Integrand size = 35, antiderivative size = 145 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e^2 \left (6 c^2 d^4-8 a c d^2 e^2+3 a^2 e^4\right ) x}{c^4 d^4}+\frac {e^3 \left (2 c d^2-a e^2\right ) x^2}{c^3 d^3}+\frac {e^4 x^3}{3 c^2 d^2}-\frac {\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5} \] Output:
e^2*(3*a^2*e^4-8*a*c*d^2*e^2+6*c^2*d^4)*x/c^4/d^4+e^3*(-a*e^2+2*c*d^2)*x^2 /c^3/d^3+1/3*e^4*x^3/c^2/d^2-(-a*e^2+c*d^2)^4/c^5/d^5/(c*d*x+a*e)+4*e*(-a* e^2+c*d^2)^3*ln(c*d*x+a*e)/c^5/d^5
Time = 0.05 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {-3 a^4 e^8+3 a^3 c d e^6 (4 d+3 e x)-6 a^2 c^2 d^2 e^4 \left (3 d^2+4 d e x-e^2 x^2\right )+2 a c^3 d^3 e^2 \left (6 d^3+9 d^2 e x-9 d e^2 x^2-e^3 x^3\right )+c^4 d^4 \left (-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right )-12 e \left (-c d^2+a e^2\right )^3 (a e+c d x) \log (a e+c d x)}{3 c^5 d^5 (a e+c d x)} \] Input:
Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
(-3*a^4*e^8 + 3*a^3*c*d*e^6*(4*d + 3*e*x) - 6*a^2*c^2*d^2*e^4*(3*d^2 + 4*d *e*x - e^2*x^2) + 2*a*c^3*d^3*e^2*(6*d^3 + 9*d^2*e*x - 9*d*e^2*x^2 - e^3*x ^3) + c^4*d^4*(-3*d^4 + 18*d^2*e^2*x^2 + 6*d*e^3*x^3 + e^4*x^4) - 12*e*(-( c*d^2) + a*e^2)^3*(a*e + c*d*x)*Log[a*e + c*d*x])/(3*c^5*d^5*(a*e + c*d*x) )
Time = 0.37 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1121, 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.
| \(\displaystyle \int \frac {(d+e x)^6}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {3 a^2 e^6-8 a c d^2 e^4+6 c^2 d^4 e^2}{c^4 d^4}+\frac {4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}+\frac {\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^2}+\frac {2 e^3 x \left (2 c d^2-a e^2\right )}{c^3 d^3}+\frac {e^4 x^2}{c^2 d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 x \left (3 a^2 e^4-8 a c d^2 e^2+6 c^2 d^4\right )}{c^4 d^4}-\frac {\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5}+\frac {e^3 x^2 \left (2 c d^2-a e^2\right )}{c^3 d^3}+\frac {e^4 x^3}{3 c^2 d^2}\) |
Input:
Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
(e^2*(6*c^2*d^4 - 8*a*c*d^2*e^2 + 3*a^2*e^4)*x)/(c^4*d^4) + (e^3*(2*c*d^2 - a*e^2)*x^2)/(c^3*d^3) + (e^4*x^3)/(3*c^2*d^2) - (c*d^2 - a*e^2)^4/(c^5*d ^5*(a*e + c*d*x)) + (4*e*(c*d^2 - a*e^2)^3*Log[a*e + c*d*x])/(c^5*d^5)
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 1.62 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.43
| method | result | size |
| default | \(\frac {e^{2} \left (\frac {1}{3} x^{3} c^{2} d^{2} e^{2}-x^{2} a c d \,e^{3}+2 x^{2} c^{2} d^{3} e +3 a^{2} e^{4} x -8 a c \,d^{2} e^{2} x +6 c^{2} d^{4} x \right )}{d^{4} c^{4}}-\frac {4 e \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \ln \left (c d x +a e \right )}{d^{5} c^{5}}-\frac {a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{d^{5} c^{5} \left (c d x +a e \right )}\) | \(208\) |
