c d f g-c d e h-2 c^2 f 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] Leaf count is larger than twice the leaf count of optimal. \(10675\) vs. \(2(1128)=2256\).
time = 39.75, size = 10675, normalized size = 9.46 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(110288\) vs.
\(2(1056)=2112\).
time = 1.11, size = 110289, normalized size = 97.77
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3425\) |
default | \(\text {Expression too large to display}\) | \(110289\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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[Out]
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {-C\,a^2+B\,a\,b+C\,b^2\,x^2+B\,b^2\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{7/2}\,\sqrt {c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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