Integrand size = 35, antiderivative size = 173 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}+\frac {c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {e^3}{\left (c d^2-a e^2\right )^4 (d+e x)}-\frac {4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \]
-1/3*c*d/(-a*e^2+c*d^2)^2/(c*d*x+a*e)^3+c*d*e/(-a*e^2+c*d^2)^3/(c*d*x+a*e) ^2-3*c*d*e^2/(-a*e^2+c*d^2)^4/(c*d*x+a*e)-e^3/(-a*e^2+c*d^2)^4/(e*x+d)-4*c *d*e^3*ln(c*d*x+a*e)/(-a*e^2+c*d^2)^5+4*c*d*e^3*ln(e*x+d)/(-a*e^2+c*d^2)^5
Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {\frac {c d \left (c d^2-a e^2\right )^3}{(a e+c d x)^3}-\frac {3 c d e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {9 c d e^2 \left (c d^2-a e^2\right )}{a e+c d x}+\frac {3 c d^2 e^3-3 a e^5}{d+e x}+12 c d e^3 \log (a e+c d x)-12 c d e^3 \log (d+e x)}{3 \left (-c d^2+a e^2\right )^5} \]
((c*d*(c*d^2 - a*e^2)^3)/(a*e + c*d*x)^3 - (3*c*d*e*(c*d^2 - a*e^2)^2)/(a* e + c*d*x)^2 + (9*c*d*e^2*(c*d^2 - a*e^2))/(a*e + c*d*x) + (3*c*d^2*e^3 - 3*a*e^5)/(d + e*x) + 12*c*d*e^3*Log[a*e + c*d*x] - 12*c*d*e^3*Log[d + e*x] )/(3*(-(c*d^2) + a*e^2)^5)
Time = 0.38 (sec) , antiderivative size = 173, 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.
| \(\displaystyle \int \frac {(d+e x)^2}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {3 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac {2 c^2 d^2 e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^3}+\frac {c^2 d^2}{\left (c d^2-a e^2\right )^2 (a e+c d x)^4}-\frac {4 c^2 d^2 e^3}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {4 c d e^4}{(d+e x) \left (c d^2-a e^2\right )^5}+\frac {e^4}{(d+e x)^2 \left (c d^2-a e^2\right )^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac {c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}-\frac {e^3}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac {4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5}\) |
-1/3*(c*d)/((c*d^2 - a*e^2)^2*(a*e + c*d*x)^3) + (c*d*e)/((c*d^2 - a*e^2)^ 3*(a*e + c*d*x)^2) - (3*c*d*e^2)/((c*d^2 - a*e^2)^4*(a*e + c*d*x)) - e^3/( (c*d^2 - a*e^2)^4*(d + e*x)) - (4*c*d*e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2 )^5 + (4*c*d*e^3*Log[d + e*x])/(c*d^2 - a*e^2)^5
3.20.4.3.1 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 = 2.46 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(-\frac {c d}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d x +a e \right )^{3}}+\frac {4 c d \,e^{3} \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{5}}-\frac {3 c d \,e^{2}}{\left (e^{2} a -c \,d^{2}\right )^{4} \left (c d x +a e \right )}-\frac {c d e}{\left (e^{2} a -c \,d^{2}\right )^{3} \left (c d x +a e \right )^{2}}-\frac {e^{3}}{\left (e^{2} a -c \,d^{2}\right )^{4} \left (e x +d \right )}-\frac {4 c d \,e^{3} \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{5}}\) | \(173\) |
| risch | \(\frac {-\frac {4 c^{3} d^{3} e^{3} x^{3}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {2 c^{2} d^{2} \left (5 e^{2} a +c \,d^{2}\right ) e^{2} x^{2}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {2 \left (11 a^{2} e^{4}+8 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) d e c x}{3 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}-\frac {3 e^{6} a^{3}+13 d^{2} e^{4} a^{2} c -5 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{3 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}}{\left (c d x +a e \right )^{2} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )}-\frac {4 d \,e^{3} c \ln \left (e x +d \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {4 d \,e^{3} c \ln \left (-c d x -a e \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}\) | \(546\) |
