What is the CDF of a linear combination of a Gamma and a Hyperbolic random variables?

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I am interested in the evaluation of the risk stemming from the combination of two weather events: heavy rain and low temperature. Rainfall $R$ can be modeled as a Gamma distribution, temperature $T$ as a hyperbolic one (or, alternately, as a Normal Inverse Gaussian). The resulting random variable is $$X = - \alpha R + \beta T$$. Rainfall and temperature are independent variables. What is its CDF? Or how can I evaluate it numerically?