Integrand size = 35, antiderivative size = 219 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {15 e^2 \left (c d^2-a e^2\right )^4 x}{c^6 d^6}-\frac {\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac {10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac {5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac {3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac {e^6 (a e+c d x)^5}{5 c^7 d^7}+\frac {6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7} \]
15*e^2*(-a*e^2+c*d^2)^4*x/c^6/d^6-(-a*e^2+c*d^2)^6/c^7/d^7/(c*d*x+a*e)+10* e^3*(-a*e^2+c*d^2)^3*(c*d*x+a*e)^2/c^7/d^7+5*e^4*(-a*e^2+c*d^2)^2*(c*d*x+a *e)^3/c^7/d^7+3/2*e^5*(-a*e^2+c*d^2)*(c*d*x+a*e)^4/c^7/d^7+1/5*e^6*(c*d*x+ a*e)^5/c^7/d^7+6*e*(-a*e^2+c*d^2)^5*ln(c*d*x+a*e)/c^7/d^7
Time = 0.07 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.55 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {-10 a^6 e^{12}+10 a^5 c d e^{10} (6 d+5 e x)-30 a^4 c^2 d^2 e^8 \left (5 d^2+8 d e x-e^2 x^2\right )+10 a^3 c^3 d^3 e^6 \left (20 d^3+45 d^2 e x-15 d e^2 x^2-e^3 x^3\right )-5 a^2 c^4 d^4 e^4 \left (30 d^4+80 d^3 e x-60 d^2 e^2 x^2-10 d e^3 x^3-e^4 x^4\right )+a c^5 d^5 e^2 \left (60 d^5+150 d^4 e x-300 d^3 e^2 x^2-100 d^2 e^3 x^3-25 d e^4 x^4-3 e^5 x^5\right )+c^6 d^6 \left (-10 d^6+150 d^4 e^2 x^2+100 d^3 e^3 x^3+50 d^2 e^4 x^4+15 d e^5 x^5+2 e^6 x^6\right )-60 e \left (-c d^2+a e^2\right )^5 (a e+c d x) \log (a e+c d x)}{10 c^7 d^7 (a e+c d x)} \]
(-10*a^6*e^12 + 10*a^5*c*d*e^10*(6*d + 5*e*x) - 30*a^4*c^2*d^2*e^8*(5*d^2 + 8*d*e*x - e^2*x^2) + 10*a^3*c^3*d^3*e^6*(20*d^3 + 45*d^2*e*x - 15*d*e^2* x^2 - e^3*x^3) - 5*a^2*c^4*d^4*e^4*(30*d^4 + 80*d^3*e*x - 60*d^2*e^2*x^2 - 10*d*e^3*x^3 - e^4*x^4) + a*c^5*d^5*e^2*(60*d^5 + 150*d^4*e*x - 300*d^3*e ^2*x^2 - 100*d^2*e^3*x^3 - 25*d*e^4*x^4 - 3*e^5*x^5) + c^6*d^6*(-10*d^6 + 150*d^4*e^2*x^2 + 100*d^3*e^3*x^3 + 50*d^2*e^4*x^4 + 15*d*e^5*x^5 + 2*e^6* x^6) - 60*e*(-(c*d^2) + a*e^2)^5*(a*e + c*d*x)*Log[a*e + c*d*x])/(10*c^7*d ^7*(a*e + c*d*x))
Time = 0.51 (sec) , antiderivative size = 219, 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.