| parallelrisch | \(-\frac {-12 x^{3} c^{7} d^{8} e^{4}-6 x^{2} c^{7} d^{9} e^{3}+2 x \,c^{7} d^{10} e^{2}+10 a^{3} c^{4} d^{5} e^{7}-18 a^{2} c^{5} d^{7} e^{5}+6 a \,c^{6} d^{9} e^{3}+3 a^{4} c^{3} d^{3} e^{9}+12 x^{3} a \,c^{6} d^{6} e^{6}+30 x^{2} a^{2} c^{5} d^{5} e^{7}-24 x^{2} a \,c^{6} d^{7} e^{5}+22 x \,a^{3} c^{4} d^{4} e^{8}-6 x \,a^{2} c^{5} d^{6} e^{6}-18 x a \,c^{6} d^{8} e^{4}+12 \ln \left (e x +d \right ) x^{4} c^{7} d^{7} e^{5}-12 \ln \left (c d x +a e \right ) x^{4} c^{7} d^{7} e^{5}+12 \ln \left (e x +d \right ) x^{3} c^{7} d^{8} e^{4}-12 \ln \left (c d x +a e \right ) x^{3} c^{7} d^{8} e^{4}+12 \ln \left (e x +d \right ) a^{3} c^{4} d^{5} e^{7}-12 \ln \left (c d x +a e \right ) a^{3} c^{4} d^{5} e^{7}-36 \ln \left (c d x +a e \right ) x^{2} a \,c^{6} d^{7} e^{5}+12 \ln \left (e x +d \right ) x \,a^{3} c^{4} d^{4} e^{8}+36 \ln \left (e x +d \right ) x \,a^{2} c^{5} d^{6} e^{6}-12 \ln \left (c d x +a e \right ) x \,a^{3} c^{4} d^{4} e^{8}-36 \ln \left (c d x +a e \right ) x \,a^{2} c^{5} d^{6} e^{6}+36 \ln \left (e x +d \right ) x^{3} a \,c^{6} d^{6} e^{6}-36 \ln \left (c d x +a e \right ) x^{3} a \,c^{6} d^{6} e^{6}+36 \ln \left (e x +d \right ) x^{2} a^{2} c^{5} d^{5} e^{7}+36 \ln \left (e x +d \right ) x^{2} a \,c^{6} d^{7} e^{5}-36 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{5} d^{5} e^{7}-c^{7} d^{11} e}{3 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right ) \left (c d x +a e \right )^{2} d^{3} e \,c^{3}}\) | \(670\) |
| norman | \(\frac {-\frac {4 c^{3} d^{3} e^{5} x^{5}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {\left (-10 a \,c^{4} d^{4} e^{8}-10 d^{6} e^{6} c^{5}\right ) x^{4}}{e^{2} d^{2} c^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {\left (-a^{3} c^{3} d^{2} e^{10}-19 a^{2} c^{4} d^{4} e^{8}-19 a \,c^{5} d^{6} e^{6}-d^{8} e^{4} c^{6}\right ) x^{2}}{e^{2} d^{2} c^{3} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {a \left (-2 a^{2} c^{3} d^{2} e^{8}-16 a \,c^{4} d^{4} e^{6}-2 e^{4} d^{6} c^{5}\right ) x}{e d \,c^{3} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {-3 c^{3} e^{6} d^{2} a^{3}-13 c^{4} d^{4} e^{4} a^{2}+5 c^{5} d^{6} e^{2} a -c^{6} d^{8}}{3 c^{3} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {\left (-22 a^{2} c^{4} d^{4} e^{10}-76 a \,c^{5} d^{6} e^{8}-22 c^{6} d^{8} e^{6}\right ) x^{3}}{3 e^{3} d^{3} c^{3} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}-\frac {4 d \,e^{3} c \ln \left (e x +d \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {4 d \,e^{3} c \ln \left (c d x +a e \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}\) | \(792\) |
-1/3*c*d/(a*e^2-c*d^2)^2/(c*d*x+a*e)^3+4*c*d/(a*e^2-c*d^2)^5*e^3*ln(c*d*x+ a*e)-3*c*d/(a*e^2-c*d^2)^4*e^2/(c*d*x+a*e)-c*d/(a*e^2-c*d^2)^3*e/(c*d*x+a* e)^2-e^3/(a*e^2-c*d^2)^4/(e*x+d)-4*c*d/(a*e^2-c*d^2)^5*e^3*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 837 vs. \(2 (171) = 342\).
