Integrand size = 25, antiderivative size = 173 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3} \, dx=\frac {1}{4374 c^6 d (c-3 d x)^2}+\frac {2}{2187 c^7 d (c-3 d x)}-\frac {1}{405 c^3 d (2 c+3 d x)^5}-\frac {1}{324 c^4 d (2 c+3 d x)^4}-\frac {2}{729 c^5 d (2 c+3 d x)^3}-\frac {5}{2187 c^6 d (2 c+3 d x)^2}-\frac {5}{2187 c^7 d (2 c+3 d x)}-\frac {7 \log (c-3 d x)}{6561 c^8 d}+\frac {7 \log (2 c+3 d x)}{6561 c^8 d} \] Output:
1/4374/c^6/d/(-3*d*x+c)^2+2/2187/c^7/d/(-3*d*x+c)-1/405/c^3/d/(3*d*x+2*c)^ 5-1/324/c^4/d/(3*d*x+2*c)^4-2/729/c^5/d/(3*d*x+2*c)^3-5/2187/c^6/d/(3*d*x+ 2*c)^2-5/2187/c^7/d/(3*d*x+2*c)-7/6561*ln(-3*d*x+c)/c^8/d+7/6561*ln(3*d*x+ 2*c)/c^8/d
Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3} \, dx=\frac {-\frac {3 c \left (1658 c^6-12663 c^5 d x-40068 c^4 d^2 x^2+23625 c^3 d^3 x^3+204120 c^2 d^4 x^4+255150 c d^5 x^5+102060 d^6 x^6\right )}{(c-3 d x)^2 (2 c+3 d x)^5}-140 \log (c-3 d x)+140 \log (2 c+3 d x)}{131220 c^8 d} \] Input:
Integrate[(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(-3),x]
Output:
((-3*c*(1658*c^6 - 12663*c^5*d*x - 40068*c^4*d^2*x^2 + 23625*c^3*d^3*x^3 + 204120*c^2*d^4*x^4 + 255150*c*d^5*x^5 + 102060*d^6*x^6))/((c - 3*d*x)^2*( 2*c + 3*d*x)^5) - 140*Log[c - 3*d*x] + 140*Log[2*c + 3*d*x])/(131220*c^8*d )
Time = 0.34 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2462, 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.
| \(\displaystyle \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {7}{2187 c^8 (2 c+3 d x)}+\frac {7}{2187 c^8 (c-3 d x)}+\frac {5}{729 c^7 (2 c+3 d x)^2}+\frac {2}{729 c^7 (c-3 d x)^2}+\frac {10}{729 c^6 (2 c+3 d x)^3}+\frac {1}{729 c^6 (c-3 d x)^3}+\frac {2}{81 c^5 (2 c+3 d x)^4}+\frac {1}{27 c^4 (2 c+3 d x)^5}+\frac {1}{27 c^3 (2 c+3 d x)^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {7 \log (c-3 d x)}{6561 c^8 d}+\frac {7 \log (2 c+3 d x)}{6561 c^8 d}+\frac {2}{2187 c^7 d (c-3 d x)}-\frac {5}{2187 c^7 d (2 c+3 d x)}+\frac {1}{4374 c^6 d (c-3 d x)^2}-\frac {5}{2187 c^6 d (2 c+3 d x)^2}-\frac {2}{729 c^5 d (2 c+3 d x)^3}-\frac {1}{324 c^4 d (2 c+3 d x)^4}-\frac {1}{405 c^3 d (2 c+3 d x)^5}\) |
Input:
Int[(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(-3),x]
Output:
1/(4374*c^6*d*(c - 3*d*x)^2) + 2/(2187*c^7*d*(c - 3*d*x)) - 1/(405*c^3*d*( 2*c + 3*d*x)^5) - 1/(324*c^4*d*(2*c + 3*d*x)^4) - 2/(729*c^5*d*(2*c + 3*d* x)^3) - 5/(2187*c^6*d*(2*c + 3*d*x)^2) - 5/(2187*c^7*d*(2*c + 3*d*x)) - (7 *Log[c - 3*d*x])/(6561*c^8*d) + (7*Log[2*c + 3*d*x])/(6561*c^8*d)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.17 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.73
| method | result | size |