| \(\displaystyle \int \frac {(d+e x)^8}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {e^6 (a e+c d x)^4}{c^6 d^6}+\frac {20 \left (c d^2 e-a e^3\right )^3 (a e+c d x)}{c^6 d^6}+\frac {6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^2}+\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6}+\frac {6 \left (c d^2 e^5-a e^7\right ) (a e+c d x)^3}{c^6 d^6}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^6 (a e+c d x)^5}{5 c^7 d^7}-\frac {\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac {6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7}+\frac {3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac {5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac {10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac {15 e^2 x \left (c d^2-a e^2\right )^4}{c^6 d^6}\) |
(15*e^2*(c*d^2 - a*e^2)^4*x)/(c^6*d^6) - (c*d^2 - a*e^2)^6/(c^7*d^7*(a*e + c*d*x)) + (10*e^3*(c*d^2 - a*e^2)^3*(a*e + c*d*x)^2)/(c^7*d^7) + (5*e^4*( c*d^2 - a*e^2)^2*(a*e + c*d*x)^3)/(c^7*d^7) + (3*e^5*(c*d^2 - a*e^2)*(a*e + c*d*x)^4)/(2*c^7*d^7) + (e^6*(a*e + c*d*x)^5)/(5*c^7*d^7) + (6*e*(c*d^2 - a*e^2)^5*Log[a*e + c*d*x])/(c^7*d^7)
3.19.74.3.1 Defintions of rubi rules used
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 = 2.35 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.84
| method | result | size |
| default | \(\frac {e^{2} \left (\frac {1}{5} c^{4} d^{4} e^{4} x^{5}-\frac {1}{2} a \,c^{3} d^{3} e^{5} x^{4}+\frac {3}{2} c^{4} d^{5} e^{3} x^{4}+a^{2} c^{2} d^{2} e^{6} x^{3}-4 a \,c^{3} d^{4} e^{4} x^{3}+5 c^{4} d^{6} e^{2} x^{3}-2 a^{3} c d \,e^{7} x^{2}+9 a^{2} c^{2} d^{3} e^{5} x^{2}-15 a \,c^{3} d^{5} e^{3} x^{2}+10 c^{4} d^{7} e \,x^{2}+5 a^{4} e^{8} x -24 a^{3} c \,d^{2} e^{6} x +45 a^{2} c^{2} d^{4} e^{4} x -40 a \,c^{3} d^{6} e^{2} x +15 c^{4} d^{8} x \right )}{c^{6} d^{6}}-\frac {6 e \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \ln \left (c d x +a e \right )}{c^{7} d^{7}}-\frac {a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}{c^{7} d^{7} \left (c d x +a e \right )}\) | \(402\) |
| norman | \(\frac {\frac {e^{3} \left (3 a^{4} e^{8}-16 a^{3} c \,d^{2} e^{6}+35 a^{2} c^{2} d^{4} e^{4}-40 a \,c^{3} d^{6} e^{2}+25 c^{4} d^{8}\right ) x^{3}}{c^{5} d^{5}}-\frac {6 a^{6} e^{12}-27 a^{5} c \,d^{2} e^{10}+45 a^{4} c^{2} d^{4} e^{8}-30 a^{3} c^{3} d^{6} e^{6}+9 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}{c^{7} d^{6}}+\frac {e^{7} x^{7}}{5 c d}-\frac {e^{4} \left (2 e^{6} a^{3}-11 d^{2} e^{4} a^{2} c +25 d^{4} e^{2} c^{2} a -30 c^{3} d^{6}\right ) x^{4}}{2 c^{4} d^{4}}+\frac {e^{5} \left (5 a^{2} e^{4}-28 a c \,d^{2} e^{2}+65 c^{2} d^{4}\right ) x^{5}}{10 c^{3} d^{3}}-\frac {e^{6} \left (3 e^{2} a -17 c \,d^{2}\right ) x^{6}}{10 c^{2} d^{2}}-\frac {\left (6 a^{6} e^{14}-27 a^{5} c \,d^{2} e^{12}+48 a^{4} c^{2} d^{4} e^{10}-45 a^{3} c^{3} d^{6} e^{8}+30 a^{2} c^{4} d^{8} e^{6}-21 a \,c^{5} d^{10} e^{4}+16 c^{6} d^{12} e^{2}\right ) x}{c^{7} d^{7} e}}{\left (c d x +a e \right ) \left (e x +d \right )}-\frac {6 e \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \ln \left (c d x +a e \right )}{c^{7} d^{7}}\) | \(486\) |
| risch | \(-\frac {e^{7} a \,x^{4}}{2 c^{3} d^{3}}-\frac {2 e^{9} a^{3} x^{2}}{c^{5} d^{5}}+\frac {9 e^{7} a^{2} x^{2}}{c^{4} d^{3}}-\frac {15 e^{5} a \,x^{2}}{c^{3} d}+\frac {5 e^{10} a^{4} x}{c^{6} d^{6}}-\frac {24 e^{8} a^{3} x}{c^{5} d^{4}}+\frac {45 e^{6} a^{2} x}{c^{4} d^{2}}-\frac {40 e^{4} a x}{c^{3}}-\frac {a^{6} e^{12}}{c^{7} d^{7} \left (c d x +a e \right )}+\frac {6 a^{5} e^{10}}{c^{6} d^{5} \left (c d x +a e \right )}-\frac {15 a^{4} e^{8}}{c^{5} d^{3} \left (c d x +a e \right )}+\frac {20 a^{3} e^{6}}{c^{4} d \left (c d x +a e \right )}-\frac {15 d \,a^{2} e^{4}}{c^{3} \left (c d x +a e \right )}+\frac {6 d^{3} a \,e^{2}}{c^{2} \left (c d x +a e \right )}-\frac {d^{5}}{c \left (c d x +a e \right )}+\frac {6 d^{3} e \ln \left (c d x +a e \right )}{c^{2}}+\frac {e^{6} x^{5}}{5 c^{2} d^{2}}+\frac {3 e^{5} x^{4}}{2 c^{2} d}+\frac {5 e^{4} x^{3}}{c^{2}}+\frac {10 e^{3} d \,x^{2}}{c^{2}}+\frac {15 e^{2} d^{2} x}{c^{2}}-\frac {6 e^{11} \ln \left (c d x +a e \right ) a^{5}}{c^{7} d^{7}}+\frac {30 e^{9} \ln \left (c d x +a e \right ) a^{4}}{c^{6} d^{5}}-\frac {60 e^{7} \ln \left (c d x +a e \right ) a^{3}}{c^{5} d^{3}}+\frac {60 e^{5} \ln \left (c d x +a e \right ) a^{2}}{c^{4} d}-\frac {30 d \,e^{3} \ln \left (c d x +a e \right ) a}{c^{3}}+\frac {e^{8} a^{2} x^{3}}{c^{4} d^{4}}-\frac {4 e^{6} a \,x^{3}}{c^{3} d^{2}}\) | \(502\) |
| parallelrisch | \(-\frac {60 a^{6} e^{12}+10 c^{6} d^{12}-60 a \,c^{5} d^{10} e^{2}-600 a^{3} c^{3} d^{6} e^{6}+300 a^{2} c^{4} d^{8} e^{4}-300 a^{5} c \,d^{2} e^{10}+600 a^{4} c^{2} d^{4} e^{8}-2 x^{6} e^{6} c^{6} d^{6}-15 x^{5} c^{6} d^{7} e^{5}-50 x^{4} c^{6} d^{8} e^{4}-100 x^{3} c^{6} d^{9} e^{3}-150 x^{2} c^{6} d^{10} e^{2}-300 \ln \left (c d x +a e \right ) x \,a^{4} c^{2} d^{3} e^{9}+600 \ln \left (c d x +a e \right ) x \,a^{3} c^{3} d^{5} e^{7}+60 \ln \left (c d x +a e \right ) a^{6} e^{12}-600 \ln \left (c d x +a e \right ) x \,a^{2} c^{4} d^{7} e^{5}+300 \ln \left (c d x +a e \right ) x a \,c^{5} d^{9} e^{3}+60 \ln \left (c d x +a e \right ) x \,a^{5} c d \,e^{11}+150 x^{2} a^{3} c^{3} d^{4} e^{8}-300 x^{2} a^{2} c^{4} d^{6} e^{6}+300 x^{2} a \,c^{5} d^{8} e^{4}-60 \ln \left (c d x +a e \right ) x \,c^{6} d^{11} e -300 \ln \left (c d x +a e \right ) a^{5} c \,d^{2} e^{10}+600 \ln \left (c d x +a e \right ) a^{4} c^{2} d^{4} e^{8}-600 \ln \left (c d x +a e \right ) a^{3} c^{3} d^{6} e^{6}+300 \ln \left (c d x +a e \right ) a^{2} c^{4} d^{8} e^{4}-60 \ln \left (c d x +a e \right ) a \,c^{5} d^{10} e^{2}+3 x^{5} a \,c^{5} d^{5} e^{7}-5 x^{4} a^{2} c^{4} d^{4} e^{8}+25 x^{4} a \,c^{5} d^{6} e^{6}+10 x^{3} a^{3} c^{3} d^{3} e^{9}-50 x^{3} a^{2} c^{4} d^{5} e^{7}+100 x^{3} a \,c^{5} d^{7} e^{5}-30 x^{2} a^{4} c^{2} d^{2} e^{10}}{10 c^{7} d^{7} \left (c d x +a e \right )}\) | \(595\) |
e^2/c^6/d^6*(1/5*c^4*d^4*e^4*x^5-1/2*a*c^3*d^3*e^5*x^4+3/2*c^4*d^5*e^3*x^4 +a^2*c^2*d^2*e^6*x^3-4*a*c^3*d^4*e^4*x^3+5*c^4*d^6*e^2*x^3-2*a^3*c*d*e^7*x ^2+9*a^2*c^2*d^3*e^5*x^2-15*a*c^3*d^5*e^3*x^2+10*c^4*d^7*e*x^2+5*a^4*e^8*x -24*a^3*c*d^2*e^6*x+45*a^2*c^2*d^4*e^4*x-40*a*c^3*d^6*e^2*x+15*c^4*d^8*x)- 6/c^7/d^7*e*(a^5*e^10-5*a^4*c*d^2*e^8+10*a^3*c^2*d^4*e^6-10*a^2*c^3*d^6*e^ 4+5*a*c^4*d^8*e^2-c^5*d^10)*ln(c*d*x+a*e)-1/c^7/d^7*(a^6*e^12-6*a^5*c*d^2* e^10+15*a^4*c^2*d^4*e^8-20*a^3*c^3*d^6*e^6+15*a^2*c^4*d^8*e^4-6*a*c^5*d^10 *e^2+c^6*d^12)/(c*d*x+a*e)
Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (215) = 430\).
Time = 0.39 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.49 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {2 \, c^{6} d^{6} e^{6} x^{6} - 10 \, c^{6} d^{12} + 60 \, a c^{5} d^{10} e^{2} - 150 \, a^{2} c^{4} d^{8} e^{4} + 200 \, a^{3} c^{3} d^{6} e^{6} - 150 \, a^{4} c^{2} d^{4} e^{8} + 60 \, a^{5} c d^{2} e^{10} - 10 \, a^{6} e^{12} + 3 \, {\left (5 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 5 \, {\left (10 \, c^{6} d^{8} e^{4} - 5 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + 10 \, {\left (10 \, c^{6} d^{9} e^{3} - 10 \, a c^{5} d^{7} e^{5} + 5 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 30 \, {\left (5 \, c^{6} d^{10} e^{2} - 10 \, a c^{5} d^{8} e^{4} + 10 \, a^{2} c^{4} d^{6} e^{6} - 5 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 10 \, {\left (15 \, a c^{5} d^{9} e^{3} - 40 \, a^{2} c^{4} d^{7} e^{5} + 45 \, a^{3} c^{3} d^{5} e^{7} - 24 \, a^{4} c^{2} d^{3} e^{9} + 5 \, a^{5} c d e^{11}\right )} x + 60 \, {\left (a c^{5} d^{10} e^{2} - 5 \, a^{2} c^{4} d^{8} e^{4} + 10 \, a^{3} c^{3} d^{6} e^{6} - 10 \, a^{4} c^{2} d^{4} e^{8} + 5 \, a^{5} c d^{2} e^{10} - a^{6} e^{12} + {\left (c^{6} d^{11} e - 5 \, a c^{5} d^{9} e^{3} + 10 \, a^{2} c^{4} d^{7} e^{5} - 10 \, a^{3} c^{3} d^{5} e^{7} + 5 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x\right )} \log \left (c d x + a e\right )}{10 \, {\left (c^{8} d^{8} x + a c^{7} d^{7} e\right )}} \]
