\(\int \frac {1}{(c+d x)^4 (a+b (c+d x)^3)^2} \, dx\) [75]

Optimal result

Integrand size = 21, antiderivative size = 80 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {1}{3 a^2 d (c+d x)^3}-\frac {b}{3 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 b \log (c+d x)}{a^3 d}+\frac {2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d} \] Output:

-1/3/a^2/d/(d*x+c)^3-1/3*b/a^2/d/(a+b*(d*x+c)^3)-2*b*ln(d*x+c)/a^3/d+2/3*b 
*ln(a+b*(d*x+c)^3)/a^3/d
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {a \left (\frac {1}{(c+d x)^3}+\frac {b}{a+b (c+d x)^3}\right )+6 b \log (c+d x)-2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d} \] Input:

Integrate[1/((c + d*x)^4*(a + b*(c + d*x)^3)^2),x]
 

Output:

-1/3*(a*((c + d*x)^(-3) + b/(a + b*(c + d*x)^3)) + 6*b*Log[c + d*x] - 2*b* 
Log[a + b*(c + d*x)^3])/(a^3*d)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89, 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.

\(\displaystyle \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx\)

\(\Big \downarrow \) 895

\(\displaystyle \frac {\int \frac {1}{(c+d x)^4 \left (b (c+d x)^3+a\right )^2}d(c+d x)}{d}\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {\int \frac {1}{(c+d x)^6 \left (b (c+d x)^3+a\right )^2}d(c+d x)^3}{3 d}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {\int \left (\frac {2 b^2}{a^3 \left (b (c+d x)^3+a\right )}+\frac {b^2}{a^2 \left (b (c+d x)^3+a\right )^2}-\frac {2 b}{a^3 (c+d x)^3}+\frac {1}{a^2 (c+d x)^6}\right )d(c+d x)^3}{3 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {2 b \log \left ((c+d x)^3\right )}{a^3}+\frac {2 b \log \left (a+b (c+d x)^3\right )}{a^3}-\frac {b}{a^2 \left (a+b (c+d x)^3\right )}-\frac {1}{a^2 (c+d x)^3}}{3 d}\)

Input:

Int[1/((c + d*x)^4*(a + b*(c + d*x)^3)^2),x]
 

Output:

(-(1/(a^2*(c + d*x)^3)) - b/(a^2*(a + b*(c + d*x)^3)) - (2*b*Log[(c + d*x) 
^3])/a^3 + (2*b*Log[a + b*(c + d*x)^3])/a^3)/(3*d)
 

Defintions of rubi rules used

rule 54

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])
 

rule 798

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]]
 

rule 895

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58

method	result	size
default	\(-\frac {1}{3 a^{2} d \left (x d +c \right )^{3}}-\frac {2 b \ln \left (x d +c \right )}{a^{3} d}+\frac {b^{2} \left (-\frac {a}{3 d b \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}+\frac {2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}{3 b d}\right )}{a^{3}}\)	\(126\)
risch	\(\frac {-\frac {2 b \,d^{2} x^{3}}{3 a^{2}}-\frac {2 b c d \,x^{2}}{a^{2}}-\frac {2 b x \,c^{2}}{a^{2}}-\frac {2 c^{3} b +a}{3 d \,a^{2}}}{\left (x d +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}-\frac {2 b \ln \left (x d +c \right )}{a^{3} d}+\frac {2 b \ln \left (-b \,d^{3} x^{3}-3 b c \,d^{2} x^{2}-3 d b x \,c^{2}-c^{3} b -a \right )}{3 a^{3} d}\)	\(157\)
norman	\(\frac {-\frac {2 b c d \,x^{2}}{a^{2}}-\frac {2 b x \,c^{2}}{a^{2}}+\frac {-2 b^{2} c^{3} d^{5}-a b \,d^{5}}{3 a^{2} d^{6} b}-\frac {2 b \,d^{2} x^{3}}{3 a^{2}}}{\left (x d +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}-\frac {2 b \ln \left (x d +c \right )}{a^{3} d}+\frac {2 b \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}{3 a^{3} d}\)	\(167\)
parallelrisch	\(-\frac {18 \ln \left (x d +c \right ) x^{2} a \,b^{2} c \,d^{7}-6 \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^{2} c \,d^{7}+18 \ln \left (x d +c \right ) x a \,b^{2} c^{2} d^{6}-6 \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^{2} c^{2} d^{6}+2 a \,b^{2} c^{3} d^{5}+2 x^{3} a \,b^{2} d^{8}+6 \ln \left (x d +c \right ) x^{6} b^{3} 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^{3} d^{11}+6 \ln \left (x d +c \right ) b^{3} 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^{3} c^{6} d^{5}+36 \ln \left (x d +c \right ) x^{5} b^{3} 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^{3} c \,d^{10}+90 \ln \left (x d +c \right ) x^{4} b^{3} 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^{3} c^{2} d^{9}+120 \ln \left (x d +c \right ) x^{3} b^{3} 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^{3} c^{3} d^{8}+90 \ln \left (x d +c \right ) x^{2} b^{3} 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^{3} c^{4} d^{7}+36 \ln \left (x d +c \right ) x \,b^{3} 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^{3} c^{5} d^{6}+6 \ln \left (x d +c \right ) x^{3} a \,b^{2} d^{8}-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^{3} a \,b^{2} d^{8}+6 \ln \left (x d +c \right ) a \,b^{2} 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 \,b^{2} c^{3} d^{5}+6 x^{2} a \,b^{2} c \,d^{7}+6 x a \,b^{2} c^{2} d^{6}+a^{2} b \,d^{5}}{3 a^{3} b \,d^{6} \left (x d +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}\)	\(829\)