| norman | \(\frac {-\frac {505}{2187 c d}+\frac {1141 d \,x^{2}}{324 c^{3}}+\frac {3871 d^{2} x^{3}}{648 c^{4}}-\frac {917 d^{3} x^{4}}{96 c^{5}}-\frac {35147 d^{4} x^{5}}{960 c^{6}}-\frac {4501 d^{5} x^{6}}{120 c^{7}}-\frac {4221 d^{6} x^{7}}{320 c^{8}}}{\left (3 d x +2 c \right )^{5} \left (-3 d x +c \right )^{2}}-\frac {7 \ln \left (-3 d x +c \right )}{6561 c^{8} d}+\frac {7 \ln \left (3 d x +2 c \right )}{6561 c^{8} d}\) | \(126\) |
| risch | \(\frac {-\frac {7 d^{5} x^{6}}{3 c^{7}}-\frac {35 d^{4} x^{5}}{6 c^{6}}-\frac {14 d^{3} x^{4}}{3 c^{5}}-\frac {175 d^{2} x^{3}}{324 c^{4}}+\frac {371 d \,x^{2}}{405 c^{3}}+\frac {469 x}{1620 c^{2}}-\frac {829}{21870 c d}}{\left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{2} \left (3 d x +2 c \right )}-\frac {7 \ln \left (-3 d x +c \right )}{6561 c^{8} d}+\frac {7 \ln \left (3 d x +2 c \right )}{6561 c^{8} d}\) | \(139\) |
| default | \(\frac {1}{4374 c^{6} d \left (-3 d x +c \right )^{2}}+\frac {2}{2187 c^{7} d \left (-3 d x +c \right )}-\frac {1}{405 c^{3} d \left (3 d x +2 c \right )^{5}}-\frac {1}{324 c^{4} d \left (3 d x +2 c \right )^{4}}-\frac {2}{729 c^{5} d \left (3 d x +2 c \right )^{3}}-\frac {5}{2187 c^{6} d \left (3 d x +2 c \right )^{2}}-\frac {5}{2187 c^{7} d \left (3 d x +2 c \right )}-\frac {7 \ln \left (-3 d x +c \right )}{6561 c^{8} d}+\frac {7 \ln \left (3 d x +2 c \right )}{6561 c^{8} d}\) | \(156\) |
| parallelrisch | \(-\frac {4898880 \ln \left (d x -\frac {c}{3}\right ) x^{7} d^{7}+13063680 \ln \left (d x -\frac {c}{3}\right ) x^{6} c \,d^{6}-13063680 \ln \left (d x +\frac {2 c}{3}\right ) x^{6} c \,d^{6}+11430720 \ln \left (d x -\frac {c}{3}\right ) x^{5} c^{2} d^{5}-11430720 \ln \left (d x +\frac {2 c}{3}\right ) x^{5} c^{2} d^{5}+1814400 \ln \left (d x -\frac {c}{3}\right ) x^{4} c^{3} d^{4}-1814400 \ln \left (d x +\frac {2 c}{3}\right ) x^{4} c^{3} d^{4}-2419200 \ln \left (d x -\frac {c}{3}\right ) x^{3} c^{4} d^{3}+2419200 \ln \left (d x +\frac {2 c}{3}\right ) x^{3} c^{4} d^{3}-967680 \ln \left (d x -\frac {c}{3}\right ) x^{2} c^{5} d^{2}+967680 \ln \left (d x +\frac {2 c}{3}\right ) x^{2} c^{5} d^{2}+107520 \ln \left (d x -\frac {c}{3}\right ) x \,c^{6} d -107520 \ln \left (d x +\frac {2 c}{3}\right ) x \,c^{6} d -5439069 d^{7} x^{7}+71680 \ln \left (d x -\frac {c}{3}\right ) c^{7}-71680 \ln \left (d x +\frac {2 c}{3}\right ) c^{7}-4898880 \ln \left (d x +\frac {2 c}{3}\right ) x^{7} d^{7}-9605304 c \,d^{6} x^{6}-443961 c^{2} d^{5} x^{5}+7783290 c^{3} d^{4} x^{4}+3819960 c^{4} d^{3} x^{3}-848880 c^{5} d^{2} x^{2}-727200 c^{6} d x}{2099520 c^{8} \left (27 d^{3} x^{3}+27 c \,d^{2} x^{2}-4 c^{3}\right )^{2} \left (3 d x +2 c \right ) d}\) | \(387\) |
Input:
int(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^3,x,method=_RETURNVERBOSE)
Output:
(-505/2187/c/d+1141/324*d/c^3*x^2+3871/648*d^2/c^4*x^3-917/96*d^3/c^5*x^4- 35147/960*d^4/c^6*x^5-4501/120*d^5/c^7*x^6-4221/320*d^6/c^8*x^7)/(3*d*x+2* c)^5/(-3*d*x+c)^2-7/6561*ln(-3*d*x+c)/c^8/d+7/6561*ln(3*d*x+2*c)/c^8/d
Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (159) = 318\).