1/10*(2*c^6*d^6*e^6*x^6 - 10*c^6*d^12 + 60*a*c^5*d^10*e^2 - 150*a^2*c^4*d^ 8*e^4 + 200*a^3*c^3*d^6*e^6 - 150*a^4*c^2*d^4*e^8 + 60*a^5*c*d^2*e^10 - 10 *a^6*e^12 + 3*(5*c^6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 5*(10*c^6*d^8*e^4 - 5* a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + 10*(10*c^6*d^9*e^3 - 10*a*c^5*d^7*e ^5 + 5*a^2*c^4*d^5*e^7 - a^3*c^3*d^3*e^9)*x^3 + 30*(5*c^6*d^10*e^2 - 10*a* c^5*d^8*e^4 + 10*a^2*c^4*d^6*e^6 - 5*a^3*c^3*d^4*e^8 + a^4*c^2*d^2*e^10)*x ^2 + 10*(15*a*c^5*d^9*e^3 - 40*a^2*c^4*d^7*e^5 + 45*a^3*c^3*d^5*e^7 - 24*a ^4*c^2*d^3*e^9 + 5*a^5*c*d*e^11)*x + 60*(a*c^5*d^10*e^2 - 5*a^2*c^4*d^8*e^ 4 + 10*a^3*c^3*d^6*e^6 - 10*a^4*c^2*d^4*e^8 + 5*a^5*c*d^2*e^10 - a^6*e^12 + (c^6*d^11*e - 5*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e^5 - 10*a^3*c^3*d^5*e^7 + 5*a^4*c^2*d^3*e^9 - a^5*c*d*e^11)*x)*log(c*d*x + a*e))/(c^8*d^8*x + a*c^ 7*d^7*e)
Time = 3.75 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.58 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=x^{4} \left (- \frac {a e^{7}}{2 c^{3} d^{3}} + \frac {3 e^{5}}{2 c^{2} d}\right ) + x^{3} \left (\frac {a^{2} e^{8}}{c^{4} d^{4}} - \frac {4 a e^{6}}{c^{3} d^{2}} + \frac {5 e^{4}}{c^{2}}\right ) + x^{2} \left (- \frac {2 a^{3} e^{9}}{c^{5} d^{5}} + \frac {9 a^{2} e^{7}}{c^{4} d^{3}} - \frac {15 a e^{5}}{c^{3} d} + \frac {10 d e^{3}}{c^{2}}\right ) + x \left (\frac {5 a^{4} e^{10}}{c^{6} d^{6}} - \frac {24 a^{3} e^{8}}{c^{5} d^{4}} + \frac {45 a^{2} e^{6}}{c^{4} d^{2}} - \frac {40 a e^{4}}{c^{3}} + \frac {15 d^{2} e^{2}}{c^{2}}\right ) + \frac {- a^{6} e^{12} + 6 a^{5} c d^{2} e^{10} - 15 a^{4} c^{2} d^{4} e^{8} + 20 a^{3} c^{3} d^{6} e^{6} - 15 a^{2} c^{4} d^{8} e^{4} + 6 a c^{5} d^{10} e^{2} - c^{6} d^{12}}{a c^{7} d^{7} e + c^{8} d^{8} x} + \frac {e^{6} x^{5}}{5 c^{2} d^{2}} - \frac {6 e \left (a e^{2} - c d^{2}\right )^{5} \log {\left (a e + c d x \right )}}{c^{7} d^{7}} \]
x**4*(-a*e**7/(2*c**3*d**3) + 3*e**5/(2*c**2*d)) + x**3*(a**2*e**8/(c**4*d **4) - 4*a*e**6/(c**3*d**2) + 5*e**4/c**2) + x**2*(-2*a**3*e**9/(c**5*d**5 ) + 9*a**2*e**7/(c**4*d**3) - 15*a*e**5/(c**3*d) + 10*d*e**3/c**2) + x*(5* a**4*e**10/(c**6*d**6) - 24*a**3*e**8/(c**5*d**4) + 45*a**2*e**6/(c**4*d** 2) - 40*a*e**4/c**3 + 15*d**2*e**2/c**2) + (-a**6*e**12 + 6*a**5*c*d**2*e* *10 - 15*a**4*c**2*d**4*e**8 + 20*a**3*c**3*d**6*e**6 - 15*a**2*c**4*d**8* e**4 + 6*a*c**5*d**10*e**2 - c**6*d**12)/(a*c**7*d**7*e + c**8*d**8*x) + e **6*x**5/(5*c**2*d**2) - 6*e*(a*e**2 - c*d**2)**5*log(a*e + c*d*x)/(c**7*d **7)
