Integrand size = 46, antiderivative size = 111 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{(c+d x)^5} \, dx=-\frac {(b c-a d)^4}{4 d^5 (c+d x)^4}+\frac {4 b (b c-a d)^3}{3 d^5 (c+d x)^3}-\frac {3 b^2 (b c-a d)^2}{d^5 (c+d x)^2}+\frac {4 b^3 (b c-a d)}{d^5 (c+d x)}+\frac {b^4 \log (c+d x)}{d^5} \] Output:
-1/4*(-a*d+b*c)^4/d^5/(d*x+c)^4+4/3*b*(-a*d+b*c)^3/d^5/(d*x+c)^3-3*b^2*(-a *d+b*c)^2/d^5/(d*x+c)^2+4*b^3*(-a*d+b*c)/d^5/(d*x+c)+b^4*ln(d*x+c)/d^5
Time = 0.01 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{(c+d x)^5} \, dx=\frac {\frac {(b c-a d) \left (3 a^3 d^3+a^2 b d^2 (7 c+16 d x)+a b^2 d \left (13 c^2+40 c d x+36 d^2 x^2\right )+b^3 \left (25 c^3+88 c^2 d x+108 c d^2 x^2+48 d^3 x^3\right )\right )}{(c+d x)^4}+12 b^4 \log (c+d x)}{12 d^5} \] Input:
Integrate[(a^4 + 4*a^3*b*x + 6*a^2*b^2*x^2 + 4*a*b^3*x^3 + b^4*x^4)/(c + d *x)^5,x]
Output:
(((b*c - a*d)*(3*a^3*d^3 + a^2*b*d^2*(7*c + 16*d*x) + a*b^2*d*(13*c^2 + 40 *c*d*x + 36*d^2*x^2) + b^3*(25*c^3 + 88*c^2*d*x + 108*c*d^2*x^2 + 48*d^3*x ^3)))/(c + d*x)^4 + 12*b^4*Log[c + d*x])/(12*d^5)
Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2006, 49, 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.
\(\displaystyle \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{(c+d x)^5} \, dx\) |
\(\Big \downarrow \) 2006 |
\(\displaystyle \int \frac {(a+b x)^4}{(c+d x)^5}dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (-\frac {4 b^3 (b c-a d)}{d^4 (c+d x)^2}+\frac {6 b^2 (b c-a d)^2}{d^4 (c+d x)^3}-\frac {4 b (b c-a d)^3}{d^4 (c+d x)^4}+\frac {(a d-b c)^4}{d^4 (c+d x)^5}+\frac {b^4}{d^4 (c+d x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 b^3 (b c-a d)}{d^5 (c+d x)}-\frac {3 b^2 (b c-a d)^2}{d^5 (c+d x)^2}+\frac {4 b (b c-a d)^3}{3 d^5 (c+d x)^3}-\frac {(b c-a d)^4}{4 d^5 (c+d x)^4}+\frac {b^4 \log (c+d x)}{d^5}\) |
Input:
Int[(a^4 + 4*a^3*b*x + 6*a^2*b^2*x^2 + 4*a*b^3*x^3 + b^4*x^4)/(c + d*x)^5, x]
Output:
-1/4*(b*c - a*d)^4/(d^5*(c + d*x)^4) + (4*b*(b*c - a*d)^3)/(3*d^5*(c + d*x )^3) - (3*b^2*(b*c - a*d)^2)/(d^5*(c + d*x)^2) + (4*b^3*(b*c - a*d))/(d^5* (c + d*x)) + (b^4*Log[c + d*x])/d^5
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Time = 0.10 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.59
method | result | size |
risch | \(\frac {-\frac {4 b^{3} \left (a d -b c \right ) x^{3}}{d^{2}}-\frac {3 b^{2} \left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right ) x^{2}}{d^{3}}-\frac {2 b \left (2 a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -11 b^{3} c^{3}\right ) x}{3 d^{4}}-\frac {3 d^{4} a^{4}+4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+12 a \,b^{3} c^{3} d -25 c^{4} b^{4}}{12 d^{5}}}{\left (x d +c \right )^{4}}+\frac {b^{4} \ln \left (x d +c \right )}{d^{5}}\) | \(176\) |
norman | \(\frac {-\frac {3 d^{4} a^{4}+4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+12 a \,b^{3} c^{3} d -25 c^{4} b^{4}}{12 d^{5}}-\frac {4 \left (a \,b^{3} d -b^{4} c \right ) x^{3}}{d^{2}}-\frac {3 \left (a^{2} b^{2} d^{2}+2 a \,b^{3} c d -3 b^{4} c^{2}\right ) x^{2}}{d^{3}}-\frac {2 \left (2 a^{3} b \,d^{3}+3 a^{2} b^{2} c \,d^{2}+6 a \,b^{3} c^{2} d -11 b^{4} c^{3}\right ) x}{3 d^{4}}}{\left (x d +c \right )^{4}}+\frac {b^{4} \ln \left (x d +c \right )}{d^{5}}\) | \(182\) |