Time = 0.28 (sec) , antiderivative size = 837, normalized size of antiderivative = 4.84 \[ \int \frac {(d+e x)^2}{\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} - 6 \, a c^{3} d^{6} e^{2} + 18 \, a^{2} c^{2} d^{4} e^{4} - 10 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 12 \, {\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (c^{4} d^{6} e^{2} + 4 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 2 \, {\left (c^{4} d^{7} e - 9 \, a c^{3} d^{5} e^{3} - 3 \, a^{2} c^{2} d^{3} e^{5} + 11 \, a^{3} c d e^{7}\right )} x + 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a^{3} c d^{2} e^{6} + {\left (c^{4} d^{5} e^{3} + 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} + {\left (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 ) - 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a^{3} c d^{2} e^{6} + {\left (c^{4} d^{5} e^{3} + 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} + {\left (3 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (a^{3} c^{5} d^{11} e^{3} - 5 \, a^{4} c^{4} d^{9} e^{5} + 10 \, a^{5} c^{3} d^{7} e^{7} - 10 \, a^{6} c^{2} d^{5} e^{9} + 5 \, a^{7} c d^{3} e^{11} - a^{8} d e^{13} + {\left (c^{8} d^{13} e - 5 \, a c^{7} d^{11} e^{3} + 10 \, a^{2} c^{6} d^{9} e^{5} - 10 \, a^{3} c^{5} d^{7} e^{7} + 5 \, a^{4} c^{4} d^{5} e^{9} - a^{5} c^{3} d^{3} e^{11}\right )} x^{4} + {\left (c^{8} d^{14} - 2 \, a c^{7} d^{12} e^{2} - 5 \, a^{2} c^{6} d^{10} e^{4} + 20 \, a^{3} c^{5} d^{8} e^{6} - 25 \, a^{4} c^{4} d^{6} e^{8} + 14 \, a^{5} c^{3} d^{4} e^{10} - 3 \, a^{6} c^{2} d^{2} e^{12}\right )} x^{3} + 3 \, {\left (a c^{7} d^{13} e - 4 \, a^{2} c^{6} d^{11} e^{3} + 5 \, a^{3} c^{5} d^{9} e^{5} - 5 \, a^{5} c^{3} d^{5} e^{9} + 4 \, a^{6} c^{2} d^{3} e^{11} - a^{7} c d e^{13}\right )} x^{2} + {\left (3 \, a^{2} c^{6} d^{12} e^{2} - 14 \, a^{3} c^{5} d^{10} e^{4} + 25 \, a^{4} c^{4} d^{8} e^{6} - 20 \, a^{5} c^{3} d^{6} e^{8} + 5 \, a^{6} c^{2} d^{4} e^{10} + 2 \, a^{7} c d^{2} e^{12} - a^{8} e^{14}\right )} x\right )}} \]
-1/3*(c^4*d^8 - 6*a*c^3*d^6*e^2 + 18*a^2*c^2*d^4*e^4 - 10*a^3*c*d^2*e^6 - 3*a^4*e^8 + 12*(c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 + 6*(c^4*d^6*e^2 + 4*a*c^ 3*d^4*e^4 - 5*a^2*c^2*d^2*e^6)*x^2 - 2*(c^4*d^7*e - 9*a*c^3*d^5*e^3 - 3*a^ 2*c^2*d^3*e^5 + 11*a^3*c*d*e^7)*x + 12*(c^4*d^4*e^4*x^4 + a^3*c*d^2*e^6 + (c^4*d^5*e^3 + 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) - 12*(c^4*d^4* e^4*x^4 + a^3*c*d^2*e^6 + (c^4*d^5*e^3 + 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(e *x + d))/(a^3*c^5*d^11*e^3 - 5*a^4*c^4*d^9*e^5 + 10*a^5*c^3*d^7*e^7 - 10*a ^6*c^2*d^5*e^9 + 5*a^7*c*d^3*e^11 - a^8*d*e^13 + (c^8*d^13*e - 5*a*c^7*d^1 1*e^3 + 10*a^2*c^6*d^9*e^5 - 10*a^3*c^5*d^7*e^7 + 5*a^4*c^4*d^5*e^9 - a^5* c^3*d^3*e^11)*x^4 + (c^8*d^14 - 2*a*c^7*d^12*e^2 - 5*a^2*c^6*d^10*e^4 + 20 *a^3*c^5*d^8*e^6 - 25*a^4*c^4*d^6*e^8 + 14*a^5*c^3*d^4*e^10 - 3*a^6*c^2*d^ 2*e^12)*x^3 + 3*(a*c^7*d^13*e - 4*a^2*c^6*d^11*e^3 + 5*a^3*c^5*d^9*e^5 - 5 *a^5*c^3*d^5*e^9 + 4*a^6*c^2*d^3*e^11 - a^7*c*d*e^13)*x^2 + (3*a^2*c^6*d^1 2*e^2 - 14*a^3*c^5*d^10*e^4 + 25*a^4*c^4*d^8*e^6 - 20*a^5*c^3*d^6*e^8 + 5* a^6*c^2*d^4*e^10 + 2*a^7*c*d^2*e^12 - a^8*e^14)*x)
Leaf count of result is larger than twice the leaf count of optimal. 1006 vs. \(2 (160) = 320\).
Time = 2.79 (sec) , antiderivative size = 1006, normalized size of antiderivative = 5.82 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=- \frac {4 c d e^{3} \log {\left (x + \frac {- \frac {4 a^{6} c d e^{15}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {24 a^{5} c^{2} d^{3} e^{13}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {60 a^{4} c^{3} d^{5} e^{11}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {80 a^{3} c^{4} d^{7} e^{9}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {60 a^{2} c^{5} d^{9} e^{7}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {24 a c^{6} d^{11} e^{5}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 a c d e^{5} - \frac {4 c^{7} d^{13} e^{3}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 c^{2} d^{3} e^{3}}{8 c^{2} d^{2} e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {4 c d e^{3} \log {\left (x + \frac {\frac {4 a^{6} c d e^{15}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {24 a^{5} c^{2} d^{3} e^{13}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {60 a^{4} c^{3} d^{5} e^{11}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {80 a^{3} c^{4} d^{7} e^{9}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {60 a^{2} c^{5} d^{9} e^{7}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {24 a c^{6} d^{11} e^{5}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 a c d e^{5} + \frac {4 c^{7} d^{13} e^{3}}{\left (a e^{2} - c d^{2}\right )^{5}} + 4 c^{2} d^{3} e^{3}}{8 c^{2} d^{2} e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {- 3 a^{3} e^{6} - 13 a^{2} c d^{2} e^{4} + 5 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 12 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 30 a c^{2} d^{2} e^{4} - 6 c^{3} d^{4} e^{2}\right ) + x \left (- 22 a^{2} c d e^{5} - 16 a c^{2} d^{3} e^{3} + 2 c^{3} d^{5} e\right )}{3 a^{7} d e^{11} - 12 a^{6} c d^{3} e^{9} + 18 a^{5} c^{2} d^{5} e^{7} - 12 a^{4} c^{3} d^{7} e^{5} + 3 a^{3} c^{4} d^{9} e^{3} + x^{4} \cdot \left (3 a^{4} c^{3} d^{3} e^{9} - 12 a^{3} c^{4} d^{5} e^{7} + 18 a^{2} c^{5} d^{7} e^{5} - 12 a c^{6} d^{9} e^{3} + 3 c^{7} d^{11} e\right ) + x^{3} \cdot \left (9 a^{5} c^{2} d^{2} e^{10} - 33 a^{4} c^{3} d^{4} e^{8} + 42 a^{3} c^{4} d^{6} e^{6} - 18 a^{2} c^{5} d^{8} e^{4} - 3 a c^{6} d^{10} e^{2} + 3 c^{7} d^{12}\right ) + x^{2} \cdot \left (9 a^{6} c d e^{11} - 27 a^{5} c^{2} d^{3} e^{9} + 18 a^{4} c^{3} d^{5} e^{7} + 18 a^{3} c^{4} d^{7} e^{5} - 27 a^{2} c^{5} d^{9} e^{3} + 9 a c^{6} d^{11} e\right ) + x \left (3 a^{7} e^{12} - 3 a^{6} c d^{2} e^{10} - 18 a^{5} c^{2} d^{4} e^{8} + 42 a^{4} c^{3} d^{6} e^{6} - 33 a^{3} c^{4} d^{8} e^{4} + 9 a^{2} c^{5} d^{10} e^{2}\right )} \]
-4*c*d*e**3*log(x + (-4*a**6*c*d*e**15/(a*e**2 - c*d**2)**5 + 24*a**5*c**2 *d**3*e**13/(a*e**2 - c*d**2)**5 - 60*a**4*c**3*d**5*e**11/(a*e**2 - c*d** 2)**5 + 80*a**3*c**4*d**7*e**9/(a*e**2 - c*d**2)**5 - 60*a**2*c**5*d**9*e* *7/(a*e**2 - c*d**2)**5 + 24*a*c**6*d**11*e**5/(a*e**2 - c*d**2)**5 + 4*a* c*d*e**5 - 4*c**7*d**13*e**3/(a*e**2 - c*d**2)**5 + 4*c**2*d**3*e**3)/(8*c **2*d**2*e**4))/(a*e**2 - c*d**2)**5 + 4*c*d*e**3*log(x + (4*a**6*c*d*e**1 5/(a*e**2 - c*d**2)**5 - 24*a**5*c**2*d**3*e**13/(a*e**2 - c*d**2)**5 + 60 *a**4*c**3*d**5*e**11/(a*e**2 - c*d**2)**5 - 80*a**3*c**4*d**7*e**9/(a*e** 2 - c*d**2)**5 + 60*a**2*c**5*d**9*e**7/(a*e**2 - c*d**2)**5 - 24*a*c**6*d **11*e**5/(a*e**2 - c*d**2)**5 + 4*a*c*d*e**5 + 4*c**7*d**13*e**3/(a*e**2 - c*d**2)**5 + 4*c**2*d**3*e**3)/(8*c**2*d**2*e**4))/(a*e**2 - c*d**2)**5 + (-3*a**3*e**6 - 13*a**2*c*d**2*e**4 + 5*a*c**2*d**4*e**2 - c**3*d**6 - 1 2*c**3*d**3*e**3*x**3 + x**2*(-30*a*c**2*d**2*e**4 - 6*c**3*d**4*e**2) + x *(-22*a**2*c*d*e**5 - 16*a*c**2*d**3*e**3 + 2*c**3*d**5*e))/(3*a**7*d*e**1 1 - 12*a**6*c*d**3*e**9 + 18*a**5*c**2*d**5*e**7 - 12*a**4*c**3*d**7*e**5 + 3*a**3*c**4*d**9*e**3 + x**4*(3*a**4*c**3*d**3*e**9 - 12*a**3*c**4*d**5* e**7 + 18*a**2*c**5*d**7*e**5 - 12*a*c**6*d**9*e**3 + 3*c**7*d**11*e) + x* *3*(9*a**5*c**2*d**2*e**10 - 33*a**4*c**3*d**4*e**8 + 42*a**3*c**4*d**6*e* *6 - 18*a**2*c**5*d**8*e**4 - 3*a*c**6*d**10*e**2 + 3*c**7*d**12) + x**2*( 9*a**6*c*d*e**11 - 27*a**5*c**2*d**3*e**9 + 18*a**4*c**3*d**5*e**7 + 18...
Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (171) = 342\).
