Integrand size = 21, antiderivative size = 82 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {1}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {1}{3 a^2 d \left (a+b (c+d x)^3\right )}+\frac {\log (c+d x)}{a^3 d}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a^3 d} \] Output:
1/6/a/d/(a+b*(d*x+c)^3)^2+1/3/a^2/d/(a+b*(d*x+c)^3)+ln(d*x+c)/a^3/d-1/3*ln (a+b*(d*x+c)^3)/a^3/d
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {\frac {a \left (a+2 \left (a+b (c+d x)^3\right )\right )}{\left (a+b (c+d x)^3\right )^2}+6 \log (c+d x)-2 \log \left (a+b (c+d x)^3\right )}{6 a^3 d} \] Input:
Integrate[1/((c + d*x)*(a + b*(c + d*x)^3)^3),x]
Output:
((a*(a + 2*(a + b*(c + d*x)^3)))/(a + b*(c + d*x)^3)^2 + 6*Log[c + d*x] - 2*Log[a + b*(c + d*x)^3])/(6*a^3*d)
Time = 0.37 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {895, 798, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 895 |
| \(\displaystyle \frac {\int \frac {1}{(c+d x) \left (b (c+d x)^3+a\right )^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {\int \frac {1}{(c+d x)^3 \left (b (c+d x)^3+a\right )^3}d(c+d x)^3}{3 d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\int \left (-\frac {b}{a^3 \left (b (c+d x)^3+a\right )}-\frac {b}{a^2 \left (b (c+d x)^3+a\right )^2}-\frac {b}{a \left (b (c+d x)^3+a\right )^3}+\frac {1}{a^3 (c+d x)^3}\right )d(c+d x)^3}{3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\log \left (a+b (c+d x)^3\right )}{a^3}+\frac {\log \left ((c+d x)^3\right )}{a^3}+\frac {1}{a^2 \left (a+b (c+d x)^3\right )}+\frac {1}{2 a \left (a+b (c+d x)^3\right )^2}}{3 d}\) |
Input:
Int[1/((c + d*x)*(a + b*(c + d*x)^3)^3),x]
Output:
(1/(2*a*(a + b*(c + d*x)^3)^2) + 1/(a^2*(a + b*(c + d*x)^3)) + Log[(c + d* x)^3]/a^3 - Log[a + b*(c + d*x)^3]/a^3)/(3*d)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff icient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ {a, b, m, n, p}, x] && LinearPairQ[u, v, x]
Time = 0.79 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.74
| method | result | size |
| risch | \(\frac {\frac {b \,d^{2} x^{3}}{3 a^{2}}+\frac {b c d \,x^{2}}{a^{2}}+\frac {b x \,c^{2}}{a^{2}}+\frac {2 c^{3} b +3 a}{6 a^{2} d}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )^{2}}+\frac {\ln \left (x d +c \right )}{a^{3} d}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}{3 a^{3} d}\) | \(143\) |
| default | \(\frac {\ln \left (x d +c \right )}{a^{3} d}-\frac {b \left (\frac {-\frac {a \,d^{2} x^{3}}{3}-a c d \,x^{2}-a \,c^{2} x -\frac {a \left (2 c^{3} b +3 a \right )}{6 d b}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )^{2}}+\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}{3 b d}\right )}{a^{3}}\) | \(144\) |
| norman | \(\frac {\frac {b c d \,x^{2}}{a^{2}}+\frac {b x \,c^{2}}{a^{2}}+\frac {2 b^{3} c^{3} d^{5}+3 a \,b^{2} d^{5}}{6 a^{2} d^{6} b^{2}}+\frac {b \,d^{2} x^{3}}{3 a^{2}}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )^{2}}+\frac {\ln \left (x d +c \right )}{a^{3} d}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}{3 a^{3} d}\) | \(157\) |
| parallelrisch | \(\frac {-12 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x^{2} a \,b^{3} c \,d^{7}-12 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x a \,b^{3} c^{2} d^{6}+36 \ln \left (x d +c \right ) x^{2} a \,b^{3} c \,d^{7}+36 \ln \left (x d +c \right ) x a \,b^{3} c^{2} d^{6}+2 a \,b^{3} c^{3} d^{5}+6 x^{2} a \,b^{3} c \,d^{7}+6 x a \,b^{3} c^{2} d^{6}+36 \ln \left (x d +c \right ) x^{5} b^{4} c \,d^{10}-12 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x^{5} b^{4} c \,d^{10}+90 \ln \left (x d +c \right ) x^{4} b^{4} c^{2} d^{9}-30 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x^{4} b^{4} c^{2} d^{9}+120 \ln \left (x d +c \right ) x^{3} b^{4} c^{3} d^{8}-40 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x^{3} b^{4} c^{3} d^{8}+90 \ln \left (x d +c \right ) x^{2} b^{4} c^{4} d^{7}-30 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x^{2} b^{4} c^{4} d^{7}+36 \ln \left (x d +c \right ) x \,b^{4} c^{5} d^{6}-12 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x \,b^{4} c^{5} d^{6}+12 \ln \left (x d +c \right ) x^{3} a \,b^{3} d^{8}-4 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x^{3} a \,b^{3} d^{8}+12 \ln \left (x d +c \right ) a \,b^{3} c^{3} d^{5}-4 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) a \,b^{3} c^{3} d^{5}-2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) a^{2} b^{2} d^{5}+2 x^{3} a \,b^{3} d^{8}+6 \ln \left (x d +c \right ) x^{6} b^{4} d^{11}-2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x^{6} b^{4} d^{11}+6 \ln \left (x d +c \right ) b^{4} c^{6} d^{5}-2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) b^{4} c^{6} d^{5}+6 \ln \left (x d +c \right ) a^{2} b^{2} d^{5}+3 a^{2} b^{2} d^{5}}{6 a^{3} b^{2} d^{6} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )^{2}}\) | \(887\) |
Input:
int(1/(d*x+c)/(a+b*(d*x+c)^3)^3,x,method=_RETURNVERBOSE)
Output:
(1/3/a^2*b*d^2*x^3+1/a^2*b*c*d*x^2+1/a^2*b*x*c^2+1/6*(2*b*c^3+3*a)/a^2/d)/ (b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2+ln(d*x+c)/a^3/d-1/3/a^3/d* ln(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)
Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (76) = 152\).