Time = 0.20 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.82 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{c^{8} d^{8} x + a c^{7} d^{7} e} + \frac {2 \, c^{4} d^{4} e^{6} x^{5} + 5 \, {\left (3 \, c^{4} d^{5} e^{5} - a c^{3} d^{3} e^{7}\right )} x^{4} + 10 \, {\left (5 \, c^{4} d^{6} e^{4} - 4 \, a c^{3} d^{4} e^{6} + a^{2} c^{2} d^{2} e^{8}\right )} x^{3} + 10 \, {\left (10 \, c^{4} d^{7} e^{3} - 15 \, a c^{3} d^{5} e^{5} + 9 \, a^{2} c^{2} d^{3} e^{7} - 2 \, a^{3} c d e^{9}\right )} x^{2} + 10 \, {\left (15 \, c^{4} d^{8} e^{2} - 40 \, a c^{3} d^{6} e^{4} + 45 \, a^{2} c^{2} d^{4} e^{6} - 24 \, a^{3} c d^{2} e^{8} + 5 \, a^{4} e^{10}\right )} x}{10 \, c^{6} d^{6}} + \frac {6 \, {\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \]
-(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)/(c^8*d^8*x + a*c^7*d^7*e ) + 1/10*(2*c^4*d^4*e^6*x^5 + 5*(3*c^4*d^5*e^5 - a*c^3*d^3*e^7)*x^4 + 10*( 5*c^4*d^6*e^4 - 4*a*c^3*d^4*e^6 + a^2*c^2*d^2*e^8)*x^3 + 10*(10*c^4*d^7*e^ 3 - 15*a*c^3*d^5*e^5 + 9*a^2*c^2*d^3*e^7 - 2*a^3*c*d*e^9)*x^2 + 10*(15*c^4 *d^8*e^2 - 40*a*c^3*d^6*e^4 + 45*a^2*c^2*d^4*e^6 - 24*a^3*c*d^2*e^8 + 5*a^ 4*e^10)*x)/(c^6*d^6) + 6*(c^5*d^10*e - 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^ 5 - 10*a^3*c^2*d^4*e^7 + 5*a^4*c*d^2*e^9 - a^5*e^11)*log(c*d*x + a*e)/(c^7 *d^7)
Time = 0.28 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.91 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {6 \, {\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{7}} - \frac {c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{{\left (c d x + a e\right )} c^{7} d^{7}} + \frac {2 \, c^{8} d^{8} e^{6} x^{5} + 15 \, c^{8} d^{9} e^{5} x^{4} - 5 \, a c^{7} d^{7} e^{7} x^{4} + 50 \, c^{8} d^{10} e^{4} x^{3} - 40 \, a c^{7} d^{8} e^{6} x^{3} + 10 \, a^{2} c^{6} d^{6} e^{8} x^{3} + 100 \, c^{8} d^{11} e^{3} x^{2} - 150 \, a c^{7} d^{9} e^{5} x^{2} + 90 \, a^{2} c^{6} d^{7} e^{7} x^{2} - 20 \, a^{3} c^{5} d^{5} e^{9} x^{2} + 150 \, c^{8} d^{12} e^{2} x - 400 \, a c^{7} d^{10} e^{4} x + 450 \, a^{2} c^{6} d^{8} e^{6} x - 240 \, a^{3} c^{5} d^{6} e^{8} x + 50 \, a^{4} c^{4} d^{4} e^{10} x}{10 \, c^{10} d^{10}} \]
6*(c^5*d^10*e - 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 - 10*a^3*c^2*d^4*e^7 + 5*a^4*c*d^2*e^9 - a^5*e^11)*log(abs(c*d*x + a*e))/(c^7*d^7) - (c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2* d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)/((c*d*x + a*e)*c^7*d^7) + 1/10*(2*c ^8*d^8*e^6*x^5 + 15*c^8*d^9*e^5*x^4 - 5*a*c^7*d^7*e^7*x^4 + 50*c^8*d^10*e^ 4*x^3 - 40*a*c^7*d^8*e^6*x^3 + 10*a^2*c^6*d^6*e^8*x^3 + 100*c^8*d^11*e^3*x ^2 - 150*a*c^7*d^9*e^5*x^2 + 90*a^2*c^6*d^7*e^7*x^2 - 20*a^3*c^5*d^5*e^9*x ^2 + 150*c^8*d^12*e^2*x - 400*a*c^7*d^10*e^4*x + 450*a^2*c^6*d^8*e^6*x - 2 40*a^3*c^5*d^6*e^8*x + 50*a^4*c^4*d^4*e^10*x)/(c^10*d^10)