Time = 0.20 (sec) , antiderivative size = 656, normalized size of antiderivative = 3.79 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {4 \, c d e^{3} \log \left (c d x + a e\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} + \frac {4 \, c d e^{3} \log \left (e x + d\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} - \frac {12 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} - 5 \, a c^{2} d^{4} e^{2} + 13 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} + 5 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} - 11 \, a^{2} c d e^{5}\right )} x}{3 \, {\left (a^{3} c^{4} d^{9} e^{3} - 4 \, a^{4} c^{3} d^{7} e^{5} + 6 \, a^{5} c^{2} d^{5} e^{7} - 4 \, a^{6} c d^{3} e^{9} + a^{7} d e^{11} + {\left (c^{7} d^{11} e - 4 \, a c^{6} d^{9} e^{3} + 6 \, a^{2} c^{5} d^{7} e^{5} - 4 \, a^{3} c^{4} d^{5} e^{7} + a^{4} c^{3} d^{3} e^{9}\right )} x^{4} + {\left (c^{7} d^{12} - a c^{6} d^{10} e^{2} - 6 \, a^{2} c^{5} d^{8} e^{4} + 14 \, a^{3} c^{4} d^{6} e^{6} - 11 \, a^{4} c^{3} d^{4} e^{8} + 3 \, a^{5} c^{2} d^{2} e^{10}\right )} x^{3} + 3 \, {\left (a c^{6} d^{11} e - 3 \, a^{2} c^{5} d^{9} e^{3} + 2 \, a^{3} c^{4} d^{7} e^{5} + 2 \, a^{4} c^{3} d^{5} e^{7} - 3 \, a^{5} c^{2} d^{3} e^{9} + a^{6} c d e^{11}\right )} x^{2} + {\left (3 \, a^{2} c^{5} d^{10} e^{2} - 11 \, a^{3} c^{4} d^{8} e^{4} + 14 \, a^{4} c^{3} d^{6} e^{6} - 6 \, a^{5} c^{2} d^{4} e^{8} - a^{6} c d^{2} e^{10} + a^{7} e^{12}\right )} x\right )}} \]
-4*c*d*e^3*log(c*d*x + a*e)/(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e ^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10) + 4*c*d*e^3*log(e*x + d)/(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10) - 1/3*(12*c^3*d^3*e^3*x^3 + c^3*d^6 - 5*a*c ^2*d^4*e^2 + 13*a^2*c*d^2*e^4 + 3*a^3*e^6 + 6*(c^3*d^4*e^2 + 5*a*c^2*d^2*e ^4)*x^2 - 2*(c^3*d^5*e - 8*a*c^2*d^3*e^3 - 11*a^2*c*d*e^5)*x)/(a^3*c^4*d^9 *e^3 - 4*a^4*c^3*d^7*e^5 + 6*a^5*c^2*d^5*e^7 - 4*a^6*c*d^3*e^9 + a^7*d*e^1 1 + (c^7*d^11*e - 4*a*c^6*d^9*e^3 + 6*a^2*c^5*d^7*e^5 - 4*a^3*c^4*d^5*e^7 + a^4*c^3*d^3*e^9)*x^4 + (c^7*d^12 - a*c^6*d^10*e^2 - 6*a^2*c^5*d^8*e^4 + 14*a^3*c^4*d^6*e^6 - 11*a^4*c^3*d^4*e^8 + 3*a^5*c^2*d^2*e^10)*x^3 + 3*(a*c ^6*d^11*e - 3*a^2*c^5*d^9*e^3 + 2*a^3*c^4*d^7*e^5 + 2*a^4*c^3*d^5*e^7 - 3* a^5*c^2*d^3*e^9 + a^6*c*d*e^11)*x^2 + (3*a^2*c^5*d^10*e^2 - 11*a^3*c^4*d^8 *e^4 + 14*a^4*c^3*d^6*e^6 - 6*a^5*c^2*d^4*e^8 - a^6*c*d^2*e^10 + a^7*e^12) *x)
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (171) = 342\).
