Integrand size = 35, antiderivative size = 146 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^4 x}{c^4 d^4}-\frac {\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}-\frac {2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac {6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}+\frac {4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5} \] Output:
e^4*x/c^4/d^4-1/3*(-a*e^2+c*d^2)^4/c^5/d^5/(c*d*x+a*e)^3-2*e*(-a*e^2+c*d^2 )^3/c^5/d^5/(c*d*x+a*e)^2-6*e^2*(-a*e^2+c*d^2)^2/c^5/d^5/(c*d*x+a*e)+4*e^3 *(-a*e^2+c*d^2)*ln(c*d*x+a*e)/c^5/d^5
Time = 0.06 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.33 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {-13 a^4 e^8+a^3 c d e^6 (22 d-27 e x)-3 a^2 c^2 d^2 e^4 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a c^3 d^3 e^2 \left (-2 d^3-18 d^2 e x+36 d e^2 x^2+9 e^3 x^3\right )-c^4 \left (d^8+6 d^7 e x+18 d^6 e^2 x^2-3 d^4 e^4 x^4\right )-12 e^3 \left (-c d^2+a e^2\right ) (a e+c d x)^3 \log (a e+c d x)}{3 c^5 d^5 (a e+c d x)^3} \] Input:
Integrate[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
Output:
(-13*a^4*e^8 + a^3*c*d*e^6*(22*d - 27*e*x) - 3*a^2*c^2*d^2*e^4*(2*d^2 - 18 *d*e*x + 3*e^2*x^2) + a*c^3*d^3*e^2*(-2*d^3 - 18*d^2*e*x + 36*d*e^2*x^2 + 9*e^3*x^3) - c^4*(d^8 + 6*d^7*e*x + 18*d^6*e^2*x^2 - 3*d^4*e^4*x^4) - 12*e ^3*(-(c*d^2) + a*e^2)*(a*e + c*d*x)^3*Log[a*e + c*d*x])/(3*c^5*d^5*(a*e + c*d*x)^3)
Time = 0.58 (sec) , antiderivative size = 146, 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.
| \(\displaystyle \int \frac {(d+e x)^8}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {6 \left (c d^2 e-a e^3\right )^2}{c^4 d^4 (a e+c d x)^2}+\frac {4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)^3}+\frac {\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^4}+\frac {4 \left (c d^2 e^3-a e^5\right )}{c^4 d^4 (a e+c d x)}+\frac {e^4}{c^4 d^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}-\frac {2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}+\frac {4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5}+\frac {e^4 x}{c^4 d^4}\) |
Input:
Int[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
Output:
(e^4*x)/(c^4*d^4) - (c*d^2 - a*e^2)^4/(3*c^5*d^5*(a*e + c*d*x)^3) - (2*e*( c*d^2 - a*e^2)^3)/(c^5*d^5*(a*e + c*d*x)^2) - (6*e^2*(c*d^2 - a*e^2)^2)/(c ^5*d^5*(a*e + c*d*x)) + (4*e^3*(c*d^2 - a*e^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 = 1.53 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.48
| method | result | size |
| risch | \(\frac {e^{4} x}{c^{4} d^{4}}+\frac {\left (-6 a^{2} e^{6} c d +12 a \,e^{4} c^{2} d^{3}-6 d^{5} c^{3} e^{2}\right ) x^{2}-2 e \left (5 e^{6} a^{3}-9 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}+d^{6} c^{3}\right ) x -\frac {13 a^{4} e^{8}-22 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{3 c d}}{d^{4} c^{4} \left (c d x +a e \right )^{3}}-\frac {4 e^{5} \ln \left (c d x +a e \right ) a}{d^{5} c^{5}}+\frac {4 e^{3} \ln \left (c d x +a e \right )}{d^{3} c^{4}}\) | \(216\) |
| default | \(\frac {e^{4} x}{c^{4} d^{4}}-\frac {a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{3 d^{5} c^{5} \left (c d x +a e \right )^{3}}-\frac {4 e^{3} \left (a \,e^{2}-c \,d^{2}\right ) \ln \left (c d x +a e \right )}{d^{5} c^{5}}+\frac {2 e \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}{d^{5} c^{5} \left (c d x +a e \right )^{2}}-\frac {6 e^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{c^{5} d^{5} \left (c d x +a e \right )}\) | \(221\) |