Time = 0.09 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.86 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3} \, dx=-\frac {306180 \, c d^{6} x^{6} + 765450 \, c^{2} d^{5} x^{5} + 612360 \, c^{3} d^{4} x^{4} + 70875 \, c^{4} d^{3} x^{3} - 120204 \, c^{5} d^{2} x^{2} - 37989 \, c^{6} d x + 4974 \, c^{7} - 140 \, {\left (2187 \, d^{7} x^{7} + 5832 \, c d^{6} x^{6} + 5103 \, c^{2} d^{5} x^{5} + 810 \, c^{3} d^{4} x^{4} - 1080 \, c^{4} d^{3} x^{3} - 432 \, c^{5} d^{2} x^{2} + 48 \, c^{6} d x + 32 \, c^{7}\right )} \log \left (3 \, d x + 2 \, c\right ) + 140 \, {\left (2187 \, d^{7} x^{7} + 5832 \, c d^{6} x^{6} + 5103 \, c^{2} d^{5} x^{5} + 810 \, c^{3} d^{4} x^{4} - 1080 \, c^{4} d^{3} x^{3} - 432 \, c^{5} d^{2} x^{2} + 48 \, c^{6} d x + 32 \, c^{7}\right )} \log \left (3 \, d x - c\right )}{131220 \, {\left (2187 \, c^{8} d^{8} x^{7} + 5832 \, c^{9} d^{7} x^{6} + 5103 \, c^{10} d^{6} x^{5} + 810 \, c^{11} d^{5} x^{4} - 1080 \, c^{12} d^{4} x^{3} - 432 \, c^{13} d^{3} x^{2} + 48 \, c^{14} d^{2} x + 32 \, c^{15} d\right )}} \] Input:
integrate(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^3,x, algorithm="fricas")
Output:
-1/131220*(306180*c*d^6*x^6 + 765450*c^2*d^5*x^5 + 612360*c^3*d^4*x^4 + 70 875*c^4*d^3*x^3 - 120204*c^5*d^2*x^2 - 37989*c^6*d*x + 4974*c^7 - 140*(218 7*d^7*x^7 + 5832*c*d^6*x^6 + 5103*c^2*d^5*x^5 + 810*c^3*d^4*x^4 - 1080*c^4 *d^3*x^3 - 432*c^5*d^2*x^2 + 48*c^6*d*x + 32*c^7)*log(3*d*x + 2*c) + 140*( 2187*d^7*x^7 + 5832*c*d^6*x^6 + 5103*c^2*d^5*x^5 + 810*c^3*d^4*x^4 - 1080* c^4*d^3*x^3 - 432*c^5*d^2*x^2 + 48*c^6*d*x + 32*c^7)*log(3*d*x - c))/(2187 *c^8*d^8*x^7 + 5832*c^9*d^7*x^6 + 5103*c^10*d^6*x^5 + 810*c^11*d^5*x^4 - 1 080*c^12*d^4*x^3 - 432*c^13*d^3*x^2 + 48*c^14*d^2*x + 32*c^15*d)
Time = 0.48 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3} \, dx=- \frac {1658 c^{6} - 12663 c^{5} d x - 40068 c^{4} d^{2} x^{2} + 23625 c^{3} d^{3} x^{3} + 204120 c^{2} d^{4} x^{4} + 255150 c d^{5} x^{5} + 102060 d^{6} x^{6}}{1399680 c^{14} d + 2099520 c^{13} d^{2} x - 18895680 c^{12} d^{3} x^{2} - 47239200 c^{11} d^{4} x^{3} + 35429400 c^{10} d^{5} x^{4} + 223205220 c^{9} d^{6} x^{5} + 255091680 c^{8} d^{7} x^{6} + 95659380 c^{7} d^{8} x^{7}} - \frac {\frac {7 \log {\left (- \frac {c}{3 d} + x \right )}}{6561} - \frac {7 \log {\left (\frac {2 c}{3 d} + x \right )}}{6561}}{c^{8} d} \] Input:
integrate(1/(-27*d**3*x**3-27*c*d**2*x**2+4*c**3)**3,x)
Output:
-(1658*c**6 - 12663*c**5*d*x - 40068*c**4*d**2*x**2 + 23625*c**3*d**3*x**3 + 204120*c**2*d**4*x**4 + 255150*c*d**5*x**5 + 102060*d**6*x**6)/(1399680 *c**14*d + 2099520*c**13*d**2*x - 18895680*c**12*d**3*x**2 - 47239200*c**1 1*d**4*x**3 + 35429400*c**10*d**5*x**4 + 223205220*c**9*d**6*x**5 + 255091 680*c**8*d**7*x**6 + 95659380*c**7*d**8*x**7) - (7*log(-c/(3*d) + x)/6561 - 7*log(2*c/(3*d) + x)/6561)/(c**8*d)