default | \(-\frac {4 b^{3} \left (a d -b c \right )}{d^{5} \left (x d +c \right )}+\frac {b^{4} \ln \left (x d +c \right )}{d^{5}}-\frac {3 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{5} \left (x d +c \right )^{2}}-\frac {d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}{4 d^{5} \left (x d +c \right )^{4}}-\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 d^{5} \left (x d +c \right )^{3}}\) | \(184\) |
parallelrisch | \(\frac {12 \ln \left (x d +c \right ) x^{4} b^{4} d^{4}+48 \ln \left (x d +c \right ) x^{3} b^{4} c \,d^{3}+72 \ln \left (x d +c \right ) x^{2} b^{4} c^{2} d^{2}-48 a \,b^{3} d^{4} x^{3}+48 b^{4} c \,d^{3} x^{3}+48 \ln \left (x d +c \right ) x \,b^{4} c^{3} d -36 a^{2} b^{2} d^{4} x^{2}-72 a \,b^{3} c \,d^{3} x^{2}+108 b^{4} c^{2} d^{2} x^{2}+12 \ln \left (x d +c \right ) b^{4} c^{4}-16 a^{3} b \,d^{4} x -24 a^{2} b^{2} c \,d^{3} x -48 a \,b^{3} c^{2} d^{2} x +88 b^{4} c^{3} d x -3 d^{4} a^{4}-4 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-12 a \,b^{3} c^{3} d +25 c^{4} b^{4}}{12 d^{5} \left (x d +c \right )^{4}}\) | \(260\) |
Input:
int((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(d*x+c)^5,x,method=_ RETURNVERBOSE)
Output:
(-4*b^3*(a*d-b*c)/d^2*x^3-3*b^2*(a^2*d^2+2*a*b*c*d-3*b^2*c^2)/d^3*x^2-2/3* b*(2*a^3*d^3+3*a^2*b*c*d^2+6*a*b^2*c^2*d-11*b^3*c^3)/d^4*x-1/12*(3*a^4*d^4 +4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2+12*a*b^3*c^3*d-25*b^4*c^4)/d^5)/(d*x+c)^4 +b^4*ln(d*x+c)/d^5
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (107) = 214\).
Time = 0.07 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.41 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{(c+d x)^5} \, dx=\frac {25 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4} + 48 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 36 \, {\left (3 \, b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} - a^{2} b^{2} d^{4}\right )} x^{2} + 8 \, {\left (11 \, b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} x + 12 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4}\right )} \log \left (d x + c\right )}{12 \, {\left (d^{9} x^{4} + 4 \, c d^{8} x^{3} + 6 \, c^{2} d^{7} x^{2} + 4 \, c^{3} d^{6} x + c^{4} d^{5}\right )}} \] Input:
integrate((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(d*x+c)^5,x, a lgorithm="fricas")
Output:
1/12*(25*b^4*c^4 - 12*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - 3* a^4*d^4 + 48*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 36*(3*b^4*c^2*d^2 - 2*a*b^3*c*d ^3 - a^2*b^2*d^4)*x^2 + 8*(11*b^4*c^3*d - 6*a*b^3*c^2*d^2 - 3*a^2*b^2*c*d^ 3 - 2*a^3*b*d^4)*x + 12*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4)*log(d*x + c))/(d^9*x^4 + 4*c*d^8*x^3 + 6*c^2*d ^7*x^2 + 4*c^3*d^6*x + c^4*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (100) = 200\).