Time = 0.10 (sec) , antiderivative size = 467, normalized size of antiderivative = 5.70 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2} - 2 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \, {\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \, {\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \, {\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \, {\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (d x + c\right )}{6 \, {\left (a^{3} b^{2} d^{7} x^{6} + 6 \, a^{3} b^{2} c d^{6} x^{5} + 15 \, a^{3} b^{2} c^{2} d^{5} x^{4} + 2 \, {\left (10 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{3} x^{2} + 6 \, {\left (a^{3} b^{2} c^{5} + a^{4} b c^{2}\right )} d^{2} x + {\left (a^{3} b^{2} c^{6} + 2 \, a^{4} b c^{3} + a^{5}\right )} d\right )}} \] Input:
integrate(1/(d*x+c)/(a+b*(d*x+c)^3)^3,x, algorithm="fricas")
Output:
1/6*(2*a*b*d^3*x^3 + 6*a*b*c*d^2*x^2 + 6*a*b*c^2*d*x + 2*a*b*c^3 + 3*a^2 - 2*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + b^2*c^6 + 2*(10*b ^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*x^2 + 6*(b ^2*c^5 + a*b*c^2)*d*x + a^2)*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a) + 6*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + b^2* c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d ^2*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x + a^2)*log(d*x + c))/(a^3*b^2*d^7*x^6 + 6*a^3*b^2*c*d^6*x^5 + 15*a^3*b^2*c^2*d^5*x^4 + 2*(10*a^3*b^2*c^3 + a^4*b) *d^4*x^3 + 3*(5*a^3*b^2*c^4 + 2*a^4*b*c)*d^3*x^2 + 6*(a^3*b^2*c^5 + a^4*b* c^2)*d^2*x + (a^3*b^2*c^6 + 2*a^4*b*c^3 + a^5)*d)
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (66) = 132\).
Time = 2.20 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.28 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {3 a + 2 b c^{3} + 6 b c^{2} d x + 6 b c d^{2} x^{2} + 2 b d^{3} x^{3}}{6 a^{4} d + 12 a^{3} b c^{3} d + 6 a^{2} b^{2} c^{6} d + 90 a^{2} b^{2} c^{2} d^{5} x^{4} + 36 a^{2} b^{2} c d^{6} x^{5} + 6 a^{2} b^{2} d^{7} x^{6} + x^{3} \cdot \left (12 a^{3} b d^{4} + 120 a^{2} b^{2} c^{3} d^{4}\right ) + x^{2} \cdot \left (36 a^{3} b c d^{3} + 90 a^{2} b^{2} c^{4} d^{3}\right ) + x \left (36 a^{3} b c^{2} d^{2} + 36 a^{2} b^{2} c^{5} d^{2}\right )} + \frac {\log {\left (\frac {c}{d} + x \right )}}{a^{3} d} - \frac {\log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{3} d} \] Input:
integrate(1/(d*x+c)/(a+b*(d*x+c)**3)**3,x)
Output:
(3*a + 2*b*c**3 + 6*b*c**2*d*x + 6*b*c*d**2*x**2 + 2*b*d**3*x**3)/(6*a**4* d + 12*a**3*b*c**3*d + 6*a**2*b**2*c**6*d + 90*a**2*b**2*c**2*d**5*x**4 + 36*a**2*b**2*c*d**6*x**5 + 6*a**2*b**2*d**7*x**6 + x**3*(12*a**3*b*d**4 + 120*a**2*b**2*c**3*d**4) + x**2*(36*a**3*b*c*d**3 + 90*a**2*b**2*c**4*d**3 ) + x*(36*a**3*b*c**2*d**2 + 36*a**2*b**2*c**5*d**2)) + log(c/d + x)/(a**3 *d) - log(3*c**2*x/d**2 + 3*c*x**2/d + x**3 + (a + b*c**3)/(b*d**3))/(3*a* *3*d)
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (76) = 152\).
