Integrand size = 35, antiderivative size = 142 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {e^3 \left (4 c d^2-3 a e^2\right ) x}{c^4 d^4}+\frac {e^4 x^2}{2 c^3 d^3}-\frac {\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}-\frac {4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}+\frac {6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5} \] Output:
e^3*(-3*a*e^2+4*c*d^2)*x/c^4/d^4+1/2*e^4*x^2/c^3/d^3-1/2*(-a*e^2+c*d^2)^4/ c^5/d^5/(c*d*x+a*e)^2-4*e*(-a*e^2+c*d^2)^3/c^5/d^5/(c*d*x+a*e)+6*e^2*(-a*e ^2+c*d^2)^2*ln(c*d*x+a*e)/c^5/d^5
Time = 0.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {7 a^4 e^8+2 a^3 c d e^6 (-10 d+e x)+a^2 c^2 d^2 e^4 \left (18 d^2-16 d e x-11 e^2 x^2\right )-4 a c^3 d^3 e^2 \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+c^4 d^4 \left (-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right )+12 e^2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2 \log (a e+c d x)}{2 c^5 d^5 (a e+c d x)^2} \] Input:
Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
(7*a^4*e^8 + 2*a^3*c*d*e^6*(-10*d + e*x) + a^2*c^2*d^2*e^4*(18*d^2 - 16*d* e*x - 11*e^2*x^2) - 4*a*c^3*d^3*e^2*(d^3 - 6*d^2*e*x - 4*d*e^2*x^2 + e^3*x ^3) + c^4*d^4*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) + 12*e^2*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(2*c^5*d^5*(a*e + c*d*x)^2)
Time = 0.57 (sec) , antiderivative size = 142, 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.
| \(\displaystyle \int \frac {(d+e x)^7}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {6 \left (c d^2 e-a e^3\right )^2}{c^4 d^4 (a e+c d x)}+\frac {4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)^2}+\frac {\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^3}+\frac {4 c d^2 e^3-3 a e^5}{c^4 d^4}+\frac {e^4 x}{c^3 d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}+\frac {6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}+\frac {e^3 x \left (4 c d^2-3 a e^2\right )}{c^4 d^4}+\frac {e^4 x^2}{2 c^3 d^3}\) |
Input:
Int[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
(e^3*(4*c*d^2 - 3*a*e^2)*x)/(c^4*d^4) + (e^4*x^2)/(2*c^3*d^3) - (c*d^2 - a *e^2)^4/(2*c^5*d^5*(a*e + c*d*x)^2) - (4*e*(c*d^2 - a*e^2)^3)/(c^5*d^5*(a* e + c*d*x)) + (6*e^2*(c*d^2 - a*e^2)^2*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 = 1.59 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(-\frac {e^{3} \left (-\frac {1}{2} c d \,x^{2} e +3 a \,e^{2} x -4 c \,d^{2} x \right )}{d^{4} c^{4}}+\frac {6 e^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}-\frac {a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{2 d^{5} c^{5} \left (c d x +a e \right )^{2}}+\frac {4 e \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}{d^{5} c^{5} \left (c d x +a e \right )}\) | \(211\) |
