I am trying to use Maple to evaluate some integrals, in the following form:
$$\int\frac{1}{(Z^2 + \alpha^2)^{5/2}}dZ = \frac{Z(3\alpha^2 + 2Z^2)}{3\alpha^4(Z^2 + \alpha^2)^{3/2}}$$
The indefinite integral given there blows up as $\alpha \to 0$; however, the definite integral, given below, is bounded as $\alpha \to 0$:
$$\int_{Z_1}^{Z_2}\frac{1}{(Z^2 + \alpha^2)^{5/2}}dZ = \frac{Z_1(3\alpha^2 + 2Z_1^2)}{3\alpha^4(Z_1^2 + \alpha^2)^{3/2}} - \frac{Z_2(3\alpha^2 + 2Z_2^2)}{3\alpha^4(Z_2^2 + \alpha^2)^{3/2}}$$
However, evaluating the definite integral above using a computer is not going to work, since the two individual terms blow up at $\alpha = 0$ (although the difference doesn't). I am hoping to use Maple to evaluate these integrals, and I am wondering if it is possible to tell Maple to combine the two terms into a single fraction, before evaluating? How can I do that?
(btw. $Z, \alpha$ here are real numbers)