transfer function - How to find analytic description of filtered signal

I am looking for an exact analytic description of a filtered signal. I have an electronic circuit whose input is a monoexponential decay. First (1) the signal gets filtered by a simple RC-Lowpass. Then (2) a second order Bessel Low Pass Filter is applied.

The respective transfer functions are: \begin{align} TF_{\rm RC}(s) &= \frac{R}{s+\frac{1}{R C}}\\ TF_{\rm Bessel}(s) &= \frac{a_1}{a_1+b_1s+s^2} \end{align}

The input signal is: $$I_{\rm in}(t) = \frac{V}{R_1 + R_2} + R_1\cdot V_{\rm cmd}\cdot e^{-t\frac{R_1 + R_2}{CR_1R_2}}\frac{1}{R_2(R_1 + R_2)}$$

My idea was to calculate the output in the $s$-domain: $$ V_{\rm out}(s) = I_{\rm in}(s)\cdot TF_{RC}(s)\cdot TF_{Bessel}(s) $$ and then apply the inverse laplace transform to obtain $V_{\rm out}$(t). The inverse transform is what troubles me. I tried to calculate it with MATLAB, but it doesn't return an analytic function. I guess this comes from the complex poles of the bessel filter. Could you give me a hint on how to calculate the output?

Edit: $$ TF(s) = \frac{6.092e12}{s^3 + 5.367e04*s^2 + 1.038e09*s + 6.074e12}$$

$$ I_{in}(t) = \frac{10e-3}{500e6 + 10e6} + 500e6*10e-3*e^{-t*\frac{500e6 + 10e6}{10e-12*500e6*10e6}}\frac{1}{10e6*(500e6 + 10e6)} $$

$$ TF(s)*I_{in}(s) = \frac{4.9e-21*(1.2e24*s + 2.5e26)}{s*(s + 1.0e4)*(s^3 + 5.4e4*s^2 + 1.0e9*s + 6.1e12)} $$

PFE yields:

$$ TF(s)*I_{in}(s) = \frac{1.1e-9 - 4.8e-11i}{s + 2.2e4 - 1.1e4i} + \frac{1.1e-9 + 4.8e-11i}{s + 2.2e4 + 1.1e4i} + \frac{6.6223e-7}{s + 10200.0} + \frac{-6.6441e-7}{s + 10235.0} + \frac{1.9666e-11}{s} $$

Inverse Laplace transform of the upper term will yield a complex result. I know that I can simplify the first two terms by bringing them to a common denominator and thereby get rid of the complex numbers. The inverse transform is however still complex.

Edit: calculation with symbolic parameters yields: $$ H(s) = \frac{a*s+b}{c*s^2+d*s+f} $$ $$ H(t) = a*dirac(1, t) + (2*b*sin((t*(- d^2 + 4*c*f)^{1/2})/(2*c))*exp(-\frac{d*t}{2*c}))\frac{1}{(4*c*f - d^2)^{1/2}} $$

I don't quite understand where the dirac comes from, but this is basically the same result as doing the calculation with complex numbers and the taking the real part.