Input:

int(1/(d*x+c)^4/(a+b*(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/3/a^2/d/(d*x+c)^3-2*b*ln(d*x+c)/a^3/d+b^2/a^3*(-1/3*a/d/b/(b*d^3*x^3+3* 
b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)+2/3/b/d*ln(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^ 
2*d*x+b*c^3+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (74) = 148\).

Time = 0.08 (sec) , antiderivative size = 431, normalized size of antiderivative = 5.39 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + a^{2} - 2 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + {\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \, {\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + {\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \, {\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (d x + c\right )}{3 \, {\left (a^{3} b d^{7} x^{6} + 6 \, a^{3} b c d^{6} x^{5} + 15 \, a^{3} b c^{2} d^{5} x^{4} + {\left (20 \, a^{3} b c^{3} + a^{4}\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{3} b c^{4} + a^{4} c\right )} d^{3} x^{2} + 3 \, {\left (2 \, a^{3} b c^{5} + a^{4} c^{2}\right )} d^{2} x + {\left (a^{3} b c^{6} + a^{4} c^{3}\right )} d\right )}} \] Input:

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3)^2,x, algorithm="fricas")
 

Output:

-1/3*(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 + 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 + (20*b^2* 
c^3 + a*b)*d^3*x^3 + a*b*c^3 + 3*(5*b^2*c^4 + a*b*c)*d^2*x^2 + 3*(2*b^2*c^ 
5 + a*b*c^2)*d*x)*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 + (20*b 
^2*c^3 + a*b)*d^3*x^3 + a*b*c^3 + 3*(5*b^2*c^4 + a*b*c)*d^2*x^2 + 3*(2*b^2 
*c^5 + a*b*c^2)*d*x)*log(d*x + c))/(a^3*b*d^7*x^6 + 6*a^3*b*c*d^6*x^5 + 15 
*a^3*b*c^2*d^5*x^4 + (20*a^3*b*c^3 + a^4)*d^4*x^3 + 3*(5*a^3*b*c^4 + a^4*c 
)*d^3*x^2 + 3*(2*a^3*b*c^5 + a^4*c^2)*d^2*x + (a^3*b*c^6 + a^4*c^3)*d)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (70) = 140\).

Time = 2.14 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.12 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=\frac {- a - 2 b c^{3} - 6 b c^{2} d x - 6 b c d^{2} x^{2} - 2 b d^{3} x^{3}}{3 a^{3} c^{3} d + 3 a^{2} b c^{6} d + 45 a^{2} b c^{2} d^{5} x^{4} + 18 a^{2} b c d^{6} x^{5} + 3 a^{2} b d^{7} x^{6} + x^{3} \cdot \left (3 a^{3} d^{4} + 60 a^{2} b c^{3} d^{4}\right ) + x^{2} \cdot \left (9 a^{3} c d^{3} + 45 a^{2} b c^{4} d^{3}\right ) + x \left (9 a^{3} c^{2} d^{2} + 18 a^{2} b c^{5} d^{2}\right )} - \frac {2 b \log {\left (\frac {c}{d} + x \right )}}{a^{3} d} + \frac {2 b \log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{3} d} \] Input:

integrate(1/(d*x+c)**4/(a+b*(d*x+c)**3)**2,x)
 

Output:

(-a - 2*b*c**3 - 6*b*c**2*d*x - 6*b*c*d**2*x**2 - 2*b*d**3*x**3)/(3*a**3*c 
**3*d + 3*a**2*b*c**6*d + 45*a**2*b*c**2*d**5*x**4 + 18*a**2*b*c*d**6*x**5 
 + 3*a**2*b*d**7*x**6 + x**3*(3*a**3*d**4 + 60*a**2*b*c**3*d**4) + x**2*(9 
*a**3*c*d**3 + 45*a**2*b*c**4*d**3) + x*(9*a**3*c**2*d**2 + 18*a**2*b*c**5 
*d**2)) - 2*b*log(c/d + x)/(a**3*d) + 2*b*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)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (74) = 148\).

