Integrand size = 18, antiderivative size = 169 \[ \int \frac {a+b x^2}{x^5 (c+d x)^3} \, dx=-\frac {a}{4 c^3 x^4}+\frac {a d}{c^4 x^3}-\frac {b c^2+6 a d^2}{2 c^5 x^2}+\frac {d \left (3 b c^2+10 a d^2\right )}{c^6 x}+\frac {d^2 \left (b c^2+a d^2\right )}{2 c^5 (c+d x)^2}+\frac {d^2 \left (3 b c^2+5 a d^2\right )}{c^6 (c+d x)}+\frac {3 d^2 \left (2 b c^2+5 a d^2\right ) \log (x)}{c^7}-\frac {3 d^2 \left (2 b c^2+5 a d^2\right ) \log (c+d x)}{c^7} \] Output:
-1/4*a/c^3/x^4+a*d/c^4/x^3-1/2*(6*a*d^2+b*c^2)/c^5/x^2+d*(10*a*d^2+3*b*c^2 )/c^6/x+1/2*d^2*(a*d^2+b*c^2)/c^5/(d*x+c)^2+d^2*(5*a*d^2+3*b*c^2)/c^6/(d*x +c)+3*d^2*(5*a*d^2+2*b*c^2)*ln(x)/c^7-3*d^2*(5*a*d^2+2*b*c^2)*ln(d*x+c)/c^ 7
Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x^2}{x^5 (c+d x)^3} \, dx=\frac {\frac {2 b c^3 x^2 \left (-c^3+4 c^2 d x+18 c d^2 x^2+12 d^3 x^3\right )+a c \left (-c^5+2 c^4 d x-5 c^3 d^2 x^2+20 c^2 d^3 x^3+90 c d^4 x^4+60 d^5 x^5\right )}{x^4 (c+d x)^2}+12 \left (2 b c^2 d^2+5 a d^4\right ) \log (x)-12 \left (2 b c^2 d^2+5 a d^4\right ) \log (c+d x)}{4 c^7} \] Input:
Integrate[(a + b*x^2)/(x^5*(c + d*x)^3),x]
Output:
((2*b*c^3*x^2*(-c^3 + 4*c^2*d*x + 18*c*d^2*x^2 + 12*d^3*x^3) + a*c*(-c^5 + 2*c^4*d*x - 5*c^3*d^2*x^2 + 20*c^2*d^3*x^3 + 90*c*d^4*x^4 + 60*d^5*x^5))/ (x^4*(c + d*x)^2) + 12*(2*b*c^2*d^2 + 5*a*d^4)*Log[x] - 12*(2*b*c^2*d^2 + 5*a*d^4)*Log[c + d*x])/(4*c^7)
Time = 0.61 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {522, 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.
| \(\displaystyle \int \frac {a+b x^2}{x^5 (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (-\frac {3 \left (5 a d^5+2 b c^2 d^3\right )}{c^7 (c+d x)}+\frac {3 \left (5 a d^4+2 b c^2 d^2\right )}{c^7 x}+\frac {-10 a d^3-3 b c^2 d}{c^6 x^2}+\frac {-5 a d^5-3 b c^2 d^3}{c^6 (c+d x)^2}+\frac {6 a d^2+b c^2}{c^5 x^3}-\frac {d^3 \left (a d^2+b c^2\right )}{c^5 (c+d x)^3}-\frac {3 a d}{c^4 x^4}+\frac {a}{c^3 x^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 d^2 \log (x) \left (5 a d^2+2 b c^2\right )}{c^7}-\frac {3 d^2 \left (5 a d^2+2 b c^2\right ) \log (c+d x)}{c^7}+\frac {d^2 \left (5 a d^2+3 b c^2\right )}{c^6 (c+d x)}+\frac {d \left (10 a d^2+3 b c^2\right )}{c^6 x}-\frac {6 a d^2+b c^2}{2 c^5 x^2}+\frac {d^2 \left (a d^2+b c^2\right )}{2 c^5 (c+d x)^2}+\frac {a d}{c^4 x^3}-\frac {a}{4 c^3 x^4}\) |