Time = 0.04 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3} \, dx=-\frac {102060 \, d^{6} x^{6} + 255150 \, c d^{5} x^{5} + 204120 \, c^{2} d^{4} x^{4} + 23625 \, c^{3} d^{3} x^{3} - 40068 \, c^{4} d^{2} x^{2} - 12663 \, c^{5} d x + 1658 \, c^{6}}{43740 \, {\left (2187 \, c^{7} d^{8} x^{7} + 5832 \, c^{8} d^{7} x^{6} + 5103 \, c^{9} d^{6} x^{5} + 810 \, c^{10} d^{5} x^{4} - 1080 \, c^{11} d^{4} x^{3} - 432 \, c^{12} d^{3} x^{2} + 48 \, c^{13} d^{2} x + 32 \, c^{14} d\right )}} + \frac {7 \, \log \left (3 \, d x + 2 \, c\right )}{6561 \, c^{8} d} - \frac {7 \, \log \left (3 \, d x - c\right )}{6561 \, c^{8} d} \] Input:
integrate(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^3,x, algorithm="maxima")
Output:
-1/43740*(102060*d^6*x^6 + 255150*c*d^5*x^5 + 204120*c^2*d^4*x^4 + 23625*c ^3*d^3*x^3 - 40068*c^4*d^2*x^2 - 12663*c^5*d*x + 1658*c^6)/(2187*c^7*d^8*x ^7 + 5832*c^8*d^7*x^6 + 5103*c^9*d^6*x^5 + 810*c^10*d^5*x^4 - 1080*c^11*d^ 4*x^3 - 432*c^12*d^3*x^2 + 48*c^13*d^2*x + 32*c^14*d) + 7/6561*log(3*d*x + 2*c)/(c^8*d) - 7/6561*log(3*d*x - c)/(c^8*d)
Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3} \, dx=\frac {7 \, \log \left ({\left | 3 \, d x + 2 \, c \right |}\right )}{6561 \, c^{8} d} - \frac {7 \, \log \left ({\left | 3 \, d x - c \right |}\right )}{6561 \, c^{8} d} - \frac {102060 \, c d^{6} x^{6} + 255150 \, c^{2} d^{5} x^{5} + 204120 \, c^{3} d^{4} x^{4} + 23625 \, c^{4} d^{3} x^{3} - 40068 \, c^{5} d^{2} x^{2} - 12663 \, c^{6} d x + 1658 \, c^{7}}{43740 \, {\left (3 \, d x + 2 \, c\right )}^{5} {\left (3 \, d x - c\right )}^{2} c^{8} d} \] Input:
integrate(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^3,x, algorithm="giac")
Output:
7/6561*log(abs(3*d*x + 2*c))/(c^8*d) - 7/6561*log(abs(3*d*x - c))/(c^8*d) - 1/43740*(102060*c*d^6*x^6 + 255150*c^2*d^5*x^5 + 204120*c^3*d^4*x^4 + 23 625*c^4*d^3*x^3 - 40068*c^5*d^2*x^2 - 12663*c^6*d*x + 1658*c^7)/((3*d*x + 2*c)^5*(3*d*x - c)^2*c^8*d)
Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3} \, dx=\frac {14\,\mathrm {atanh}\left (\frac {2\,d\,x}{c}+\frac {1}{3}\right )}{6561\,c^8\,d}-\frac {\frac {829}{21870\,c\,d}-\frac {469\,x}{1620\,c^2}-\frac {371\,d\,x^2}{405\,c^3}+\frac {175\,d^2\,x^3}{324\,c^4}+\frac {14\,d^3\,x^4}{3\,c^5}+\frac {35\,d^4\,x^5}{6\,c^6}+\frac {7\,d^5\,x^6}{3\,c^7}}{32\,c^7+48\,c^6\,d\,x-432\,c^5\,d^2\,x^2-1080\,c^4\,d^3\,x^3+810\,c^3\,d^4\,x^4+5103\,c^2\,d^5\,x^5+5832\,c\,d^6\,x^6+2187\,d^7\,x^7} \] Input:
int(-1/(27*d^3*x^3 - 4*c^3 + 27*c*d^2*x^2)^3,x)
Output:
(14*atanh((2*d*x)/c + 1/3))/(6561*c^8*d) - (829/(21870*c*d) - (469*x)/(162 0*c^2) - (371*d*x^2)/(405*c^3) + (175*d^2*x^3)/(324*c^4) + (14*d^3*x^4)/(3 *c^5) + (35*d^4*x^5)/(6*c^6) + (7*d^5*x^6)/(3*c^7))/(32*c^7 + 2187*d^7*x^7 + 5832*c*d^6*x^6 - 432*c^5*d^2*x^2 - 1080*c^4*d^3*x^3 + 810*c^3*d^4*x^4 + 5103*c^2*d^5*x^5 + 48*c^6*d*x)
Time = 0.17 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.45 \[ \int \frac {1}{\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3} \, dx=\frac {8960 \,\mathrm {log}\left (3 d x +2 c \right ) c^{7}+13440 \,\mathrm {log}\left (3 d x +2 c \right ) c^{6} d x -120960 \,\mathrm {log}\left (3 d x +2 c \right ) c^{5} d^{2} x^{2}-302400 \,\mathrm {log}\left (3 d x +2 c \right ) c^{4} d^{3} x^{3}+226800 \,\mathrm {log}\left (3 d x +2 c \right ) c^{3} d^{4} x^{4}+1428840 \,\mathrm {log}\left (3 d x +2 c \right ) c^{2} d^{5} x^{5}+1632960 \,\mathrm {log}\left (3 d x +2 c \right ) c \,d^{6} x^{6}+612360 \,\mathrm {log}\left (3 d x +2 c \right ) d^{7} x^{7}-8960 \,\mathrm {log}\left (-3 d x +c \right ) c^{7}-13440 \,\mathrm {log}\left (-3 d x +c \right ) c^{6} d x +120960 \,\mathrm {log}\left (-3 d x +c \right ) c^{5} d^{2} x^{2}+302400 \,\mathrm {log}\left (-3 d x +c \right ) c^{4} d^{3} x^{3}-226800 \,\mathrm {log}\left (-3 d x +c \right ) c^{3} d^{4} x^{4}-1428840 \,\mathrm {log}\left (-3 d x +c \right ) c^{2} d^{5} x^{5}-1632960 \,\mathrm {log}\left (-3 d x +c \right ) c \,d^{6} x^{6}-612360 \,\mathrm {log}\left (-3 d x +c \right ) d^{7} x^{7}-6588 c^{7}+81018 c^{6} d x +195048 c^{5} d^{2} x^{2}-255150 c^{4} d^{3} x^{3}-1139670 c^{3} d^{4} x^{4}-995085 c^{2} d^{5} x^{5}+229635 d^{7} x^{7}}{262440 c^{8} d \left (2187 d^{7} x^{7}+5832 c \,d^{6} x^{6}+5103 c^{2} d^{5} x^{5}+810 c^{3} d^{4} x^{4}-1080 c^{4} d^{3} x^{3}-432 c^{5} d^{2} x^{2}+48 c^{6} d x +32 c^{7}\right )} \] Input:
int(1/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^3,x)
Output:
(8960*log(2*c + 3*d*x)*c**7 + 13440*log(2*c + 3*d*x)*c**6*d*x - 120960*log (2*c + 3*d*x)*c**5*d**2*x**2 - 302400*log(2*c + 3*d*x)*c**4*d**3*x**3 + 22 6800*log(2*c + 3*d*x)*c**3*d**4*x**4 + 1428840*log(2*c + 3*d*x)*c**2*d**5* x**5 + 1632960*log(2*c + 3*d*x)*c*d**6*x**6 + 612360*log(2*c + 3*d*x)*d**7 *x**7 - 8960*log(c - 3*d*x)*c**7 - 13440*log(c - 3*d*x)*c**6*d*x + 120960* log(c - 3*d*x)*c**5*d**2*x**2 + 302400*log(c - 3*d*x)*c**4*d**3*x**3 - 226 800*log(c - 3*d*x)*c**3*d**4*x**4 - 1428840*log(c - 3*d*x)*c**2*d**5*x**5 - 1632960*log(c - 3*d*x)*c*d**6*x**6 - 612360*log(c - 3*d*x)*d**7*x**7 - 6 588*c**7 + 81018*c**6*d*x + 195048*c**5*d**2*x**2 - 255150*c**4*d**3*x**3 - 1139670*c**3*d**4*x**4 - 995085*c**2*d**5*x**5 + 229635*d**7*x**7)/(2624 40*c**8*d*(32*c**7 + 48*c**6*d*x - 432*c**5*d**2*x**2 - 1080*c**4*d**3*x** 3 + 810*c**3*d**4*x**4 + 5103*c**2*d**5*x**5 + 5832*c*d**6*x**6 + 2187*d** 7*x**7))