Time = 0.04 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.78 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + a}{3 \, {\left (a^{2} b d^{7} x^{6} + 6 \, a^{2} b c d^{6} x^{5} + 15 \, a^{2} b c^{2} d^{5} x^{4} + {\left (20 \, a^{2} b c^{3} + a^{3}\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{2} b c^{4} + a^{3} c\right )} d^{3} x^{2} + 3 \, {\left (2 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d^{2} x + {\left (a^{2} b c^{6} + a^{3} c^{3}\right )} d\right )}} + \frac {2 \, b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d} - \frac {2 \, b \log \left (d x + c\right )}{a^{3} d} \] Input:

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3)^2,x, algorithm="maxima")
 

Output:

-1/3*(2*b*d^3*x^3 + 6*b*c*d^2*x^2 + 6*b*c^2*d*x + 2*b*c^3 + a)/(a^2*b*d^7* 
x^6 + 6*a^2*b*c*d^6*x^5 + 15*a^2*b*c^2*d^5*x^4 + (20*a^2*b*c^3 + a^3)*d^4* 
x^3 + 3*(5*a^2*b*c^4 + a^3*c)*d^3*x^2 + 3*(2*a^2*b*c^5 + a^3*c^2)*d^2*x + 
(a^2*b*c^6 + a^3*c^3)*d) + 2/3*b*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) - 2*b*log(d*x + c)/(a^3*d)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=\frac {2 \, b \log \left ({\left | -b - \frac {a}{{\left (d x + c\right )}^{3}} \right |}\right )}{3 \, a^{3} d} + \frac {b^{2}}{3 \, a^{3} {\left (b + \frac {a}{{\left (d x + c\right )}^{3}}\right )} d} - \frac {1}{3 \, {\left (d x + c\right )}^{3} a^{2} d} \] Input:

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3)^2,x, algorithm="giac")
 

Output:

2/3*b*log(abs(-b - a/(d*x + c)^3))/(a^3*d) + 1/3*b^2/(a^3*(b + a/(d*x + c) 
^3)*d) - 1/3/((d*x + c)^3*a^2*d)
 

Mupad [B] (verification not implemented)

Time = 1.15 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.64 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=\frac {2\,b\,\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^3\,d}-\frac {\frac {2\,b\,c^3+a}{3\,a^2\,d}+\frac {2\,b\,d^2\,x^3}{3\,a^2}+\frac {2\,b\,c^2\,x}{a^2}+\frac {2\,b\,c\,d\,x^2}{a^2}}{x\,\left (6\,b\,d\,c^5+3\,a\,d\,c^2\right )+x^3\,\left (20\,b\,c^3\,d^3+a\,d^3\right )+a\,c^3+b\,c^6+x^2\,\left (15\,b\,c^4\,d^2+3\,a\,c\,d^2\right )+b\,d^6\,x^6+15\,b\,c^2\,d^4\,x^4+6\,b\,c\,d^5\,x^5}-\frac {2\,b\,\ln \left (c+d\,x\right )}{a^3\,d} \] Input:

int(1/((a + b*(c + d*x)^3)^2*(c + d*x)^4),x)
 

Output:

(2*b*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) - 
 ((a + 2*b*c^3)/(3*a^2*d) + (2*b*d^2*x^3)/(3*a^2) + (2*b*c^2*x)/a^2 + (2*b 
*c*d*x^2)/a^2)/(x*(3*a*c^2*d + 6*b*c^5*d) + x^3*(a*d^3 + 20*b*c^3*d^3) + a 
*c^3 + b*c^6 + x^2*(15*b*c^4*d^2 + 3*a*c*d^2) + b*d^6*x^6 + 15*b*c^2*d^4*x 
^4 + 6*b*c*d^5*x^5) - (2*b*log(c + d*x))/(a^3*d)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1293, normalized size of antiderivative = 16.16 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(d*x+c)^4/(a+b*(d*x+c)^3)^2,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*b*c**3 + 6*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 + 6*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 + 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 
*b*d**3*x**3 + 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)*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*d**5*x**5 + 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)*...