import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad

def F(x) :
    def g(t) :
        return (1-t)/(1-x*t**3)
    return quad(g,0,1)[0]

X = np.linspace(-1,1,100)
Y = []
for x in X :
    Y.append(F(x))
plt.plot(X,Y,color='k')


def S(n,x) :
    r = 0
    for k in range(0,n+1) :
        r += x**k/((3*k+1)*(3*k+2))
    return r

for n in [2,5,8] :
    Z = []
    for x in X :
        Z.append(S(n,x))
    plt.plot(X,Z)

plt.show()