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For such an easy example, Sage is able to do it:

sage: var('x,c,m,t')
(x, c, m, t)
sage: f(x) = m*x + c
sage: solve([f.integral(x,0,t) == 0],t)
[t == -2*c/m, t == 0]

For such an easy example, Sage is able to do it:

sage: var('x,c,m,t')
(x, c, m, t)
sage: f(x) = m*x + c
sage: solve([f.integral(x,0,t) == 0],t)
0], t)
[t == -2*c/m, t == 0]

For such an easy example, Sage is able to do it:it in the following way:

sage: var('x,c,m,t')
(x, c, m, t)
sage: f(x) = m*x + c
sage: solve([f.integral(x,0,t) == 0], t)
[t == -2*c/m, t == 0]

For such an easy example, Sage is able to do it in the following way:

sage: var('x,c,m,t')
(x, c, m, t)
sage: f(x) = m*x + c
 sage: solve([f.integral(x,0,t) == 0], t)
[t == -2*c/m, t == 0]

sage: solve([f.integral(x,0,t) == 10],t)
[t == -(c + sqrt(c^2 + 20*m))/m, t == -(c - sqrt(c^2 + 20*m))/m]

For such an easy example, Sage is able to do it in the following way:

sage: var('x,c,m,t')
(x, c, m, t)
sage: f(x) = m*x + c

sage: solve([f.integral(x,0,t) == 0], t)
[t == -2*c/m, t == 0]

sage: solve([f.integral(x,0,t) == 10],t)
10], t)
[t == -(c + sqrt(c^2 + 20*m))/m, t == -(c - sqrt(c^2 + 20*m))/m]