| risch | \(\frac {e^{4} x^{3}}{3 c^{2} d^{2}}-\frac {e^{5} x^{2} a}{d^{3} c^{3}}+\frac {2 e^{3} x^{2}}{d \,c^{2}}+\frac {3 e^{6} a^{2} x}{d^{4} c^{4}}-\frac {8 e^{4} a x}{d^{2} c^{3}}+\frac {6 e^{2} x}{c^{2}}-\frac {4 e^{7} \ln \left (c d x +a e \right ) a^{3}}{d^{5} c^{5}}+\frac {12 e^{5} \ln \left (c d x +a e \right ) a^{2}}{d^{3} c^{4}}-\frac {12 e^{3} \ln \left (c d x +a e \right ) a}{d \,c^{3}}+\frac {4 d e \ln \left (c d x +a e \right )}{c^{2}}-\frac {a^{4} e^{8}}{d^{5} c^{5} \left (c d x +a e \right )}+\frac {4 a^{3} e^{6}}{d^{3} c^{4} \left (c d x +a e \right )}-\frac {6 a^{2} e^{4}}{d \,c^{3} \left (c d x +a e \right )}+\frac {4 d a \,e^{2}}{c^{2} \left (c d x +a e \right )}-\frac {d^{3}}{c \left (c d x +a e \right )}\) | \(275\) |
| norman | \(\frac {-\frac {4 a^{4} e^{8}-10 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{c^{5} d^{4}}+\frac {e^{5} x^{5}}{3 c d}-\frac {\left (4 a^{4} e^{10}-10 a^{3} c \,d^{2} e^{8}+8 a^{2} c^{2} d^{4} e^{6}-4 a \,c^{3} d^{6} e^{4}+7 c^{4} d^{8} e^{2}\right ) x}{d^{5} c^{5} e}+\frac {2 e^{3} \left (3 a^{2} e^{4}-10 a c \,d^{2} e^{2}+12 c^{2} d^{4}\right ) x^{3}}{3 d^{3} c^{3}}-\frac {e^{4} \left (2 a \,e^{2}-7 c \,d^{2}\right ) x^{4}}{3 c^{2} d^{2}}}{\left (e x +d \right ) \left (c d x +a e \right )}-\frac {4 e \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \ln \left (c d x +a e \right )}{d^{5} c^{5}}\) | \(293\) |
| parallelrisch | \(-\frac {-c^{4} d^{4} e^{4} x^{4}+2 a \,c^{3} d^{3} e^{5} x^{3}-6 c^{4} d^{5} e^{3} x^{3}+12 \ln \left (c d x +a e \right ) x \,a^{3} c d \,e^{7}-36 \ln \left (c d x +a e \right ) x \,a^{2} c^{2} d^{3} e^{5}+36 \ln \left (c d x +a e \right ) x a \,c^{3} d^{5} e^{3}-12 \ln \left (c d x +a e \right ) x \,c^{4} d^{7} e -6 a^{2} c^{2} d^{2} e^{6} x^{2}+18 a \,c^{3} d^{4} e^{4} x^{2}-18 c^{4} d^{6} e^{2} x^{2}+12 \ln \left (c d x +a e \right ) a^{4} e^{8}-36 \ln \left (c d x +a e \right ) a^{3} c \,d^{2} e^{6}+36 \ln \left (c d x +a e \right ) a^{2} c^{2} d^{4} e^{4}-12 \ln \left (c d x +a e \right ) a \,c^{3} d^{6} e^{2}+12 a^{4} e^{8}-36 a^{3} c \,d^{2} e^{6}+36 a^{2} c^{2} d^{4} e^{4}-12 a \,c^{3} d^{6} e^{2}+3 c^{4} d^{8}}{3 d^{5} c^{5} \left (c d x +a e \right )}\) | \(330\) |
Input:
int((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2,x,method=_RETURNVERBOSE)
Output:
e^2/d^4/c^4*(1/3*x^3*c^2*d^2*e^2-x^2*a*c*d*e^3+2*x^2*c^2*d^3*e+3*a^2*e^4*x -8*a*c*d^2*e^2*x+6*c^2*d^4*x)-4/d^5*e/c^5*(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2 *d^4*e^2-c^3*d^6)*ln(c*d*x+a*e)-(a^4*e^8-4*a^3*c*d^2*e^6+6*a^2*c^2*d^4*e^4 -4*a*c^3*d^6*e^2+c^4*d^8)/d^5/c^5/(c*d*x+a*e)
Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (143) = 286\).
