Optimal. Leaf size=832 \[ \frac{2 b p q \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) r^2}{h (b g-a h)}+\frac{2 d p q \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) r^2}{h (d g-c h)}-\frac{2 d p q \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right ) r^2}{h (d g-c h)}-\frac{2 b p q \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right ) r^2}{h (b g-a h)}-\frac{2 b p^2 \log (a+b x) \log \left (\frac{b g-a h}{h (a+b x)}+1\right ) r^2}{h (b g-a h)}-\frac{2 d q^2 \log (c+d x) \log \left (\frac{d g-c h}{h (c+d x)}+1\right ) r^2}{h (d g-c h)}+\frac{2 b p^2 \text{PolyLog}\left (2,-\frac{b g-a h}{h (a+b x)}\right ) r^2}{h (b g-a h)}+\frac{2 d p q \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) r^2}{h (d g-c h)}-\frac{2 d p q \text{PolyLog}\left (2,-\frac{h (a+b x)}{b g-a h}\right ) r^2}{h (d g-c h)}+\frac{2 d q^2 \text{PolyLog}\left (2,-\frac{d g-c h}{h (c+d x)}\right ) r^2}{h (d g-c h)}+\frac{2 b p q \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) r^2}{h (b g-a h)}-\frac{2 b p q \text{PolyLog}\left (2,-\frac{h (c+d x)}{d g-c h}\right ) r^2}{h (b g-a h)}-\frac{2 b p \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) r}{h (b g-a h)}-\frac{2 d q \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) r}{h (d g-c h)}+\frac{2 b p \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x) r}{h (b g-a h)}+\frac{2 d q \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x) r}{h (d g-c h)}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)} \]
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Rubi [A] time = 0.93174, antiderivative size = 878, normalized size of antiderivative = 1.06, number of steps used = 35, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387, Rules used = {2498, 2513, 2411, 2344, 2301, 2317, 2391, 2418, 2394, 2393, 36, 31} \[ \frac{b p^2 \log ^2(a+b x) r^2}{h (b g-a h)}+\frac{d q^2 \log ^2(c+d x) r^2}{h (d g-c h)}+\frac{2 b p q \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) r^2}{h (b g-a h)}+\frac{2 d p q \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) r^2}{h (d g-c h)}-\frac{2 b p^2 \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right ) r^2}{h (b g-a h)}-\frac{2 d p q \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right ) r^2}{h (d g-c h)}-\frac{2 d q^2 \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right ) r^2}{h (d g-c h)}-\frac{2 b p q \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right ) r^2}{h (b g-a h)}+\frac{2 d p q \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) r^2}{h (d g-c h)}-\frac{2 b p^2 \text{PolyLog}\left (2,-\frac{h (a+b x)}{b g-a h}\right ) r^2}{h (b g-a h)}-\frac{2 d p q \text{PolyLog}\left (2,-\frac{h (a+b x)}{b g-a h}\right ) r^2}{h (d g-c h)}+\frac{2 b p q \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) r^2}{h (b g-a h)}-\frac{2 d q^2 \text{PolyLog}\left (2,-\frac{h (c+d x)}{d g-c h}\right ) r^2}{h (d g-c h)}-\frac{2 b p q \text{PolyLog}\left (2,-\frac{h (c+d x)}{d g-c h}\right ) r^2}{h (b g-a h)}-\frac{2 b p \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) r}{h (b g-a h)}-\frac{2 d q \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) r}{h (d g-c h)}+\frac{2 b p \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x) r}{h (b g-a h)}+\frac{2 d q \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x) r}{h (d g-c h)}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)} \]
Antiderivative was successfully verified.
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Rule 2498
Rule 2513
Rule 2411
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2418
Rule 2394
Rule 2393
Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^2} \, dx &=-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac{(2 b p r) \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x) (g+h x)} \, dx}{h}+\frac{(2 d q r) \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(c+d x) (g+h x)} \, dx}{h}\\ &=-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac{\left (2 b p^2 r^2\right ) \int \frac{\log (a+b x)}{(a+b x) (g+h x)} \, dx}{h}+\frac{\left (2 b p q r^2\right ) \int \frac{\log (c+d x)}{(a+b x) (g+h x)} \, dx}{h}+\frac{\left (2 d p q r^2\right ) \int \frac{\log (a+b x)}{(c+d x) (g+h x)} \, dx}{h}+\frac{\left (2 d q^2 r^2\right ) \int \frac{\log (c+d x)}{(c+d x) (g+h x)} \, dx}{h}-\frac{\left (2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{1}{(a+b x) (g+h x)} \, dx}{h}-\frac{\left (2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{1}{(c+d x) (g+h x)} \, dx}{h}\\ &=-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac{\left (2 p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (\frac{b g-a h}{b}+\frac{h x}{b}\right )} \, dx,x,a+b x\right )}{h}+\frac{\left (2 b p q r^2\right ) \int \left (\frac{b \log (c+d x)}{(b g-a h) (a+b x)}-\frac{h \log (c+d x)}{(b g-a h) (g+h x)}\right ) \, dx}{h}+\frac{\left (2 d p q r^2\right ) \int \left (\frac{d \log (a+b x)}{(d g-c h) (c+d x)}-\frac{h \log (a+b x)}{(d g-c h) (g+h x)}\right ) \, dx}{h}+\frac{\left (2 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (\frac{d g-c h}{d}+\frac{h x}{d}\right )} \, dx,x,c+d x\right )}{h}+\frac{\left (2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{1}{g+h x} \, dx}{b g-a h}-\frac{\left (2 b^2 p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{1}{a+b x} \, dx}{h (b g-a h)}+\frac{\left (2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{1}{g+h x} \, dx}{d g-c h}-\frac{\left (2 d^2 q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac{1}{c+d x} \, dx}{h (d g-c h)}\\ &=-\frac{2 b p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (b g-a h)}-\frac{2 d q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (d g-c h)}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac{2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (b g-a h)}+\frac{2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (d g-c h)}-\frac{\left (2 p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b