Time = 0.11 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.85 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=x^4\,\left (\frac {3\,e^5}{2\,c^2\,d}-\frac {a\,e^7}{2\,c^3\,d^3}\right )+x^2\,\left (\frac {10\,d\,e^3}{c^2}+\frac {a\,e\,\left (\frac {a^2\,e^8}{c^4\,d^4}-\frac {15\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c\,d}\right )}{c\,d}-\frac {a^2\,e^2\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{2\,c^2\,d^2}\right )-x^3\,\left (\frac {a^2\,e^8}{3\,c^4\,d^4}-\frac {5\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{3\,c\,d}\right )+x\,\left (\frac {15\,d^2\,e^2}{c^2}+\frac {a^2\,e^2\,\left (\frac {a^2\,e^8}{c^4\,d^4}-\frac {15\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c\,d}\right )}{c^2\,d^2}-\frac {2\,a\,e\,\left (\frac {20\,d\,e^3}{c^2}+\frac {2\,a\,e\,\left (\frac {a^2\,e^8}{c^4\,d^4}-\frac {15\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c\,d}\right )}{c\,d}-\frac {a^2\,e^2\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c^2\,d^2}\right )}{c\,d}\right )-\frac {a^6\,e^{12}-6\,a^5\,c\,d^2\,e^{10}+15\,a^4\,c^2\,d^4\,e^8-20\,a^3\,c^3\,d^6\,e^6+15\,a^2\,c^4\,d^8\,e^4-6\,a\,c^5\,d^{10}\,e^2+c^6\,d^{12}}{c\,d\,\left (x\,c^7\,d^7+a\,e\,c^6\,d^6\right )}+\frac {e^6\,x^5}{5\,c^2\,d^2}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (6\,a^5\,e^{11}-30\,a^4\,c\,d^2\,e^9+60\,a^3\,c^2\,d^4\,e^7-60\,a^2\,c^3\,d^6\,e^5+30\,a\,c^4\,d^8\,e^3-6\,c^5\,d^{10}\,e\right )}{c^7\,d^7} \]
x^4*((3*e^5)/(2*c^2*d) - (a*e^7)/(2*c^3*d^3)) + x^2*((10*d*e^3)/c^2 + (a*e *((a^2*e^8)/(c^4*d^4) - (15*e^4)/c^2 + (2*a*e*((6*e^5)/(c^2*d) - (2*a*e^7) /(c^3*d^3)))/(c*d)))/(c*d) - (a^2*e^2*((6*e^5)/(c^2*d) - (2*a*e^7)/(c^3*d^ 3)))/(2*c^2*d^2)) - x^3*((a^2*e^8)/(3*c^4*d^4) - (5*e^4)/c^2 + (2*a*e*((6* e^5)/(c^2*d) - (2*a*e^7)/(c^3*d^3)))/(3*c*d)) + x*((15*d^2*e^2)/c^2 + (a^2 *e^2*((a^2*e^8)/(c^4*d^4) - (15*e^4)/c^2 + (2*a*e*((6*e^5)/(c^2*d) - (2*a* e^7)/(c^3*d^3)))/(c*d)))/(c^2*d^2) - (2*a*e*((20*d*e^3)/c^2 + (2*a*e*((a^2 *e^8)/(c^4*d^4) - (15*e^4)/c^2 + (2*a*e*((6*e^5)/(c^2*d) - (2*a*e^7)/(c^3* d^3)))/(c*d)))/(c*d) - (a^2*e^2*((6*e^5)/(c^2*d) - (2*a*e^7)/(c^3*d^3)))/( c^2*d^2)))/(c*d)) - (a^6*e^12 + c^6*d^12 - 6*a*c^5*d^10*e^2 - 6*a^5*c*d^2* e^10 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8)/(c*d* (c^7*d^7*x + a*c^6*d^6*e)) + (e^6*x^5)/(5*c^2*d^2) - (log(a*e + c*d*x)*(6* a^5*e^11 - 6*c^5*d^10*e + 30*a*c^4*d^8*e^3 - 30*a^4*c*d^2*e^9 - 60*a^2*c^3 *d^6*e^5 + 60*a^3*c^2*d^4*e^7))/(c^7*d^7)