Optimal. Leaf size=449 \[ \frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac {8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt {g+h x}}-\frac {8 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 h (f g-e h)^{3/2}}-\frac {8 b^2 f^{3/2} p^2 q^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 h (f g-e h)^{3/2}} \]
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Rubi [A]
time = 1.60, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 15, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used =
{2445, 2458, 2389, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 2495}
\begin {gather*} -\frac {8 b^2 f^{3/2} p^2 q^2 \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 h (f g-e h)^{3/2}}-\frac {8 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h \sqrt {g+h x} (f g-e h)}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\frac {8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}-\frac {16 b^2 f^{3/2} p^2 q^2 \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 1601
Rule 2352
Rule 2356
Rule 2389
Rule 2390
Rule 2445
Rule 2449
Rule 2458
Rule 2495
Rule 6055
Rule 6131
Rule 6873
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{5/2}} \, dx &=\text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(g+h x)^{5/2}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\text {Subst}\left (\frac {(4 b f p q) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) (g+h x)^{3/2}} \, dx}{3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\text {Subst}\left (\frac {(4 b p q) \text {Subst}\left (\int \frac {a+b \log \left (c d^q x^{p q}\right )}{x \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^{3/2}} \, dx,x,e+f x\right )}{3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\text {Subst}\left (\frac {(4 b p q) \text {Subst}\left (\int \frac {a+b \log \left (c d^q x^{p q}\right )}{\left (\frac {f g-e h}{f}+\frac {h x}{f}\right )^{3/2}} \, dx,x,e+f x\right )}{3 (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(4 b f p q) \text {Subst}\left (\int \frac {a+b \log \left (c d^q x^{p q}\right )}{x \sqrt {\frac {f g-e h}{f}+\frac {h x}{f}}} \, dx,x,e+f x\right )}{3 h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt {g+h x}}-\frac {8 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\text {Subst}\left (\frac {\left (4 b^2 f p^2 q^2\right ) \text {Subst}\left (\int -\frac {2 \sqrt {f} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h x}{f}}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h} x} \, dx,x,e+f x\right )}{3 h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (8 b^2 f p^2 q^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {f g-e h}{f}+\frac {h x}{f}}} \, dx,x,e+f x\right )}{3 h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt {g+h x}}-\frac {8 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\text {Subst}\left (\frac {\left (8 b^2 f^{3/2} p^2 q^2\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g-\frac {e h}{f}+\frac {h x}{f}}}{\sqrt {f g-e h}}\right )}{x} \, dx,x,e+f x\right )}{3 h (f g-e h)^{3/2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (16 b^2 f^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {f g-e h}{h}+\frac {f x^2}{h}} \, dx,x,\sqrt {g+h x}\right )}{3 h^2 (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt {g+h x}}-\frac {8 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\text {Subst}\left (\frac {\left (16 b^2 f^{5/2} p^2 q^2\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {f g-e h}}\right )}{e h+f \left (-g+x^2\right )} \, dx,x,\sqrt {g+h x}\right )}{3 h (f g-e h)^{3/2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt {g+h x}}-\frac {8 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\text {Subst}\left (\frac {\left (16 b^2 f^{5/2} p^2 q^2\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {f g-e h}}\right )}{-f g+e h+f x^2} \, dx,x,\sqrt {g+h x}\right )}{3 h (f g-e h)^{3/2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac {8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt {g+h x}}-\frac {8 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\text {Subst}\left (\frac {\left (16 b^2 f^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {f g-e h}}\right )}{1-\frac {\sqrt {f} x}{\sqrt {f g-e h}}} \, dx,x,\sqrt {g+h x}\right )}{3 h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac {8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt {g+h x}}-\frac {8 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 h (f g-e h)^{3/2}}+\text {Subst}\left (\frac {\left (16 b^2 f^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {f} x}{\sqrt {f g-e h}}}\right )}{1-\frac {f x^2}{f g-e h}} \, dx,x,\sqrt {g+h x}\right )}{3 h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac {8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt {g+h x}}-\frac {8 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 h (f g-e h)^{3/2}}-\text {Subst}\left (\frac {\left (16 b^2 f^{3/2} p^2 q^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 h (f g-e h)^{3/2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac {8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt {g+h x}}-\frac {8 b f^{3/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\frac {16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 h (f g-e h)^{3/2}}-\frac {8 b^2 f^{3/2} p^2 q^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 h (f g-e h)^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 4.85, size = 657, normalized size = 1.46 \begin {gather*} \frac {2 \left (-4 a b f^{3/2} \sqrt {f g-e h} p q (g+h x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )+3 b^2 f h p^2 q^2 (e+f x) (g+h x) \sqrt {\frac {f (g+h x)}{f g-e h}} \, _4F_3\left (1,1,1,\frac {5}{2};2,2,2;\frac {h (e+f x)}{-f g+e h}\right )+4 b^2 f^{3/2} \sqrt {f g-e h} p^2 q^2 (g+h x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log (e+f x)+2 a b (f g-e h) p q (2 f (g+h x)+(-f g+e h) \log (e+f x))-2 b^2 (f g-e h) p^2 q^2 \log (e+f x) (2 f (g+h x)+(-f g+e h) \log (e+f x))-4 b^2 f^{3/2} \sqrt {f g-e h} p q (g+h x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+2 b^2 (f g-e h) p q (2 f (g+h x)+(-f g+e h) \log (e+f x)) \log \left (c \left (d (e+f x)^p\right )^q\right )-(f g-e h)^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+b^2 (f g-e h) p^2 q^2 \log (e+f x) \left (\left (e h+f h x \sqrt {\frac {f (g+h x)}{f g-e h}}+f g \left (-1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right ) \log (e+f x)-4 f (g+h x) \left (-1+\sqrt {\frac {f (g+h x)}{f g-e h}}+\sqrt {\frac {f (g+h x)}{f g-e h}} \log \left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )\right )\right )\right )}{3 h (f g-e h)^2 (g+h x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{2}}{\left (h x +g \right )^{\frac {5}{2}}}\, dx\]
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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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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\left (g + h x\right )^{\frac {5}{2}}}\, dx \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 {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{{\left (g+h\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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