| parallelrisch | \(-\frac {22 a^{4} e^{8}-22 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}-12 \ln \left (c d x +a e \right ) a^{3} c \,d^{2} e^{6}-54 a^{2} c^{2} d^{3} e^{5} x +18 a \,c^{3} d^{5} e^{3} x -12 \ln \left (c d x +a e \right ) x^{3} c^{4} d^{5} e^{3}+12 \ln \left (c d x +a e \right ) x^{3} a \,c^{3} d^{3} e^{5}+36 \ln \left (c d x +a e \right ) x \,a^{3} c d \,e^{7}-36 \ln \left (c d x +a e \right ) x \,a^{2} c^{2} d^{3} e^{5}+36 a^{2} c^{2} d^{2} e^{6} x^{2}-36 a \,c^{3} d^{4} e^{4} x^{2}+54 a^{3} c d \,e^{7} x -3 c^{4} d^{4} e^{4} x^{4}+18 c^{4} d^{6} e^{2} x^{2}+6 c^{4} d^{7} e x +12 \ln \left (c d x +a e \right ) a^{4} e^{8}+36 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{2} d^{2} e^{6}-36 \ln \left (c d x +a e \right ) x^{2} a \,c^{3} d^{4} e^{4}}{3 d^{5} c^{5} \left (c d x +a e \right )^{3}}\) | \(361\) |
| norman | \(\frac {\frac {e^{7} x^{7}}{c d}-\frac {22 a^{4} e^{8}-13 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{3 c^{5} d^{2}}-\frac {\left (22 a^{4} e^{14}+149 a^{3} c \,d^{2} e^{12}+33 a^{2} c^{2} d^{4} e^{10}+29 a \,c^{3} d^{6} e^{8}+82 c^{4} d^{8} e^{6}\right ) x^{3}}{3 d^{5} c^{5} e^{3}}-\frac {\left (22 a^{4} e^{12}+41 a^{3} c \,d^{2} e^{10}-9 a^{2} c^{2} d^{4} e^{8}+17 a \,c^{3} d^{6} e^{6}+13 c^{4} d^{8} e^{4}\right ) x^{2}}{d^{4} c^{5} e^{2}}-\frac {\left (18 a^{3} e^{12}+27 a^{2} c \,d^{2} e^{10}-3 d^{4} a \,c^{2} e^{8}+28 d^{6} e^{6} c^{3}\right ) x^{4}}{d^{4} c^{4} e^{2}}-\frac {\left (22 a^{4} e^{10}+5 d^{2} e^{8} c \,a^{3}-3 e^{6} d^{4} c^{2} a^{2}+8 c^{3} d^{6} e^{4} a +3 e^{2} d^{8} c^{4}\right ) x}{d^{3} c^{5} e}-\frac {3 \left (4 a^{2} e^{10}-a \,d^{2} c \,e^{8}+4 e^{6} d^{4} c^{2}\right ) x^{5}}{d^{3} c^{3} e}}{\left (e x +d \right )^{3} \left (c d x +a e \right )^{3}}-\frac {4 e^{3} \left (a \,e^{2}-c \,d^{2}\right ) \ln \left (c d x +a e \right )}{d^{5} c^{5}}\) | \(444\) |
Input:
int((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^4,x,method=_RETURNVERBOSE)
Output:
e^4*x/c^4/d^4+((-6*a^2*c*d*e^6+12*a*c^2*d^3*e^4-6*c^3*d^5*e^2)*x^2-2*e*(5* a^3*e^6-9*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2+c^3*d^6)*x-1/3*(13*a^4*e^8-22*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/d)/d^4/c^4/(c*d*x+a *e)^3-4*e^5/d^5/c^5*ln(c*d*x+a*e)*a+4*e^3/d^3/c^4*ln(c*d*x+a*e)
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (144) = 288\).
Time = 0.08 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.38 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {3 \, c^{4} d^{4} e^{4} x^{4} + 9 \, a c^{3} d^{3} e^{5} x^{3} - c^{4} d^{8} - 2 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 22 \, a^{3} c d^{2} e^{6} - 13 \, a^{4} e^{8} - 9 \, {\left (2 \, c^{4} d^{6} e^{2} - 4 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 3 \, {\left (2 \, c^{4} d^{7} e + 6 \, a c^{3} d^{5} e^{3} - 18 \, a^{2} c^{2} d^{3} e^{5} + 9 \, a^{3} c d e^{7}\right )} x + 12 \, {\left (a^{3} c d^{2} e^{6} - a^{4} e^{8} + {\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (a c^{3} d^{4} e^{4} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 3 \, {\left (a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{3 \, {\left (c^{8} d^{8} x^{3} + 3 \, a c^{7} d^{7} e x^{2} + 3 \, a^{2} c^{6} d^{6} e^{2} x + a^{3} c^{5} d^{5} e^{3}\right )}} \] Input:
integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fric as")
