Suma de la serie (n-2)^3*(x+3)^(2*n)/2*n+3

     /          2 /            6             2             4\                                                                                                                                                                                               
     |   (3 + x) *\-1 - (3 + x)  - 11*(3 + x)  - 11*(3 + x) /        |       2|                                                                                                                                                                             
     |---------------------------------------------------------  for |(3 + x) | < 1                                                                                                                                                                         
     |              2                                                                                                                                                                                                                                       
     |/           2\  /            6            4            2\                                                                                                                                                                                             
     |\1 - (3 + x) / *\-1 + (3 + x)  - 3*(3 + x)  + 3*(3 + x) /                                                                                                                                                                                             
     |                                                                                                                                                                                                                                                      
     <                     oo                                                                                                                                                                                                                               
     |                    ___                                                           //            2                         \     //           2 /           4            2\                       \     //       2 /            2\                    \
     |                    \  `                                                          ||     (3 + x)            |       2|    |     ||    (3 + x) *\1 + (3 + x)  + 4*(3 + x) /         |       2|    |     ||(3 + x) *\-1 - (3 + x) /      |       2|    |
     |                     \    4        2*n                                            || ---------------    for |(3 + x) | < 1|     ||-------------------------------------------  for |(3 + x) | < 1|     ||------------------------  for |(3 + x) | < 1|
     |                     /   n *(3 + x)                            otherwise          ||               2                      |     ||              2                                                |     ||                   3                        |
     |                    /__,                                                          || /           2\                       |     ||/           2\  /           4            2\                    |     ||    /            2\                         |
     |                   n = 1                                                          || \1 - (3 + x) /                       |     ||\1 - (3 + x) / *\1 + (3 + x)  - 2*(3 + x) /                    |     ||    \-1 + (3 + x) /                         |
     \                                                                                  ||                                      |     ||                                                               |     ||                                            |
oo + ------------------------------------------------------------------------------ - 4*|<  oo                                  | - 3*|<              oo                                               | + 6*|<    oo                                      |
                                           2                                            || ___                                  |     ||             ___                                               |     ||   ___                                      |
                                                                                        || \  `                                 |     ||             \  `                                              |     ||   \  `                                     |
                                                                                        ||  \            2*n                    |     ||              \    3        2*n                                |     ||    \    2        2*n                       |
                                                                                        ||  /   n*(3 + x)         otherwise     |     ||              /   n *(3 + x)                     otherwise     |     ||    /   n *(3 + x)            otherwise     |
                                                                                        || /__,                                 |     ||             /__,                                              |     ||   /__,                                     |
                                                                                        ||n = 1                                 |     ||            n = 1                                              |     ||  n = 1                                     |
                                                                                        \\                                      /     \\                                                               /     \\                                            /

$$- 4 \left(\begin{cases} \frac{\left(x + 3\right)^{2}}{\left(1 - \left(x + 3\right)^{2}\right)^{2}} & \text{for}\: \left|{\left(x + 3\right)^{2}}\right| < 1 \\\sum_{n=1}^{\infty} n \left(x + 3\right)^{2 n} & \text{otherwise} \end{cases}\right) + 6 \left(\begin{cases} \frac{\left(x + 3\right)^{2} \left(- \left(x + 3\right)^{2} - 1\right)}{\left(\left(x + 3\right)^{2} - 1\right)^{3}} & \text{for}\: \left|{\left(x + 3\right)^{2}}\right| < 1 \\\sum_{n=1}^{\infty} n^{2} \left(x + 3\right)^{2 n} & \text{otherwise} \end{cases}\right) - 3 \left(\begin{cases} \frac{\left(x + 3\right)^{2} \left(\left(x + 3\right)^{4} + 4 \left(x + 3\right)^{2} + 1\right)}{\left(1 - \left(x + 3\right)^{2}\right)^{2} \left(\left(x + 3\right)^{4} - 2 \left(x + 3\right)^{2} + 1\right)} & \text{for}\: \left|{\left(x + 3\right)^{2}}\right| < 1 \\\sum_{n=1}^{\infty} n^{3} \left(x + 3\right)^{2 n} & \text{otherwise} \end{cases}\right) + \frac{\begin{cases} \frac{\left(x + 3\right)^{2} \left(- \left(x + 3\right)^{6} - 11 \left(x + 3\right)^{4} - 11 \left(x + 3\right)^{2} - 1\right)}{\left(1 - \left(x + 3\right)^{2}\right)^{2} \left(\left(x + 3\right)^{6} - 3 \left(x + 3\right)^{4} + 3 \left(x + 3\right)^{2} - 1\right)} & \text{for}\: \left|{\left(x + 3\right)^{2}}\right| < 1 \\\sum_{n=1}^{\infty} n^{4} \left(x + 3\right)^{2 n} & \text{otherwise} \end{cases}}{2} + \infty$$

oo + Piecewise(((3 + x)^2*(-1 - (3 + x)^6 - 11*(3 + x)^2 - 11*(3 + x)^4)/((1 - (3 + x)^2)^2*(-1 + (3 + x)^6 - 3*(3 + x)^4 + 3*(3 + x)^2)), Abs((3 + x)^2) < 1), (Sum(n^4*(3 + x)^(2*n), (n, 1, oo)), True))/2 - 4*Piecewise(((3 + x)^2/(1 - (3 + x)^2)^2, Abs((3 + x)^2) < 1), (Sum(n*(3 + x)^(2*n), (n, 1, oo)), True)) - 3*Piecewise(((3 + x)^2*(1 + (3 + x)^4 + 4*(3 + x)^2)/((1 - (3 + x)^2)^2*(1 + (3 + x)^4 - 2*(3 + x)^2)), Abs((3 + x)^2) < 1), (Sum(n^3*(3 + x)^(2*n), (n, 1, oo)), True)) + 6*Piecewise(((3 + x)^2*(-1 - (3 + x)^2)/(-1 + (3 + x)^2)^3, Abs((3 + x)^2) < 1), (Sum(n^2*(3 + x)^(2*n), (n, 1, oo)), True))