Q:

Calculation of one constant similar to MZV

The series arose in the calculation of Mean value of a function associated with continued fractions:
$$C=\sum_{1\le b\le d<\infty}\frac{1}{b(b+d)d^2}.$$
Obviously 
$C=C_1-C_2,$
where
$$C_1=\sum_{1\le b\le d<\infty}\frac{1}{bd^3},\quad C_2=\sum_{1\le b\le d<\infty}\frac{1}{(b+d)d^3}.$$
Mathematica gives
$C_1=\frac{\pi^4}{ 72},$
and $C_2$ can be simplifyied to
$$C_2=2\int_{0}^{1}\frac{\mathrm{Li}_2(t)}{t}\log(1+t)dt$$
using the following trick:
$$\frac{1}{n+1}+\ldots+\frac{1}{2n}=1-\frac12+\frac13-\frac14+\ldots-\frac{1}{2n}=\int_0^1\frac{1-t^{2n}}{1+t}dt.$$

Q: Is it possible to find closed form for $C$?

A:

In terms of alternating MZV's, one has (experimentally but it should be easy to prove) $$C_2=2\zeta(-3,1)+(3/4)\zeta(4)$$
where $\zeta(-3,1)=\sum_{1\le b<d}(-1)^d/(d^3b)$.
It is almost certain (proved?) that $\zeta(-3,1)$ does not have a "closed" expression.