Time = 0.28 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.23 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {4 \, c^{2} d^{2} e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{6} d^{11} - 5 \, a c^{5} d^{9} e^{2} + 10 \, a^{2} c^{4} d^{7} e^{4} - 10 \, a^{3} c^{3} d^{5} e^{6} + 5 \, a^{4} c^{2} d^{3} e^{8} - a^{5} c d e^{10}} + \frac {4 \, c d e^{4} \log \left ({\left | e x + d \right |}\right )}{c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}} - \frac {c^{4} d^{8} - 6 \, a c^{3} d^{6} e^{2} + 18 \, a^{2} c^{2} d^{4} e^{4} - 10 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 12 \, {\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (c^{4} d^{6} e^{2} + 4 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 2 \, {\left (c^{4} d^{7} e - 9 \, a c^{3} d^{5} e^{3} - 3 \, a^{2} c^{2} d^{3} e^{5} + 11 \, a^{3} c d e^{7}\right )} x}{3 \, {\left (c d^{2} - a e^{2}\right )}^{5} {\left (c d x + a e\right )}^{3} {\left (e x + d\right )}} \]
-4*c^2*d^2*e^3*log(abs(c*d*x + a*e))/(c^6*d^11 - 5*a*c^5*d^9*e^2 + 10*a^2* c^4*d^7*e^4 - 10*a^3*c^3*d^5*e^6 + 5*a^4*c^2*d^3*e^8 - a^5*c*d*e^10) + 4*c *d*e^4*log(abs(e*x + d))/(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) - 1/3*(c^4*d^8 - 6*a* c^3*d^6*e^2 + 18*a^2*c^2*d^4*e^4 - 10*a^3*c*d^2*e^6 - 3*a^4*e^8 + 12*(c^4* d^5*e^3 - a*c^3*d^3*e^5)*x^3 + 6*(c^4*d^6*e^2 + 4*a*c^3*d^4*e^4 - 5*a^2*c^ 2*d^2*e^6)*x^2 - 2*(c^4*d^7*e - 9*a*c^3*d^5*e^3 - 3*a^2*c^2*d^3*e^5 + 11*a ^3*c*d*e^7)*x)/((c*d^2 - a*e^2)^5*(c*d*x + a*e)^3*(e*x + d))
Time = 10.26 (sec) , antiderivative size = 617, normalized size of antiderivative = 3.57 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {8\,c\,d\,e^3\,\mathrm {atanh}\left (\frac {a^5\,e^{10}-3\,a^4\,c\,d^2\,e^8+2\,a^3\,c^2\,d^4\,e^6+2\,a^2\,c^3\,d^6\,e^4-3\,a\,c^4\,d^8\,e^2+c^5\,d^{10}}{{\left (a\,e^2-c\,d^2\right )}^5}+\frac {2\,c\,d\,e\,x\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{{\left (a\,e^2-c\,d^2\right )}^5}\right )}{{\left (a\,e^2-c\,d^2\right )}^5}-\frac {\frac {3\,a^3\,e^6+13\,a^2\,c\,d^2\,e^4-5\,a\,c^2\,d^4\,e^2+c^3\,d^6}{3\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {2\,e\,x\,\left (11\,a^2\,c\,d\,e^4+8\,a\,c^2\,d^3\,e^2-c^3\,d^5\right )}{3\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {2\,e^2\,x^2\,\left (c^3\,d^4+5\,a\,c^2\,d^2\,e^2\right )}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}+\frac {4\,c^3\,d^3\,e^3\,x^3}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}}{x\,\left (a^3\,e^4+3\,c\,a^2\,d^2\,e^2\right )+x^3\,\left (c^3\,d^4+3\,a\,c^2\,d^2\,e^2\right )+x^2\,\left (3\,a^2\,c\,d\,e^3+3\,a\,c^2\,d^3\,e\right )+a^3\,d\,e^3+c^3\,d^3\,e\,x^4} \]
(8*c*d*e^3*atanh((a^5*e^10 + c^5*d^10 - 3*a*c^4*d^8*e^2 - 3*a^4*c*d^2*e^8 + 2*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^6)/(a*e^2 - c*d^2)^5 + (2*c*d*e*x*(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)) /(a*e^2 - c*d^2)^5))/(a*e^2 - c*d^2)^5 - ((3*a^3*e^6 + c^3*d^6 - 5*a*c^2*d ^4*e^2 + 13*a^2*c*d^2*e^4)/(3*(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)) + (2*e*x*(8*a*c^2*d^3*e^2 - c^3*d^5 + 11* a^2*c*d*e^4))/(3*(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)) + (2*e^2*x^2*(c^3*d^4 + 5*a*c^2*d^2*e^2))/(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) + (4*c^3*d ^3*e^3*x^3)/(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))/(x*(a^3*e^4 + 3*a^2*c*d^2*e^2) + x^3*(c^3*d^4 + 3*a*c^2*d^2 *e^2) + x^2*(3*a*c^2*d^3*e + 3*a^2*c*d*e^3) + a^3*d*e^3 + c^3*d^3*e*x^4)