Output:
1/3*(3*c^4*d^4*e^4*x^4 + 9*a*c^3*d^3*e^5*x^3 - c^4*d^8 - 2*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 + 22*a^3*c*d^2*e^6 - 13*a^4*e^8 - 9*(2*c^4*d^6*e^2 - 4* a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 - 3*(2*c^4*d^7*e + 6*a*c^3*d^5*e^3 - 18*a^2*c^2*d^3*e^5 + 9*a^3*c*d*e^7)*x + 12*(a^3*c*d^2*e^6 - a^4*e^8 + (c^4 *d^5*e^3 - a*c^3*d^3*e^5)*x^3 + 3*(a*c^3*d^4*e^4 - a^2*c^2*d^2*e^6)*x^2 + 3*(a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*log(c*d*x + a*e))/(c^8*d^8*x^3 + 3*a* c^7*d^7*e*x^2 + 3*a^2*c^6*d^6*e^2*x + a^3*c^5*d^5*e^3)
Time = 6.34 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.76 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {- 13 a^{4} e^{8} + 22 a^{3} c d^{2} e^{6} - 6 a^{2} c^{2} d^{4} e^{4} - 2 a c^{3} d^{6} e^{2} - c^{4} d^{8} + x^{2} \left (- 18 a^{2} c^{2} d^{2} e^{6} + 36 a c^{3} d^{4} e^{4} - 18 c^{4} d^{6} e^{2}\right ) + x \left (- 30 a^{3} c d e^{7} + 54 a^{2} c^{2} d^{3} e^{5} - 18 a c^{3} d^{5} e^{3} - 6 c^{4} d^{7} e\right )}{3 a^{3} c^{5} d^{5} e^{3} + 9 a^{2} c^{6} d^{6} e^{2} x + 9 a c^{7} d^{7} e x^{2} + 3 c^{8} d^{8} x^{3}} + \frac {e^{4} x}{c^{4} d^{4}} - \frac {4 e^{3} \left (a e^{2} - c d^{2}\right ) \log {\left (a e + c d x \right )}}{c^{5} d^{5}} \] Input:
integrate((e*x+d)**8/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)
Output:
(-13*a**4*e**8 + 22*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 + x**2*(-18*a**2*c**2*d**2*e**6 + 36*a*c**3*d**4*e**4 - 18*c**4*d**6*e**2) + x*(-30*a**3*c*d*e**7 + 54*a**2*c**2*d**3*e**5 - 18* a*c**3*d**5*e**3 - 6*c**4*d**7*e))/(3*a**3*c**5*d**5*e**3 + 9*a**2*c**6*d* *6*e**2*x + 9*a*c**7*d**7*e*x**2 + 3*c**8*d**8*x**3) + e**4*x/(c**4*d**4) - 4*e**3*(a*e**2 - c*d**2)*log(a*e + c*d*x)/(c**5*d**5)
Time = 0.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.66 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {c^{4} d^{8} + 2 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a^{3} c d^{2} e^{6} + 13 \, a^{4} e^{8} + 18 \, {\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 6 \, {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + 5 \, a^{3} c d e^{7}\right )} x}{3 \, {\left (c^{8} d^{8} x^{3} + 3 \, a c^{7} d^{7} e x^{2} + 3 \, a^{2} c^{6} d^{6} e^{2} x + a^{3} c^{5} d^{5} e^{3}\right )}} + \frac {e^{4} x}{c^{4} d^{4}} + \frac {4 \, {\left (c d^{2} e^{3} - a e^{5}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \] Input:
integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxi ma")
Output:
-1/3*(c^4*d^8 + 2*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 22*a^3*c*d^2*e^6 + 1 3*a^4*e^8 + 18*(c^4*d^6*e^2 - 2*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 6*( c^4*d^7*e + 3*a*c^3*d^5*e^3 - 9*a^2*c^2*d^3*e^5 + 5*a^3*c*d*e^7)*x)/(c^8*d ^8*x^3 + 3*a*c^7*d^7*e*x^2 + 3*a^2*c^6*d^6*e^2*x + a^3*c^5*d^5*e^3) + e^4* x/(c^4*d^4) + 4*(c*d^2*e^3 - a*e^5)*log(c*d*x + a*e)/(c^5*d^5)
Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^{4} x}{c^{4} d^{4}} + \frac {4 \, {\left (c d^{2} e^{3} - a e^{5}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{5}} - \frac {c^{4} d^{8} + 2 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a^{3} c d^{2} e^{6} + 13 \, a^{4} e^{8} + 18 \, {\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 6 \, {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + 5 \, a^{3} c d e^{7}\right )} x}{3 \, {\left (c d x + a e\right )}^{3} c^{5} d^{5}} \] Input:
integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac ")
Output:
e^4*x/(c^4*d^4) + 4*(c*d^2*e^3 - a*e^5)*log(abs(c*d*x + a*e))/(c^5*d^5) - 1/3*(c^4*d^8 + 2*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 22*a^3*c*d^2*e^6 + 13 *a^4*e^8 + 18*(c^4*d^6*e^2 - 2*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 6*(c ^4*d^7*e + 3*a*c^3*d^5*e^3 - 9*a^2*c^2*d^3*e^5 + 5*a^3*c*d*e^7)*x)/((c*d*x + a*e)^3*c^5*d^5)
Time = 5.28 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.68 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^4\,x}{c^4\,d^4}-\frac {x\,\left (10\,a^3\,e^7-18\,a^2\,c\,d^2\,e^5+6\,a\,c^2\,d^4\,e^3+2\,c^3\,d^6\,e\right )+x^2\,\left (6\,a^2\,c\,d\,e^6-12\,a\,c^2\,d^3\,e^4+6\,c^3\,d^5\,e^2\right )+\frac {13\,a^4\,e^8-22\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4+2\,a\,c^3\,d^6\,e^2+c^4\,d^8}{3\,c\,d}}{a^3\,c^4\,d^4\,e^3+3\,a^2\,c^5\,d^5\,e^2\,x+3\,a\,c^6\,d^6\,e\,x^2+c^7\,d^7\,x^3}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (4\,a\,e^5-4\,c\,d^2\,e^3\right )}{c^5\,d^5} \] Input:
int((d + e*x)^8/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)
Output:
(e^4*x)/(c^4*d^4) - (x*(10*a^3*e^7 + 2*c^3*d^6*e + 6*a*c^2*d^4*e^3 - 18*a^ 2*c*d^2*e^5) + x^2*(6*c^3*d^5*e^2 - 12*a*c^2*d^3*e^4 + 6*a^2*c*d*e^6) + (1 3*a^4*e^8 + c^4*d^8 + 2*a*c^3*d^6*e^2 - 22*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e ^4)/(3*c*d))/(c^7*d^7*x^3 + a^3*c^4*d^4*e^3 + 3*a*c^6*d^6*e*x^2 + 3*a^2*c^ 5*d^5*e^2*x) - (log(a*e + c*d*x)*(4*a*e^5 - 4*c*d^2*e^3))/(c^5*d^5)
Time = 0.22 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.60 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {-12 \,\mathrm {log}\left (c d x +a e \right ) a^{5} e^{8}+12 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c \,d^{2} e^{6}-36 \,\mathrm {log}\left (c d x +a e \right ) a^{4} c d \,e^{7} x +36 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{2} d^{3} e^{5} x -36 \,\mathrm {log}\left (c d x +a e \right ) a^{3} c^{2} d^{2} e^{6} x^{2}+36 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{3} d^{4} e^{4} x^{2}-12 \,\mathrm {log}\left (c d x +a e \right ) a^{2} c^{3} d^{3} e^{5} x^{3}+12 \,\mathrm {log}\left (c d x +a e \right ) a \,c^{4} d^{5} e^{3} x^{3}-10 a^{5} e^{8}+10 a^{4} c \,d^{2} e^{6}-18 a^{4} c d \,e^{7} x +18 a^{3} c^{2} d^{3} e^{5} x -2 a^{2} c^{3} d^{6} e^{2}+12 a^{2} c^{3} d^{3} e^{5} x^{3}-a \,c^{4} d^{8}-6 a \,c^{4} d^{7} e x -12 a \,c^{4} d^{5} e^{3} x^{3}+3 a \,c^{4} d^{4} e^{4} x^{4}+6 c^{5} d^{7} e \,x^{3}}{3 a \,c^{5} d^{5} \left (c^{3} d^{3} x^{3}+3 a \,c^{2} d^{2} e \,x^{2}+3 a^{2} c d \,e^{2} x +a^{3} e^{3}\right )} \] Input:
int((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)
Output:
( - 12*log(a*e + c*d*x)*a**5*e**8 + 12*log(a*e + c*d*x)*a**4*c*d**2*e**6 - 36*log(a*e + c*d*x)*a**4*c*d*e**7*x + 36*log(a*e + c*d*x)*a**3*c**2*d**3* e**5*x - 36*log(a*e + c*d*x)*a**3*c**2*d**2*e**6*x**2 + 36*log(a*e + c*d*x )*a**2*c**3*d**4*e**4*x**2 - 12*log(a*e + c*d*x)*a**2*c**3*d**3*e**5*x**3 + 12*log(a*e + c*d*x)*a*c**4*d**5*e**3*x**3 - 10*a**5*e**8 + 10*a**4*c*d** 2*e**6 - 18*a**4*c*d*e**7*x + 18*a**3*c**2*d**3*e**5*x - 2*a**2*c**3*d**6* e**2 + 12*a**2*c**3*d**3*e**5*x**3 - a*c**4*d**8 - 6*a*c**4*d**7*e*x - 12* a*c**4*d**5*e**3*x**3 + 3*a*c**4*d**4*e**4*x**4 + 6*c**5*d**7*e*x**3)/(3*a *c**5*d**5*(a**3*e**3 + 3*a**2*c*d*e**2*x + 3*a*c**2*d**2*e*x**2 + c**3*d* *3*x**3))