| risch | \(\frac {e^{4} x^{2}}{2 c^{3} d^{3}}-\frac {3 e^{5} a x}{d^{4} c^{4}}+\frac {4 e^{3} x}{d^{2} c^{3}}+\frac {\left (4 a^{3} e^{7}-12 a^{2} e^{5} c \,d^{2}+12 a \,e^{3} c^{2} d^{4}-4 d^{6} c^{3} e \right ) x +\frac {7 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}+18 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}-c^{4} d^{8}}{2 c d}}{d^{4} c^{4} \left (c d x +a e \right )^{2}}+\frac {6 e^{6} \ln \left (c d x +a e \right ) a^{2}}{c^{5} d^{5}}-\frac {12 e^{4} \ln \left (c d x +a e \right ) a}{c^{4} d^{3}}+\frac {6 e^{2} \ln \left (c d x +a e \right )}{c^{3} d}\) | \(230\) |
| parallelrisch | \(\frac {c^{4} d^{4} e^{4} x^{4}+12 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{2} d^{2} e^{6}-24 \ln \left (c d x +a e \right ) x^{2} a \,c^{3} d^{4} e^{4}+12 \ln \left (c d x +a e \right ) x^{2} c^{4} d^{6} e^{2}-4 a \,c^{3} d^{3} e^{5} x^{3}+8 c^{4} d^{5} e^{3} x^{3}+24 \ln \left (c d x +a e \right ) x \,a^{3} c d \,e^{7}-48 \ln \left (c d x +a e \right ) x \,a^{2} c^{2} d^{3} e^{5}+24 \ln \left (c d x +a e \right ) x a \,c^{3} d^{5} e^{3}+12 \ln \left (c d x +a e \right ) a^{4} e^{8}-24 \ln \left (c d x +a e \right ) a^{3} c \,d^{2} e^{6}+12 \ln \left (c d x +a e \right ) a^{2} c^{2} d^{4} e^{4}+24 a^{3} c d \,e^{7} x -48 a^{2} c^{2} d^{3} e^{5} x +24 a \,c^{3} d^{5} e^{3} x -8 c^{4} d^{7} e x +18 a^{4} e^{8}-36 a^{3} c \,d^{2} e^{6}+18 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}-c^{4} d^{8}}{2 d^{5} c^{5} \left (c d x +a e \right )^{2}}\) | \(365\) |
| norman | \(\frac {\frac {\left (18 a^{4} e^{10}-16 a^{3} c \,d^{2} e^{8}-15 a^{2} d^{4} e^{6} c^{2}-9 a \,d^{6} e^{4} c^{3}-5 c^{4} d^{8} e^{2}\right ) x}{d^{4} c^{5} e}+\frac {\left (12 a^{3} e^{10}-16 a^{2} c \,d^{2} e^{8}+d^{4} a \,c^{2} e^{6}-17 c^{3} d^{6} e^{4}\right ) x^{3}}{d^{4} c^{4} e}+\frac {18 a^{4} e^{8}-28 a^{3} c \,d^{2} e^{6}+a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}-c^{4} d^{8}}{2 d^{3} c^{5}}+\frac {e^{6} x^{6}}{2 c d}+\frac {\left (18 a^{4} e^{12}+20 a^{3} c \,d^{2} e^{10}-63 a^{2} c^{2} d^{4} e^{8}-16 a \,c^{3} d^{6} e^{6}-34 c^{4} d^{8} e^{4}\right ) x^{2}}{2 d^{5} c^{5} e^{2}}-\frac {e^{5} \left (2 a \,e^{2}-5 c \,d^{2}\right ) x^{5}}{c^{2} d^{2}}}{\left (e x +d \right )^{2} \left (c d x +a e \right )^{2}}+\frac {6 e^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}\) | \(366\) |
Input:
int((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x,method=_RETURNVERBOSE)
Output:
-e^3/d^4/c^4*(-1/2*c*d*x^2*e+3*a*e^2*x-4*c*d^2*x)+6*e^2/c^5/d^5*(a^2*e^4-2 *a*c*d^2*e^2+c^2*d^4)*ln(c*d*x+a*e)-1/2*(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)/d^5/c^5/(c*d*x+a*e)^2+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)/(c*d*x+a*e)
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (138) = 276\).