g-a h}{b}+\frac{h x}{b}} \, dx,x,a+b x\right )}{b g-a h}+\frac{\left (2 b p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{h (b g-a h)}-\frac{\left (2 b p q r^2\right ) \int \frac{\log (c+d x)}{g+h x} \, dx}{b g-a h}+\frac{\left (2 b^2 p q r^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{h (b g-a h)}-\frac{\left (2 d p q r^2\right ) \int \frac{\log (a+b x)}{g+h x} \, dx}{d g-c h}+\frac{\left (2 d^2 p q r^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{h (d g-c h)}-\frac{\left (2 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{d g-c h}{d}+\frac{h x}{d}} \, dx,x,c+d x\right )}{d g-c h}+\frac{\left (2 d q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{h (d g-c h)}\\ &=\frac{b p^2 r^2 \log ^2(a+b x)}{h (b g-a h)}+\frac{2 b p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{h (b g-a h)}+\frac{d q^2 r^2 \log ^2(c+d x)}{h (d g-c h)}+\frac{2 d p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{h (d g-c h)}-\frac{2 b p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (b g-a h)}-\frac{2 d q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (d g-c h)}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac{2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (b g-a h)}+\frac{2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (d g-c h)}-\frac{2 b p^2 r^2 \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right )}{h (b g-a h)}-\frac{2 d p q r^2 \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right )}{h (d g-c h)}-\frac{2 b p q r^2 \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h (b g-a h)}-\frac{2 d q^2 r^2 \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h (d g-c h)}+\frac{\left (2 b p^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{h x}{b g-a h}\right )}{x} \, dx,x,a+b x\right )}{h (b g-a h)}-\frac{\left (2 b d p q r^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{h (b g-a h)}+\frac{\left (2 b d p q r^2\right ) \int \frac{\log \left (\frac{d (g+h x)}{d g-c h}\right )}{c+d x} \, dx}{h (b g-a h)}-\frac{\left (2 b d p q r^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{h (d g-c h)}+\frac{\left (2 b d p q r^2\right ) \int \frac{\log \left (\frac{b (g+h x)}{b g-a h}\right )}{a+b x} \, dx}{h (d g-c h)}+\frac{\left (2 d q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{h x}{d g-c h}\right )}{x} \, dx,x,c+d x\right )}{h (d g-c h)}\\ &=\frac{b p^2 r^2 \log ^2(a+b x)}{h (b g-a h)}+\frac{2 b p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{h (b g-a h)}+\frac{d q^2 r^2 \log ^2(c+d x)}{h (d g-c h)}+\frac{2 d p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{h (d g-c h)}-\frac{2 b p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (b g-a h)}-\frac{2 d q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (d g-c h)}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac{2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (b g-a h)}+\frac{2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (d g-c h)}-\frac{2 b p^2 r^2 \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right )}{h (b g-a h)}-\frac{2 d p q r^2 \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right )}{h (d g-c h)}-\frac{2 b p q r^2 \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h (b g-a h)}-\frac{2 d q^2 r^2 \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h (d g-c h)}-\frac{2 b p^2 r^2 \text{Li}_2\left (-\frac{h (a+b x)}{b g-a h}\right )}{h (b g-a h)}-\frac{2 d q^2 r^2 \text{Li}_2\left (-\frac{h (c+d x)}{d g-c h}\right )}{h (d g-c h)}-\frac{\left (2 b p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{h (b g-a h)}+\frac{\left (2 b p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{h x}{d g-c h}\right )}{x} \, dx,x,c+d x\right )}{h (b g-a h)}-\frac{\left (2 d p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{h (d g-c h)}+\frac{\left (2 d p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{h x}{b g-a h}\right )}{x} \, dx,x,a+b x\right )}{h (d g-c h)}\\ &=\frac{b p^2 r^2 \log ^2(a+b x)}{h (b g-a h)}+\frac{2 b p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{h (b g-a h)}+\frac{d q^2 r^2 \log ^2(c+d x)}{h (d g-c h)}+\frac{2 d p q r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{h (d g-c h)}-\frac{2 b p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (b g-a h)}-\frac{2 d q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (d g-c h)}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac{2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (b g-a h)}+\frac{2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (d g-c h)}-\frac{2 b p^2 r^2 \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right )}{h (b g-a h)}-\frac{2 d p q r^2 \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right )}{h (d g-c h)}-\frac{2 b p q r^2 \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h (b g-a h)}-\frac{2 d q^2 r^2 \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h (d g-c h)}+\frac{2 d p q r^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{h (d g-c h)}-\frac{2 b p^2 r^2 \text{Li}_2\left (-\frac{h (a+b x)}{b g-a h}\right )}{h (b g-a h)}-\frac{2 d p q r^2 \text{Li}_2\left (-\frac{h (a+b x)}{b g-a h}\right )}{h (d g-c h)}+\frac{2 b p q r^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{h (b g-a h)}-\frac{2 b p q r^2 \text{Li}_2\left (-\frac{h (c+d x)}{d g-c h}\right )}{h (b g-a h)}-\frac{2 d q^2 r^2 \text{Li}_2\left (-\frac{h (c+d x)}{d g-c h}\right )}{h (d g-c h)}\\ \end{align*}
Mathematica [B] time = 2.67099, size = 2930, normalized size = 3.52 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.498, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \right ) ^{2}}{ \left ( hx+g \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57674, size = 1006, normalized size = 1.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h^{2} x^{2} + 2 \, g h x + g^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (h x + g\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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