Time = 0.08 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.10 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {c^{4} d^{4} e^{4} x^{4} - 3 \, c^{4} d^{8} + 12 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 2 \, {\left (3 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (3 \, c^{4} d^{6} e^{2} - 3 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 3 \, {\left (6 \, a c^{3} d^{5} e^{3} - 8 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a^{3} c d e^{7}\right )} x + 12 \, {\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} + {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{3 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}} \] Input:
integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fric as")
Output:
1/3*(c^4*d^4*e^4*x^4 - 3*c^4*d^8 + 12*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 - 3*a^4*e^8 + 2*(3*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 + 6* (3*c^4*d^6*e^2 - 3*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 3*(6*a*c^3*d^5*e ^3 - 8*a^2*c^2*d^3*e^5 + 3*a^3*c*d*e^7)*x + 12*(a*c^3*d^6*e^2 - 3*a^2*c^2* d^4*e^4 + 3*a^3*c*d^2*e^6 - a^4*e^8 + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2 *c^2*d^3*e^5 - a^3*c*d*e^7)*x)*log(c*d*x + a*e))/(c^6*d^6*x + a*c^5*d^5*e)
Time = 0.44 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=x^{2} \left (- \frac {a e^{5}}{c^{3} d^{3}} + \frac {2 e^{3}}{c^{2} d}\right ) + x \left (\frac {3 a^{2} e^{6}}{c^{4} d^{4}} - \frac {8 a e^{4}}{c^{3} d^{2}} + \frac {6 e^{2}}{c^{2}}\right ) + \frac {- a^{4} e^{8} + 4 a^{3} c d^{2} e^{6} - 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{6} e^{2} - c^{4} d^{8}}{a c^{5} d^{5} e + c^{6} d^{6} x} + \frac {e^{4} x^{3}}{3 c^{2} d^{2}} - \frac {4 e \left (a e^{2} - c d^{2}\right )^{3} \log {\left (a e + c d x \right )}}{c^{5} d^{5}} \] Input:
integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
Output:
x**2*(-a*e**5/(c**3*d**3) + 2*e**3/(c**2*d)) + x*(3*a**2*e**6/(c**4*d**4) - 8*a*e**4/(c**3*d**2) + 6*e**2/c**2) + (-a**4*e**8 + 4*a**3*c*d**2*e**6 - 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**6*e**2 - c**4*d**8)/(a*c**5*d**5*e + c**6*d**6*x) + e**4*x**3/(3*c**2*d**2) - 4*e*(a*e**2 - c*d**2)**3*log(a*e + c*d*x)/(c**5*d**5)
Time = 0.04 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.48 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{c^{6} d^{6} x + a c^{5} d^{5} e} + \frac {c^{2} d^{2} e^{4} x^{3} + 3 \, {\left (2 \, c^{2} d^{3} e^{3} - a c d e^{5}\right )} x^{2} + 3 \, {\left (6 \, c^{2} d^{4} e^{2} - 8 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x}{3 \, c^{4} d^{4}} + \frac {4 \, {\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \] Input:
integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxi ma")
Output:
-(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^ 8)/(c^6*d^6*x + a*c^5*d^5*e) + 1/3*(c^2*d^2*e^4*x^3 + 3*(2*c^2*d^3*e^3 - a *c*d*e^5)*x^2 + 3*(6*c^2*d^4*e^2 - 8*a*c*d^2*e^4 + 3*a^2*e^6)*x)/(c^4*d^4) + 4*(c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - a^3*e^7)*log(c*d*x + a*e)/(c^5*d^5)
Time = 0.14 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {4 \, {\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{5}} - \frac {c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{{\left (c d x + a e\right )} c^{5} d^{5}} + \frac {c^{4} d^{4} e^{4} x^{3} + 6 \, c^{4} d^{5} e^{3} x^{2} - 3 \, a c^{3} d^{3} e^{5} x^{2} + 18 \, c^{4} d^{6} e^{2} x - 24 \, a c^{3} d^{4} e^{4} x + 9 \, a^{2} c^{2} d^{2} e^{6} x}{3 \, c^{6} d^{6}} \] Input:
integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac ")
Output:
4*(c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - a^3*e^7)*log(abs(c*d*x + a*e))/(c^5*d^5) - (c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3 *c*d^2*e^6 + a^4*e^8)/((c*d*x + a*e)*c^5*d^5) + 1/3*(c^4*d^4*e^4*x^3 + 6*c ^4*d^5*e^3*x^2 - 3*a*c^3*d^3*e^5*x^2 + 18*c^4*d^6*e^2*x - 24*a*c^3*d^4*e^4 *x + 9*a^2*c^2*d^2*e^6*x)/(c^6*d^6)
Time = 0.05 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.67 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=x^2\,\left (\frac {2\,e^3}{c^2\,d}-\frac {a\,e^5}{c^3\,d^3}\right )-x\,\left (\frac {a^2\,e^6}{c^4\,d^4}-\frac {6\,e^2}{c^2}+\frac {2\,a\,e\,\left (\frac {4\,e^3}{c^2\,d}-\frac {2\,a\,e^5}{c^3\,d^3}\right )}{c\,d}\right )-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (4\,a^3\,e^7-12\,a^2\,c\,d^2\,e^5+12\,a\,c^2\,d^4\,e^3-4\,c^3\,d^6\,e\right )}{c^5\,d^5}-\frac {a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}{c\,d\,\left (x\,c^5\,d^5+a\,e\,c^4\,d^4\right )}+\frac {e^4\,x^3}{3\,c^2\,d^2} \] Input:
int((d + e*x)^6/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)
Output:
x^2*((2*e^3)/(c^2*d) - (a*e^5)/(c^3*d^3)) - x*((a^2*e^6)/(c^4*d^4) - (6*e^ 2)/c^2 + (2*a*e*((4*e^3)/(c^2*d) - (2*a*e^5)/(c^3*d^3)))/(c*d)) - (log(a*e + c*d*x)*(4*a^3*e^7 - 4*c^3*d^6*e + 12*a*c^2*d^4*e^3 - 12*a^2*c*d^2*e^5)) /(c^5*d^5) - (a^4*e^8 + c^4*d^8 - 4*a*c^3*d^6*e^2 - 4*a^3*c*d^2*e^6 + 6*a^ 2*c^2*d^4*e^4)/(c*d*(c^5*d^5*x + a*c^4*d^4*e)) + (e^4*x^3)/(3*c^2*d^2)
Time = 0.23 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.46 \[ \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {-12 \,\mathrm {log}\left (c d x +a e \right ) a^{5} e^{9}+36 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c \,d^{2} e^{7}-12 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c d \,e^{8} x -36 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{2} d^{4} e^{5}+36 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{2} d^{3} e^{6} x +12 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{3} d^{6} e^{3}-36 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{3} d^{5} e^{4} x +12 \,\mathrm {log}\left (c d x +a e \right ) a \,c^{4} d^{7} e^{2} x +12 a^{4} c d \,e^{8} x -36 a^{3} c^{2} d^{3} e^{6} x +6 a^{3} c^{2} d^{2} e^{7} x^{2}+36 a^{2} c^{3} d^{5} e^{4} x -18 a^{2} c^{3} d^{4} e^{5} x^{2}-2 a^{2} c^{3} d^{3} e^{6} x^{3}-12 a \,c^{4} d^{7} e^{2} x +18 a \,c^{4} d^{6} e^{3} x^{2}+6 a \,c^{4} d^{5} e^{4} x^{3}+a \,c^{4} d^{4} e^{5} x^{4}+3 c^{5} d^{9} x}{3 a \,c^{5} d^{5} e \left (c d x +a e \right )} \] Input:
int((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
Output:
( - 12*log(a*e + c*d*x)*a**5*e**9 + 36*log(a*e + c*d*x)*a**4*c*d**2*e**7 - 12*log(a*e + c*d*x)*a**4*c*d*e**8*x - 36*log(a*e + c*d*x)*a**3*c**2*d**4* e**5 + 36*log(a*e + c*d*x)*a**3*c**2*d**3*e**6*x + 12*log(a*e + c*d*x)*a** 2*c**3*d**6*e**3 - 36*log(a*e + c*d*x)*a**2*c**3*d**5*e**4*x + 12*log(a*e + c*d*x)*a*c**4*d**7*e**2*x + 12*a**4*c*d*e**8*x - 36*a**3*c**2*d**3*e**6* x + 6*a**3*c**2*d**2*e**7*x**2 + 36*a**2*c**3*d**5*e**4*x - 18*a**2*c**3*d **4*e**5*x**2 - 2*a**2*c**3*d**3*e**6*x**3 - 12*a*c**4*d**7*e**2*x + 18*a* c**4*d**6*e**3*x**2 + 6*a*c**4*d**5*e**4*x**3 + a*c**4*d**4*e**5*x**4 + 3* c**5*d**9*x)/(3*a*c**5*d**5*e*(a*e + c*d*x))