Time = 0.04 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.99 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + 3 \, a}{6 \, {\left (a^{2} b^{2} d^{7} x^{6} + 6 \, a^{2} b^{2} c d^{6} x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 2 \, {\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} x^{2} + 6 \, {\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} x + {\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d\right )}} - \frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d} + \frac {\log \left (d x + c\right )}{a^{3} d} \] Input:
integrate(1/(d*x+c)/(a+b*(d*x+c)^3)^3,x, algorithm="maxima")
Output:
1/6*(2*b*d^3*x^3 + 6*b*c*d^2*x^2 + 6*b*c^2*d*x + 2*b*c^3 + 3*a)/(a^2*b^2*d ^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 15*a^2*b^2*c^2*d^5*x^4 + 2*(10*a^2*b^2*c^3 + a^3*b)*d^4*x^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c)*d^3*x^2 + 6*(a^2*b^2*c^5 + a^3*b*c^2)*d^2*x + (a^2*b^2*c^6 + 2*a^3*b*c^3 + a^4)*d) - 1/3*log(b*d^3* x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)/(a^3*d) + log(d*x + c)/(a^3 *d)
Time = 0.14 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {\log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{3} d} + \frac {\log \left ({\left | d x + c \right |}\right )}{a^{3} d} + \frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2}}{6 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a^{3} d} \] Input:
integrate(1/(d*x+c)/(a+b*(d*x+c)^3)^3,x, algorithm="giac")
Output:
-1/3*log(abs(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a))/(a^3*d) + log(abs(d*x + c))/(a^3*d) + 1/6*(2*a*b*d^3*x^3 + 6*a*b*c*d^2*x^2 + 6*a* b*c^2*d*x + 2*a*b*c^3 + 3*a^2)/((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)^2*a^3*d)
Time = 1.19 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.80 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {\frac {2\,b\,c^3+3\,a}{6\,a^2\,d}+\frac {b\,d^2\,x^3}{3\,a^2}+\frac {b\,c^2\,x}{a^2}+\frac {b\,c\,d\,x^2}{a^2}}{x^3\,\left (20\,b^2\,c^3\,d^3+2\,a\,b\,d^3\right )+x^2\,\left (15\,b^2\,c^4\,d^2+6\,a\,b\,c\,d^2\right )+a^2+x\,\left (6\,d\,b^2\,c^5+6\,a\,d\,b\,c^2\right )+b^2\,c^6+b^2\,d^6\,x^6+2\,a\,b\,c^3+6\,b^2\,c\,d^5\,x^5+15\,b^2\,c^2\,d^4\,x^4}-\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^3\,d}+\frac {\ln \left (c+d\,x\right )}{a^3\,d} \] Input:
int(1/((a + b*(c + d*x)^3)^3*(c + d*x)),x)
Output:
((3*a + 2*b*c^3)/(6*a^2*d) + (b*d^2*x^3)/(3*a^2) + (b*c^2*x)/a^2 + (b*c*d* x^2)/a^2)/(x^3*(20*b^2*c^3*d^3 + 2*a*b*d^3) + x^2*(15*b^2*c^4*d^2 + 6*a*b* c*d^2) + a^2 + x*(6*b^2*c^5*d + 6*a*b*c^2*d) + b^2*c^6 + b^2*d^6*x^6 + 2*a *b*c^3 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4) - log(a + b*c^3 + b*d^3*x^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2)/(3*a^3*d) + log(c + d*x)/(a^3*d)
Time = 0.22 (sec) , antiderivative size = 1402, normalized size of antiderivative = 17.10 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(d*x+c)/(a+b*(d*x+c)^3)^3,x)
Output:
( - 2*log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a**(1/3)*d*x + b**(2/3 )*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*a**2 - 4*log(a**(2/3) - b* *(1/3)*a**(1/3)*c - b**(1/3)*a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d *x + b**(2/3)*d**2*x**2)*a*b*c**3 - 12*log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2 *x**2)*a*b*c**2*d*x - 12*log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a** (1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*a*b*c*d **2*x**2 - 4*log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*a*b*d**3*x**3 - 2*l og(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*b**2*c**6 - 12*log(a**(2/3) - b** (1/3)*a**(1/3)*c - b**(1/3)*a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d* x + b**(2/3)*d**2*x**2)*b**2*c**5*d*x - 30*log(a**(2/3) - b**(1/3)*a**(1/3 )*c - b**(1/3)*a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)* d**2*x**2)*b**2*c**4*d**2*x**2 - 40*log(a**(2/3) - b**(1/3)*a**(1/3)*c - b **(1/3)*a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x* *2)*b**2*c**3*d**3*x**3 - 30*log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3) *a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*b** 2*c**2*d**4*x**4 - 12*log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a**(1/ 3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*b**2*c*...