Optimal. Leaf size=421 \[ \frac{16 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{4 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{4 \sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{5/2} e^3 n^{5/2}}+\frac{8 (d+e x) (f+g x) (e f-d g)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]
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Rubi [A] time = 1.39878, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 41, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310} \[ \frac{16 \sqrt{2 \pi } g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{4 \sqrt{\pi } e^{-\frac{a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{4 \sqrt{3 \pi } g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{5/2} e^3 n^{5/2}}+\frac{8 (d+e x) (f+g x) (e f-d g)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2400
Rule 2401
Rule 2389
Rule 2300
Rule 2180
Rule 2204
Rule 2390
Rule 2310
Rubi steps
\begin{align*} \int \frac{(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx &=-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{2 \int \frac{(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx}{b n}-\frac{(4 (e f-d g)) \int \frac{f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx}{3 b e n}\\ &=-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{12 \int \frac{(f+g x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 n^2}-\frac{(16 (e f-d g)) \int \frac{f+g x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e n^2}-\frac{(8 (e f-d g)) \int \frac{f+g x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e n^2}+\frac{\left (8 (e f-d g)^2\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^2 n^2}\\ &=-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{12 \int \left (\frac{(e f-d g)^2}{e^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{2 g (e f-d g) (d+e x)}{e^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g^2 (d+e x)^2}{e^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{b^2 n^2}-\frac{(16 (e f-d g)) \int \left (\frac{e f-d g}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g (d+e x)}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{3 b^2 e n^2}-\frac{(8 (e f-d g)) \int \left (\frac{e f-d g}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{g (d+e x)}{e \sqrt{a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{b^2 e n^2}+\frac{\left (8 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^3 n^2}\\ &=-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (12 g^2\right ) \int \frac{(d+e x)^2}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}-\frac{(16 g (e f-d g)) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^2 n^2}-\frac{(8 g (e f-d g)) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}+\frac{(24 g (e f-d g)) \int \frac{d+e x}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}-\frac{\left (16 (e f-d g)^2\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e^2 n^2}-\frac{\left (8 (e f-d g)^2\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}+\frac{\left (12 (e f-d g)^2\right ) \int \frac{1}{\sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{b^2 e^2 n^2}+\frac{\left (8 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^3 n^3}\\ &=-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (12 g^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac{(16 g (e f-d g)) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^3 n^2}-\frac{(8 g (e f-d g)) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac{(24 g (e f-d g)) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac{\left (16 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^3 n^2}-\frac{\left (8 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac{\left (12 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac{\left (16 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^3 n^3}\\ &=\frac{8 e^{-\frac{a}{b n}} (e f-d g)^2 \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (12 g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac{\left (16 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^3 n^3}-\frac{\left (8 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}+\frac{\left (24 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac{\left (16 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^3 n^3}-\frac{\left (8 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}+\frac{\left (12 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}\\ &=\frac{8 e^{-\frac{a}{b n}} (e f-d g)^2 \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}+\frac{\left (24 g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b n}+\frac{3 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^3 e^3 n^3}-\frac{\left (32 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^3 n^3}-\frac{\left (16 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^3 e^3 n^3}+\frac{\left (48 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^3 e^3 n^3}-\frac{\left (32 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^3 n^3}-\frac{\left (16 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^3 e^3 n^3}+\frac{\left (24 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{b^3 e^3 n^3}\\ &=\frac{4 e^{-\frac{a}{b n}} (e f-d g)^2 \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{16 e^{-\frac{2 a}{b n}} g (e f-d g) \sqrt{2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac{4 e^{-\frac{3 a}{b n}} g^2 \sqrt{3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{b^{5/2} e^3 n^{5/2}}-\frac{2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac{8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}-\frac{4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}\\ \end{align*}
Mathematica [A] time = 4.52702, size = 527, normalized size = 1.25 \[ -\frac{2 e^{-\frac{3 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (\sqrt{b} \sqrt{n} e^{\frac{2 a}{b n}} \left (c (d+e x)^n\right )^{2/n} \left (2 b n \left (2 d^2 g^2+6 d e f g+e^2 f^2\right ) \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )+e e^{\frac{a}{b n}} (f+g x) \left (c (d+e x)^n\right )^{\frac{1}{n}} \left (2 a (2 d g+e f+3 e g x)+2 b (2 d g+e (f+3 g x)) \log \left (c (d+e x)^n\right )+b e n (f+g x)\right )\right )+2 \sqrt{\pi } d g e^{\frac{2 a}{b n}} (d g+8 e f) \left (c (d+e x)^n\right )^{2/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+8 \sqrt{2 \pi } g e^{\frac{a}{b n}} (d+e x) (d g-e f) \left (c (d+e x)^n\right )^{\frac{1}{n}} \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )-6 \sqrt{3 \pi } g^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )\right )}{3 b^{5/2} e^3 n^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.681, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{2} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}^{2}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (f + g x\right )^{2}}{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac{5}{2}}}\, dx \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{{\left (g x + f\right )}^{2}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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