Input:
Int[(a + b*x^2)/(x^5*(c + d*x)^3),x]
Output:
-1/4*a/(c^3*x^4) + (a*d)/(c^4*x^3) - (b*c^2 + 6*a*d^2)/(2*c^5*x^2) + (d*(3 *b*c^2 + 10*a*d^2))/(c^6*x) + (d^2*(b*c^2 + a*d^2))/(2*c^5*(c + d*x)^2) + (d^2*(3*b*c^2 + 5*a*d^2))/(c^6*(c + d*x)) + (3*d^2*(2*b*c^2 + 5*a*d^2)*Log [x])/c^7 - (3*d^2*(2*b*c^2 + 5*a*d^2)*Log[c + d*x])/c^7
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.21 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {a}{4 c^{3} x^{4}}+\frac {a d}{c^{4} x^{3}}-\frac {6 a \,d^{2}+b \,c^{2}}{2 c^{5} x^{2}}+\frac {d \left (10 a \,d^{2}+3 b \,c^{2}\right )}{c^{6} x}+\frac {d^{2} \left (a \,d^{2}+b \,c^{2}\right )}{2 c^{5} \left (d x +c \right )^{2}}+\frac {d^{2} \left (5 a \,d^{2}+3 b \,c^{2}\right )}{c^{6} \left (d x +c \right )}+\frac {3 d^{2} \left (5 a \,d^{2}+2 b \,c^{2}\right ) \ln \left (x \right )}{c^{7}}-\frac {3 d^{2} \left (5 a \,d^{2}+2 b \,c^{2}\right ) \ln \left (d x +c \right )}{c^{7}}\) | \(164\) |
| norman | \(\frac {\frac {d \left (5 a \,d^{2}+2 b \,c^{2}\right ) x^{3}}{c^{4}}-\frac {a}{4 c}-\frac {\left (5 a \,d^{2}+2 b \,c^{2}\right ) x^{2}}{4 c^{3}}+\frac {a d x}{2 c^{2}}-\frac {2 d \left (15 a \,d^{4}+6 b \,c^{2} d^{2}\right ) x^{5}}{c^{6}}-\frac {d^{2} \left (45 a \,d^{4}+18 b \,c^{2} d^{2}\right ) x^{6}}{2 c^{7}}}{x^{4} \left (d x +c \right )^{2}}+\frac {3 d^{2} \left (5 a \,d^{2}+2 b \,c^{2}\right ) \ln \left (x \right )}{c^{7}}-\frac {3 d^{2} \left (5 a \,d^{2}+2 b \,c^{2}\right ) \ln \left (d x +c \right )}{c^{7}}\) | \(172\) |
| risch | \(\frac {\frac {3 d^{3} \left (5 a \,d^{2}+2 b \,c^{2}\right ) x^{5}}{c^{6}}+\frac {9 d^{2} \left (5 a \,d^{2}+2 b \,c^{2}\right ) x^{4}}{2 c^{5}}+\frac {d \left (5 a \,d^{2}+2 b \,c^{2}\right ) x^{3}}{c^{4}}-\frac {\left (5 a \,d^{2}+2 b \,c^{2}\right ) x^{2}}{4 c^{3}}+\frac {a d x}{2 c^{2}}-\frac {a}{4 c}}{x^{4} \left (d x +c \right )^{2}}+\frac {15 d^{4} \ln \left (-x \right ) a}{c^{7}}+\frac {6 d^{2} \ln \left (-x \right ) b}{c^{5}}-\frac {15 d^{4} \ln \left (d x +c \right ) a}{c^{7}}-\frac {6 d^{2} \ln \left (d x +c \right ) b}{c^{5}}\) | \(174\) |