Time = 0.08 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.38 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {c^{4} d^{4} e^{4} x^{4} - c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 18 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a^{3} c d^{2} e^{6} + 7 \, a^{4} e^{8} + 4 \, {\left (2 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + {\left (16 \, a c^{3} d^{4} e^{4} - 11 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 2 \, {\left (4 \, c^{4} d^{7} e - 12 \, a c^{3} d^{5} e^{3} + 8 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x + 12 \, {\left (a^{2} c^{2} d^{4} e^{4} - 2 \, a^{3} c d^{2} e^{6} + a^{4} e^{8} + {\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (a c^{3} d^{5} e^{3} - 2 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{2 \, {\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} e x + a^{2} c^{5} d^{5} e^{2}\right )}} \] Input:
integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fric as")
Output:
1/2*(c^4*d^4*e^4*x^4 - c^4*d^8 - 4*a*c^3*d^6*e^2 + 18*a^2*c^2*d^4*e^4 - 20 *a^3*c*d^2*e^6 + 7*a^4*e^8 + 4*(2*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 + (16*a *c^3*d^4*e^4 - 11*a^2*c^2*d^2*e^6)*x^2 - 2*(4*c^4*d^7*e - 12*a*c^3*d^5*e^3 + 8*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x + 12*(a^2*c^2*d^4*e^4 - 2*a^3*c*d^2* e^6 + a^4*e^8 + (c^4*d^6*e^2 - 2*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 2* (a*c^3*d^5*e^3 - 2*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*log(c*d*x + a*e))/(c^ 7*d^7*x^2 + 2*a*c^6*d^6*e*x + a^2*c^5*d^5*e^2)
Time = 1.21 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.59 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=x \left (- \frac {3 a e^{5}}{c^{4} d^{4}} + \frac {4 e^{3}}{c^{3} d^{2}}\right ) + \frac {7 a^{4} e^{8} - 20 a^{3} c d^{2} e^{6} + 18 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} - c^{4} d^{8} + x \left (8 a^{3} c d e^{7} - 24 a^{2} c^{2} d^{3} e^{5} + 24 a c^{3} d^{5} e^{3} - 8 c^{4} d^{7} e\right )}{2 a^{2} c^{5} d^{5} e^{2} + 4 a c^{6} d^{6} e x + 2 c^{7} d^{7} x^{2}} + \frac {e^{4} x^{2}}{2 c^{3} d^{3}} + \frac {6 e^{2} \left (a e^{2} - c d^{2}\right )^{2} \log {\left (a e + c d x \right )}}{c^{5} d^{5}} \] Input:
integrate((e*x+d)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
Output:
x*(-3*a*e**5/(c**4*d**4) + 4*e**3/(c**3*d**2)) + (7*a**4*e**8 - 20*a**3*c* d**2*e**6 + 18*a**2*c**2*d**4*e**4 - 4*a*c**3*d**6*e**2 - c**4*d**8 + x*(8 *a**3*c*d*e**7 - 24*a**2*c**2*d**3*e**5 + 24*a*c**3*d**5*e**3 - 8*c**4*d** 7*e))/(2*a**2*c**5*d**5*e**2 + 4*a*c**6*d**6*e*x + 2*c**7*d**7*x**2) + e** 4*x**2/(2*c**3*d**3) + 6*e**2*(a*e**2 - c*d**2)**2*log(a*e + c*d*x)/(c**5* d**5)
Time = 0.05 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.58 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 7 \, a^{4} e^{8} + 8 \, {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x}{2 \, {\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} e x + a^{2} c^{5} d^{5} e^{2}\right )}} + \frac {c d e^{4} x^{2} + 2 \, {\left (4 \, c d^{2} e^{3} - 3 \, a e^{5}\right )} x}{2 \, c^{4} d^{4}} + \frac {6 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \] Input:
integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxi ma")
Output:
-1/2*(c^4*d^8 + 4*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 7*a^4*e^8 + 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)/(c^7*d^7*x^2 + 2*a*c^6*d^6*e*x + a^2*c^5*d^5*e^2) + 1/2*(c*d*e^4*x^ 2 + 2*(4*c*d^2*e^3 - 3*a*e^5)*x)/(c^4*d^4) + 6*(c^2*d^4*e^2 - 2*a*c*d^2*e^ 4 + a^2*e^6)*log(c*d*x + a*e)/(c^5*d^5)