Time = 1.22 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.07 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{(c+d x)^5} \, dx=\frac {b^{4} \log {\left (c + d x \right )}}{d^{5}} + \frac {- 3 a^{4} d^{4} - 4 a^{3} b c d^{3} - 6 a^{2} b^{2} c^{2} d^{2} - 12 a b^{3} c^{3} d + 25 b^{4} c^{4} + x^{3} \left (- 48 a b^{3} d^{4} + 48 b^{4} c d^{3}\right ) + x^{2} \left (- 36 a^{2} b^{2} d^{4} - 72 a b^{3} c d^{3} + 108 b^{4} c^{2} d^{2}\right ) + x \left (- 16 a^{3} b d^{4} - 24 a^{2} b^{2} c d^{3} - 48 a b^{3} c^{2} d^{2} + 88 b^{4} c^{3} d\right )}{12 c^{4} d^{5} + 48 c^{3} d^{6} x + 72 c^{2} d^{7} x^{2} + 48 c d^{8} x^{3} + 12 d^{9} x^{4}} \] Input:
integrate((b**4*x**4+4*a*b**3*x**3+6*a**2*b**2*x**2+4*a**3*b*x+a**4)/(d*x+ c)**5,x)
Output:
b**4*log(c + d*x)/d**5 + (-3*a**4*d**4 - 4*a**3*b*c*d**3 - 6*a**2*b**2*c** 2*d**2 - 12*a*b**3*c**3*d + 25*b**4*c**4 + x**3*(-48*a*b**3*d**4 + 48*b**4 *c*d**3) + x**2*(-36*a**2*b**2*d**4 - 72*a*b**3*c*d**3 + 108*b**4*c**2*d** 2) + x*(-16*a**3*b*d**4 - 24*a**2*b**2*c*d**3 - 48*a*b**3*c**2*d**2 + 88*b **4*c**3*d))/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d** 8*x**3 + 12*d**9*x**4)
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (107) = 214\).
Time = 0.05 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.98 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{(c+d x)^5} \, dx=\frac {25 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4} + 48 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 36 \, {\left (3 \, b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} - a^{2} b^{2} d^{4}\right )} x^{2} + 8 \, {\left (11 \, b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} x}{12 \, {\left (d^{9} x^{4} + 4 \, c d^{8} x^{3} + 6 \, c^{2} d^{7} x^{2} + 4 \, c^{3} d^{6} x + c^{4} d^{5}\right )}} + \frac {b^{4} \log \left (d x + c\right )}{d^{5}} \] Input:
integrate((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(d*x+c)^5,x, a lgorithm="maxima")
Output:
1/12*(25*b^4*c^4 - 12*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - 3* a^4*d^4 + 48*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 36*(3*b^4*c^2*d^2 - 2*a*b^3*c*d ^3 - a^2*b^2*d^4)*x^2 + 8*(11*b^4*c^3*d - 6*a*b^3*c^2*d^2 - 3*a^2*b^2*c*d^ 3 - 2*a^3*b*d^4)*x)/(d^9*x^4 + 4*c*d^8*x^3 + 6*c^2*d^7*x^2 + 4*c^3*d^6*x + c^4*d^5) + b^4*log(d*x + c)/d^5
Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (107) = 214\).
Time = 0.13 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.53 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{(c+d x)^5} \, dx=-\frac {b^{4} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{5}} + \frac {\frac {48 \, b^{4} c d^{15}}{d x + c} - \frac {36 \, b^{4} c^{2} d^{15}}{{\left (d x + c\right )}^{2}} + \frac {16 \, b^{4} c^{3} d^{15}}{{\left (d x + c\right )}^{3}} - \frac {3 \, b^{4} c^{4} d^{15}}{{\left (d x + c\right )}^{4}} - \frac {48 \, a b^{3} d^{16}}{d x + c} + \frac {72 \, a b^{3} c d^{16}}{{\left (d x + c\right )}^{2}} - \frac {48 \, a b^{3} c^{2} d^{16}}{{\left (d x + c\right )}^{3}} + \frac {12 \, a b^{3} c^{3} d^{16}}{{\left (d x + c\right )}^{4}} - \frac {36 \, a^{2} b^{2} d^{17}}{{\left (d x + c\right )}^{2}} + \frac {48 \, a^{2} b^{2} c d^{17}}{{\left (d x + c\right )}^{3}} - \frac {18 \, a^{2} b^{2} c^{2} d^{17}}{{\left (d x + c\right )}^{4}} - \frac {16 \, a^{3} b d^{18}}{{\left (d x + c\right )}^{3}} + \frac {12 \, a^{3} b c d^{18}}{{\left (d x + c\right )}^{4}} - \frac {3 \, a^{4} d^{19}}{{\left (d x + c\right )}^{4}}}{12 \, d^{20}} \] Input:
integrate((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(d*x+c)^5,x, a lgorithm="giac")
Output:
-b^4*log(abs(d*x + c)/((d*x + c)^2*abs(d)))/d^5 + 1/12*(48*b^4*c*d^15/(d*x + c) - 36*b^4*c^2*d^15/(d*x + c)^2 + 16*b^4*c^3*d^15/(d*x + c)^3 - 3*b^4* c^4*d^15/(d*x + c)^4 - 48*a*b^3*d^16/(d*x + c) + 72*a*b^3*c*d^16/(d*x + c) ^2 - 48*a*b^3*c^2*d^16/(d*x + c)^3 + 12*a*b^3*c^3*d^16/(d*x + c)^4 - 36*a^ 2*b^2*d^17/(d*x + c)^2 + 48*a^2*b^2*c*d^17/(d*x + c)^3 - 18*a^2*b^2*c^2*d^ 17/(d*x + c)^4 - 16*a^3*b*d^18/(d*x + c)^3 + 12*a^3*b*c*d^18/(d*x + c)^4 - 3*a^4*d^19/(d*x + c)^4)/d^20
Time = 0.07 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.92 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{(c+d x)^5} \, dx=\frac {b^4\,\ln \left (c+d\,x\right )}{d^5}-\frac {\frac {3\,a^4\,d^4+4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+12\,a\,b^3\,c^3\,d-25\,b^4\,c^4}{12\,d^5}+\frac {3\,x^2\,\left (a^2\,b^2\,d^2+2\,a\,b^3\,c\,d-3\,b^4\,c^2\right )}{d^3}+\frac {2\,x\,\left (2\,a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2+6\,a\,b^3\,c^2\,d-11\,b^4\,c^3\right )}{3\,d^4}+\frac {4\,b^3\,x^3\,\left (a\,d-b\,c\right )}{d^2}}{c^4+4\,c^3\,d\,x+6\,c^2\,d^2\,x^2+4\,c\,d^3\,x^3+d^4\,x^4} \] Input:
int((a^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x)/(c + d*x)^5, x)
Output:
(b^4*log(c + d*x))/d^5 - ((3*a^4*d^4 - 25*b^4*c^4 + 6*a^2*b^2*c^2*d^2 + 12 *a*b^3*c^3*d + 4*a^3*b*c*d^3)/(12*d^5) + (3*x^2*(a^2*b^2*d^2 - 3*b^4*c^2 + 2*a*b^3*c*d))/d^3 + (2*x*(2*a^3*b*d^3 - 11*b^4*c^3 + 3*a^2*b^2*c*d^2 + 6* a*b^3*c^2*d))/(3*d^4) + (4*b^3*x^3*(a*d - b*c))/d^2)/(c^4 + d^4*x^4 + 4*c* d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x)
Time = 0.16 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.42 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{(c+d x)^5} \, dx=\frac {12 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{5}+48 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{4} d x +72 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{3} d^{2} x^{2}+48 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{2} d^{3} x^{3}+12 \,\mathrm {log}\left (d x +c \right ) b^{4} c \,d^{4} x^{4}-3 a^{4} c \,d^{4}-4 a^{3} b \,c^{2} d^{3}-16 a^{3} b c \,d^{4} x -6 a^{2} b^{2} c^{3} d^{2}-24 a^{2} b^{2} c^{2} d^{3} x -36 a^{2} b^{2} c \,d^{4} x^{2}+12 a \,b^{3} d^{5} x^{4}+13 b^{4} c^{5}+40 b^{4} c^{4} d x +36 b^{4} c^{3} d^{2} x^{2}-12 b^{4} c \,d^{4} x^{4}}{12 c \,d^{5} \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}\right )} \] Input:
int((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(d*x+c)^5,x)
Output:
(12*log(c + d*x)*b**4*c**5 + 48*log(c + d*x)*b**4*c**4*d*x + 72*log(c + d* x)*b**4*c**3*d**2*x**2 + 48*log(c + d*x)*b**4*c**2*d**3*x**3 + 12*log(c + d*x)*b**4*c*d**4*x**4 - 3*a**4*c*d**4 - 4*a**3*b*c**2*d**3 - 16*a**3*b*c*d **4*x - 6*a**2*b**2*c**3*d**2 - 24*a**2*b**2*c**2*d**3*x - 36*a**2*b**2*c* d**4*x**2 + 12*a*b**3*d**5*x**4 + 13*b**4*c**5 + 40*b**4*c**4*d*x + 36*b** 4*c**3*d**2*x**2 - 12*b**4*c*d**4*x**4)/(12*c*d**5*(c**4 + 4*c**3*d*x + 6* c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4))