| parallelrisch | \(\frac {-90 x^{6} a \,d^{6}-2 x^{2} b \,c^{6}+24 \ln \left (x \right ) x^{6} b \,c^{2} d^{4}-24 \ln \left (d x +c \right ) x^{6} b \,c^{2} d^{4}+120 \ln \left (x \right ) x^{5} a c \,d^{5}+48 \ln \left (x \right ) x^{5} b \,c^{3} d^{3}-120 \ln \left (d x +c \right ) x^{5} a c \,d^{5}-48 \ln \left (d x +c \right ) x^{5} b \,c^{3} d^{3}+60 \ln \left (x \right ) x^{4} a \,c^{2} d^{4}+24 \ln \left (x \right ) x^{4} b \,c^{4} d^{2}-60 \ln \left (d x +c \right ) x^{4} a \,c^{2} d^{4}-24 \ln \left (d x +c \right ) x^{4} b \,c^{4} d^{2}-36 x^{6} b \,c^{2} d^{4}-120 x^{5} a c \,d^{5}-48 x^{5} b \,c^{3} d^{3}+20 x^{3} a \,c^{3} d^{3}+8 x^{3} b \,c^{5} d -5 x^{2} a \,c^{4} d^{2}+2 a d x \,c^{5}+60 \ln \left (x \right ) x^{6} a \,d^{6}-60 \ln \left (d x +c \right ) x^{6} a \,d^{6}-a \,c^{6}}{4 c^{7} x^{4} \left (d x +c \right )^{2}}\) | \(299\) |
Input:
int((b*x^2+a)/x^5/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-1/4*a/c^3/x^4+a*d/c^4/x^3-1/2*(6*a*d^2+b*c^2)/c^5/x^2+d*(10*a*d^2+3*b*c^2 )/c^6/x+1/2*d^2*(a*d^2+b*c^2)/c^5/(d*x+c)^2+d^2*(5*a*d^2+3*b*c^2)/c^6/(d*x +c)+3*d^2*(5*a*d^2+2*b*c^2)*ln(x)/c^7-3*d^2*(5*a*d^2+2*b*c^2)*ln(d*x+c)/c^ 7
Time = 0.10 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.65 \[ \int \frac {a+b x^2}{x^5 (c+d x)^3} \, dx=\frac {2 \, a c^{5} d x - a c^{6} + 12 \, {\left (2 \, b c^{3} d^{3} + 5 \, a c d^{5}\right )} x^{5} + 18 \, {\left (2 \, b c^{4} d^{2} + 5 \, a c^{2} d^{4}\right )} x^{4} + 4 \, {\left (2 \, b c^{5} d + 5 \, a c^{3} d^{3}\right )} x^{3} - {\left (2 \, b c^{6} + 5 \, a c^{4} d^{2}\right )} x^{2} - 12 \, {\left ({\left (2 \, b c^{2} d^{4} + 5 \, a d^{6}\right )} x^{6} + 2 \, {\left (2 \, b c^{3} d^{3} + 5 \, a c d^{5}\right )} x^{5} + {\left (2 \, b c^{4} d^{2} + 5 \, a c^{2} d^{4}\right )} x^{4}\right )} \log \left (d x + c\right ) + 12 \, {\left ({\left (2 \, b c^{2} d^{4} + 5 \, a d^{6}\right )} x^{6} + 2 \, {\left (2 \, b c^{3} d^{3} + 5 \, a c d^{5}\right )} x^{5} + {\left (2 \, b c^{4} d^{2} + 5 \, a c^{2} d^{4}\right )} x^{4}\right )} \log \left (x\right )}{4 \, {\left (c^{7} d^{2} x^{6} + 2 \, c^{8} d x^{5} + c^{9} x^{4}\right )}} \] Input:
integrate((b*x^2+a)/x^5/(d*x+c)^3,x, algorithm="fricas")
Output:
1/4*(2*a*c^5*d*x - a*c^6 + 12*(2*b*c^3*d^3 + 5*a*c*d^5)*x^5 + 18*(2*b*c^4* d^2 + 5*a*c^2*d^4)*x^4 + 4*(2*b*c^5*d + 5*a*c^3*d^3)*x^3 - (2*b*c^6 + 5*a* c^4*d^2)*x^2 - 12*((2*b*c^2*d^4 + 5*a*d^6)*x^6 + 2*(2*b*c^3*d^3 + 5*a*c*d^ 5)*x^5 + (2*b*c^4*d^2 + 5*a*c^2*d^4)*x^4)*log(d*x + c) + 12*((2*b*c^2*d^4 + 5*a*d^6)*x^6 + 2*(2*b*c^3*d^3 + 5*a*c*d^5)*x^5 + (2*b*c^4*d^2 + 5*a*c^2* d^4)*x^4)*log(x))/(c^7*d^2*x^6 + 2*c^8*d*x^5 + c^9*x^4)
Time = 0.47 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.74 \[ \int \frac {a+b x^2}{x^5 (c+d x)^3} \, dx=\frac {- a c^{5} + 2 a c^{4} d x + x^{5} \cdot \left (60 a d^{5} + 24 b c^{2} d^{3}\right ) + x^{4} \cdot \left (90 a c d^{4} + 36 b c^{3} d^{2}\right ) + x^{3} \cdot \left (20 a c^{2} d^{3} + 8 b c^{4} d\right ) + x^{2} \left (- 5 a c^{3} d^{2} - 2 b c^{5}\right )}{4 c^{8} x^{4} + 8 c^{7} d x^{5} + 4 c^{6} d^{2} x^{6}} + \frac {3 d^{2} \cdot \left (5 a d^{2} + 2 b c^{2}\right ) \log {\left (x + \frac {15 a c d^{4} + 6 b c^{3} d^{2} - 3 c d^{2} \cdot \left (5 a d^{2} + 2 b c^{2}\right )}{30 a d^{5} + 12 b c^{2} d^{3}} \right )}}{c^{7}} - \frac {3 d^{2} \cdot \left (5 a d^{2} + 2 b c^{2}\right ) \log {\left (x + \frac {15 a c d^{4} + 6 b c^{3} d^{2} + 3 c d^{2} \cdot \left (5 a d^{2} + 2 b c^{2}\right )}{30 a d^{5} + 12 b c^{2} d^{3}} \right )}}{c^{7}} \] Input:
integrate((b*x**2+a)/x**5/(d*x+c)**3,x)
Output:
(-a*c**5 + 2*a*c**4*d*x + x**5*(60*a*d**5 + 24*b*c**2*d**3) + x**4*(90*a*c *d**4 + 36*b*c**3*d**2) + x**3*(20*a*c**2*d**3 + 8*b*c**4*d) + x**2*(-5*a* c**3*d**2 - 2*b*c**5))/(4*c**8*x**4 + 8*c**7*d*x**5 + 4*c**6*d**2*x**6) + 3*d**2*(5*a*d**2 + 2*b*c**2)*log(x + (15*a*c*d**4 + 6*b*c**3*d**2 - 3*c*d* *2*(5*a*d**2 + 2*b*c**2))/(30*a*d**5 + 12*b*c**2*d**3))/c**7 - 3*d**2*(5*a *d**2 + 2*b*c**2)*log(x + (15*a*c*d**4 + 6*b*c**3*d**2 + 3*c*d**2*(5*a*d** 2 + 2*b*c**2))/(30*a*d**5 + 12*b*c**2*d**3))/c**7
Time = 0.03 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x^2}{x^5 (c+d x)^3} \, dx=\frac {2 \, a c^{4} d x - a c^{5} + 12 \, {\left (2 \, b c^{2} d^{3} + 5 \, a d^{5}\right )} x^{5} + 18 \, {\left (2 \, b c^{3} d^{2} + 5 \, a c d^{4}\right )} x^{4} + 4 \, {\left (2 \, b c^{4} d + 5 \, a c^{2} d^{3}\right )} x^{3} - {\left (2 \, b c^{5} + 5 \, a c^{3} d^{2}\right )} x^{2}}{4 \, {\left (c^{6} d^{2} x^{6} + 2 \, c^{7} d x^{5} + c^{8} x^{4}\right )}} - \frac {3 \, {\left (2 \, b c^{2} d^{2} + 5 \, a d^{4}\right )} \log \left (d x + c\right )}{c^{7}} + \frac {3 \, {\left (2 \, b c^{2} d^{2} + 5 \, a d^{4}\right )} \log \left (x\right )}{c^{7}} \] Input:
integrate((b*x^2+a)/x^5/(d*x+c)^3,x, algorithm="maxima")
Output:
1/4*(2*a*c^4*d*x - a*c^5 + 12*(2*b*c^2*d^3 + 5*a*d^5)*x^5 + 18*(2*b*c^3*d^ 2 + 5*a*c*d^4)*x^4 + 4*(2*b*c^4*d + 5*a*c^2*d^3)*x^3 - (2*b*c^5 + 5*a*c^3* d^2)*x^2)/(c^6*d^2*x^6 + 2*c^7*d*x^5 + c^8*x^4) - 3*(2*b*c^2*d^2 + 5*a*d^4 )*log(d*x + c)/c^7 + 3*(2*b*c^2*d^2 + 5*a*d^4)*log(x)/c^7
Time = 0.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.04 \[ \int \frac {a+b x^2}{x^5 (c+d x)^3} \, dx=\frac {3 \, {\left (2 \, b c^{2} d^{2} + 5 \, a d^{4}\right )} \log \left ({\left | x \right |}\right )}{c^{7}} - \frac {3 \, {\left (2 \, b c^{2} d^{3} + 5 \, a d^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{c^{7} d} + \frac {2 \, a c^{5} d x - a c^{6} + 12 \, {\left (2 \, b c^{3} d^{3} + 5 \, a c d^{5}\right )} x^{5} + 18 \, {\left (2 \, b c^{4} d^{2} + 5 \, a c^{2} d^{4}\right )} x^{4} + 4 \, {\left (2 \, b c^{5} d + 5 \, a c^{3} d^{3}\right )} x^{3} - {\left (2 \, b c^{6} + 5 \, a c^{4} d^{2}\right )} x^{2}}{4 \, {\left (d x + c\right )}^{2} c^{7} x^{4}} \] Input:
integrate((b*x^2+a)/x^5/(d*x+c)^3,x, algorithm="giac")
Output:
3*(2*b*c^2*d^2 + 5*a*d^4)*log(abs(x))/c^7 - 3*(2*b*c^2*d^3 + 5*a*d^5)*log( abs(d*x + c))/(c^7*d) + 1/4*(2*a*c^5*d*x - a*c^6 + 12*(2*b*c^3*d^3 + 5*a*c *d^5)*x^5 + 18*(2*b*c^4*d^2 + 5*a*c^2*d^4)*x^4 + 4*(2*b*c^5*d + 5*a*c^3*d^ 3)*x^3 - (2*b*c^6 + 5*a*c^4*d^2)*x^2)/((d*x + c)^2*c^7*x^4)
Time = 0.15 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2}{x^5 (c+d x)^3} \, dx=\frac {\frac {9\,d^2\,x^4\,\left (2\,b\,c^2+5\,a\,d^2\right )}{2\,c^5}-\frac {x^2\,\left (2\,b\,c^2+5\,a\,d^2\right )}{4\,c^3}-\frac {a}{4\,c}+\frac {3\,d^3\,x^5\,\left (2\,b\,c^2+5\,a\,d^2\right )}{c^6}+\frac {a\,d\,x}{2\,c^2}+\frac {d\,x^3\,\left (2\,b\,c^2+5\,a\,d^2\right )}{c^4}}{c^2\,x^4+2\,c\,d\,x^5+d^2\,x^6}-\frac {6\,d^2\,\mathrm {atanh}\left (\frac {3\,d^2\,\left (2\,b\,c^2+5\,a\,d^2\right )\,\left (c+2\,d\,x\right )}{c\,\left (6\,b\,c^2\,d^2+15\,a\,d^4\right )}\right )\,\left (2\,b\,c^2+5\,a\,d^2\right )}{c^7} \] Input:
int((a + b*x^2)/(x^5*(c + d*x)^3),x)
Output:
((9*d^2*x^4*(5*a*d^2 + 2*b*c^2))/(2*c^5) - (x^2*(5*a*d^2 + 2*b*c^2))/(4*c^ 3) - a/(4*c) + (3*d^3*x^5*(5*a*d^2 + 2*b*c^2))/c^6 + (a*d*x)/(2*c^2) + (d* x^3*(5*a*d^2 + 2*b*c^2))/c^4)/(c^2*x^4 + d^2*x^6 + 2*c*d*x^5) - (6*d^2*ata nh((3*d^2*(5*a*d^2 + 2*b*c^2)*(c + 2*d*x))/(c*(15*a*d^4 + 6*b*c^2*d^2)))*( 5*a*d^2 + 2*b*c^2))/c^7
Time = 0.20 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.84 \[ \int \frac {a+b x^2}{x^5 (c+d x)^3} \, dx=\frac {-60 \,\mathrm {log}\left (d x +c \right ) a \,c^{2} d^{4} x^{4}-120 \,\mathrm {log}\left (d x +c \right ) a c \,d^{5} x^{5}-60 \,\mathrm {log}\left (d x +c \right ) a \,d^{6} x^{6}-24 \,\mathrm {log}\left (d x +c \right ) b \,c^{4} d^{2} x^{4}-48 \,\mathrm {log}\left (d x +c \right ) b \,c^{3} d^{3} x^{5}-24 \,\mathrm {log}\left (d x +c \right ) b \,c^{2} d^{4} x^{6}+60 \,\mathrm {log}\left (x \right ) a \,c^{2} d^{4} x^{4}+120 \,\mathrm {log}\left (x \right ) a c \,d^{5} x^{5}+60 \,\mathrm {log}\left (x \right ) a \,d^{6} x^{6}+24 \,\mathrm {log}\left (x \right ) b \,c^{4} d^{2} x^{4}+48 \,\mathrm {log}\left (x \right ) b \,c^{3} d^{3} x^{5}+24 \,\mathrm {log}\left (x \right ) b \,c^{2} d^{4} x^{6}-a \,c^{6}+2 a \,c^{5} d x -5 a \,c^{4} d^{2} x^{2}+20 a \,c^{3} d^{3} x^{3}+60 a \,c^{2} d^{4} x^{4}-30 a \,d^{6} x^{6}-2 b \,c^{6} x^{2}+8 b \,c^{5} d \,x^{3}+24 b \,c^{4} d^{2} x^{4}-12 b \,c^{2} d^{4} x^{6}}{4 c^{7} x^{4} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:
int((b*x^2+a)/x^5/(d*x+c)^3,x)
Output:
( - 60*log(c + d*x)*a*c**2*d**4*x**4 - 120*log(c + d*x)*a*c*d**5*x**5 - 60 *log(c + d*x)*a*d**6*x**6 - 24*log(c + d*x)*b*c**4*d**2*x**4 - 48*log(c + d*x)*b*c**3*d**3*x**5 - 24*log(c + d*x)*b*c**2*d**4*x**6 + 60*log(x)*a*c** 2*d**4*x**4 + 120*log(x)*a*c*d**5*x**5 + 60*log(x)*a*d**6*x**6 + 24*log(x) *b*c**4*d**2*x**4 + 48*log(x)*b*c**3*d**3*x**5 + 24*log(x)*b*c**2*d**4*x** 6 - a*c**6 + 2*a*c**5*d*x - 5*a*c**4*d**2*x**2 + 20*a*c**3*d**3*x**3 + 60* a*c**2*d**4*x**4 - 30*a*d**6*x**6 - 2*b*c**6*x**2 + 8*b*c**5*d*x**3 + 24*b *c**4*d**2*x**4 - 12*b*c**2*d**4*x**6)/(4*c**7*x**4*(c**2 + 2*c*d*x + d**2 *x**2))