How can one prove that $$I\left( a,b \right)= \int_{0}^{\infty }\frac{x^{a-\frac{3}{2}}dx}{\left[ x^2+\left( b^2-2 \right)x+1 \right]^a}=b^{1-2a}\frac{\Gamma \left( \frac{1}{2} \right)\Gamma \left( a-\frac{1}{2} \right)}{\Gamma \left( a \right)},\ $$ where $a>\frac12,\ b\in \mathbb{R}^+$?
2026-04-09 02:05:53.1775700353
Prove that $\int_{0}^{\infty }\frac{x^{a-3/2}dx}{[ x^2+( b^2-2)x+1]^a}=b^{1-2a}\frac{\Gamma(1/2)\Gamma(a-1/2)}{\Gamma(a)}$
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