Time = 0.12 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.51 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {6 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{5}} + \frac {c^{3} d^{3} e^{4} x^{2} + 8 \, c^{3} d^{4} e^{3} x - 6 \, a c^{2} d^{2} e^{5} x}{2 \, c^{6} d^{6}} - \frac {c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 7 \, a^{4} e^{8} + 8 \, {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x}{2 \, {\left (c d x + a e\right )}^{2} c^{5} d^{5}} \] Input:
integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac ")
Output:
6*(c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*log(abs(c*d*x + a*e))/(c^5*d^5) + 1/2*(c^3*d^3*e^4*x^2 + 8*c^3*d^4*e^3*x - 6*a*c^2*d^2*e^5*x)/(c^6*d^6) - 1/2*(c^4*d^8 + 4*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 7 *a^4*e^8 + 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)/((c*d*x + a*e)^2*c^5*d^5)
Time = 5.27 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {x\,\left (4\,a^3\,e^7-12\,a^2\,c\,d^2\,e^5+12\,a\,c^2\,d^4\,e^3-4\,c^3\,d^6\,e\right )-\frac {-7\,a^4\,e^8+20\,a^3\,c\,d^2\,e^6-18\,a^2\,c^2\,d^4\,e^4+4\,a\,c^3\,d^6\,e^2+c^4\,d^8}{2\,c\,d}}{a^2\,c^4\,d^4\,e^2+2\,a\,c^5\,d^5\,e\,x+c^6\,d^6\,x^2}+x\,\left (\frac {4\,e^3}{c^3\,d^2}-\frac {3\,a\,e^5}{c^4\,d^4}\right )+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (6\,a^2\,e^6-12\,a\,c\,d^2\,e^4+6\,c^2\,d^4\,e^2\right )}{c^5\,d^5}+\frac {e^4\,x^2}{2\,c^3\,d^3} \] Input:
int((d + e*x)^7/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)
Output:
(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^4* d^8 - 7*a^4*e^8 + 4*a*c^3*d^6*e^2 + 20*a^3*c*d^2*e^6 - 18*a^2*c^2*d^4*e^4) /(2*c*d))/(c^6*d^6*x^2 + a^2*c^4*d^4*e^2 + 2*a*c^5*d^5*e*x) + x*((4*e^3)/( c^3*d^2) - (3*a*e^5)/(c^4*d^4)) + (log(a*e + c*d*x)*(6*a^2*e^6 + 6*c^2*d^4 *e^2 - 12*a*c*d^2*e^4))/(c^5*d^5) + (e^4*x^2)/(2*c^3*d^3)
Time = 0.23 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.77 \[ \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {12 \,\mathrm {log}\left (c d x +a e \right ) a^{5} e^{8}-24 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c \,d^{2} e^{6}+24 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c d \,e^{7} x +12 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{2} d^{4} e^{4}-48 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{2} d^{3} e^{5} x +12 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{2} d^{2} e^{6} x^{2}+24 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{3} d^{5} e^{3} x -24 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{3} d^{4} e^{4} x^{2}+12 \,\mathrm {log}\left (c d x +a e \right ) a \,c^{4} d^{6} e^{2} x^{2}+6 a^{5} e^{8}-12 a^{4} c \,d^{2} e^{6}+6 a^{3} c^{2} d^{4} e^{4}-12 a^{3} c^{2} d^{2} e^{6} x^{2}+24 a^{2} c^{3} d^{4} e^{4} x^{2}-4 a^{2} c^{3} d^{3} e^{5} x^{3}-a \,c^{4} d^{8}-12 a \,c^{4} d^{6} e^{2} x^{2}+8 a \,c^{4} d^{5} e^{3} x^{3}+a \,c^{4} d^{4} e^{4} x^{4}+4 c^{5} d^{8} x^{2}}{2 a \,c^{5} d^{5} \left (c^{2} d^{2} x^{2}+2 a c d e x +a^{2} e^{2}\right )} \] Input:
int((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
Output:
(12*log(a*e + c*d*x)*a**5*e**8 - 24*log(a*e + c*d*x)*a**4*c*d**2*e**6 + 24 *log(a*e + c*d*x)*a**4*c*d*e**7*x + 12*log(a*e + c*d*x)*a**3*c**2*d**4*e** 4 - 48*log(a*e + c*d*x)*a**3*c**2*d**3*e**5*x + 12*log(a*e + c*d*x)*a**3*c **2*d**2*e**6*x**2 + 24*log(a*e + c*d*x)*a**2*c**3*d**5*e**3*x - 24*log(a* e + c*d*x)*a**2*c**3*d**4*e**4*x**2 + 12*log(a*e + c*d*x)*a*c**4*d**6*e**2 *x**2 + 6*a**5*e**8 - 12*a**4*c*d**2*e**6 + 6*a**3*c**2*d**4*e**4 - 12*a** 3*c**2*d**2*e**6*x**2 + 24*a**2*c**3*d**4*e**4*x**2 - 4*a**2*c**3*d**3*e** 5*x**3 - a*c**4*d**8 - 12*a*c**4*d**6*e**2*x**2 + 8*a*c**4*d**5*e**3*x**3 + a*c**4*d**4*e**4*x**4 + 4*c**5*d**8*x**2)/(2*a*c**5*d**5*(a**2*e**2 + 2* a*c*d*e*x + c**2*d**2*x**2))