Integrand size = 27, antiderivative size = 191 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {c d^2+a e^2+2 c d e x}{2 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}+\frac {3 c d e \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}+\frac {6 c^2 d^2 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}-\frac {6 c^2 d^2 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \]
1/2*(-2*c*d*e*x-a*e^2-c*d^2)/(-a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e *x^2)^2+3*c*d*e*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d ^2)*x+c*d*e*x^2)+6*c^2*d^2*e^2*ln(c*d*x+a*e)/(-a*e^2+c*d^2)^5-6*c^2*d^2*e^ 2*ln(e*x+d)/(-a*e^2+c*d^2)^5
Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {\frac {c^2 d^2 \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {6 c^2 d^2 e \left (-c d^2+a e^2\right )}{a e+c d x}-\frac {\left (c d^2 e-a e^3\right )^2}{(d+e x)^2}+\frac {6 c d e^2 \left (-c d^2+a e^2\right )}{d+e x}-12 c^2 d^2 e^2 \log (a e+c d x)+12 c^2 d^2 e^2 \log (d+e x)}{2 \left (-c d^2+a e^2\right )^5} \]
((c^2*d^2*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (6*c^2*d^2*e*(-(c*d^2) + a* e^2))/(a*e + c*d*x) - (c*d^2*e - a*e^3)^2/(d + e*x)^2 + (6*c*d*e^2*(-(c*d^ 2) + a*e^2))/(d + e*x) - 12*c^2*d^2*e^2*Log[a*e + c*d*x] + 12*c^2*d^2*e^2* Log[d + e*x])/(2*(-(c*d^2) + a*e^2)^5)
Time = 0.46 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1084, 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.
| \(\displaystyle \int \frac {1}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1084 |
| \(\displaystyle c^3 d^3 e^3 \int \left (-\frac {6}{c d \left (c d^2-a e^2\right )^5 (d+e x)}-\frac {3}{c^2 d^2 \left (c d^2-a e^2\right )^4 (d+e x)^2}-\frac {1}{c^3 d^3 \left (c d^2-a e^2\right )^3 (d+e x)^3}+\frac {6}{e \left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac {3}{e^2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac {1}{e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c^3 d^3 e^3 \left (\frac {1}{2 c^3 d^3 e (d+e x)^2 \left (c d^2-a e^2\right )^3}+\frac {3}{c^2 d^2 e (d+e x) \left (c d^2-a e^2\right )^4}+\frac {3}{c d e^2 \left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac {6 \log (a e+c d x)}{c d e \left (c d^2-a e^2\right )^5}-\frac {6 \log (d+e x)}{c d e \left (c d^2-a e^2\right )^5}-\frac {1}{2 c d e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}\right )\) |
c^3*d^3*e^3*(-1/2*1/(c*d*e^3*(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)) + 1/(2*c^3*d^3*e*(c*d^2 - a*e^2)^3*(d + e*x)^2) + 3/(c^2*d^2*e*(c*d^2 - a*e^2)^4*(d + e*x)) + (6*Log[a*e + c*d*x ])/(c*d*e*(c*d^2 - a*e^2)^5) - (6*Log[d + e*x])/(c*d*e*(c*d^2 - a*e^2)^5))
3.19.94.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[(b/2 - q/2 + c*x)^p*(b/2 + q /2 + c*x)^p, x], x], x] /; !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c}, x] && IntegerQ[p] && NiceSqrtQ[b^2 - 4*a*c]
Time = 2.44 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(\frac {c^{2} d^{2}}{2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (c d x +a e \right )^{2}}-\frac {6 c^{2} d^{2} e^{2} \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{5}}+\frac {3 c^{2} d^{2} e}{\left (e^{2} a -c \,d^{2}\right )^{4} \left (c d x +a e \right )}-\frac {e^{2}}{2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{2}}+\frac {6 c^{2} d^{2} e^{2} \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{5}}+\frac {3 e^{2} c d}{\left (e^{2} a -c \,d^{2}\right )^{4} \left (e x +d \right )}\) | \(186\) |
| risch | \(\frac {\frac {6 c^{3} d^{3} e^{3} x^{3}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {9 c^{2} d^{2} e^{2} \left (e^{2} a +c \,d^{2}\right ) x^{2}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {2 \left (a^{2} e^{4}+7 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c d e x}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {e^{6} a^{3}-7 d^{2} e^{4} a^{2} c -7 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{2}}-\frac {6 c^{2} d^{2} e^{2} \ln \left (c d x +a e \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {6 c^{2} d^{2} e^{2} \ln \left (-e x -d \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}\) | \(541\) |
| norman | \(\frac {\frac {\left (9 a \,c^{4} d^{4} e^{6}+9 c^{5} d^{6} e^{4}\right ) x^{2}}{e^{2} d^{2} c^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {-a^{3} c^{2} e^{6}+7 a^{2} c^{3} d^{2} e^{4}+7 a \,c^{4} d^{4} e^{2}-c^{5} d^{6}}{2 c^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {6 c^{3} d^{3} e^{3} x^{3}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {2 \left (a^{2} c^{3} d^{2} e^{6}+7 a \,c^{4} d^{4} e^{4}+c^{5} d^{6} e^{2}\right ) x}{e d \,c^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}+\frac {6 c^{2} d^{2} e^{2} \ln \left (e x +d \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}-\frac {6 c^{2} d^{2} e^{2} \ln \left (c d x +a e \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}\) | \(568\) |
| parallelrisch | \(\frac {12 \ln \left (e x +d \right ) x^{2} a^{2} c^{4} d^{4} e^{8}+48 \ln \left (e x +d \right ) x^{2} a \,c^{5} d^{6} e^{6}+24 \ln \left (e x +d \right ) x^{3} a \,c^{5} d^{5} e^{7}-24 \ln \left (c d x +a e \right ) x^{3} a \,c^{5} d^{5} e^{7}-24 \ln \left (c d x +a e \right ) x a \,c^{5} d^{7} e^{5}-12 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{4} d^{4} e^{8}-48 \ln \left (c d x +a e \right ) x^{2} a \,c^{5} d^{6} e^{6}+24 \ln \left (e x +d \right ) x \,a^{2} c^{4} d^{5} e^{7}+24 \ln \left (e x +d \right ) x a \,c^{5} d^{7} e^{5}-24 \ln \left (c d x +a e \right ) x \,a^{2} c^{4} d^{5} e^{7}-a^{4} c^{2} d^{2} e^{10}+8 a^{3} c^{3} d^{4} e^{8}-8 a \,c^{5} d^{8} e^{4}+12 x^{3} a \,c^{5} d^{5} e^{7}+18 x^{2} a^{2} c^{4} d^{4} e^{8}+4 x \,a^{3} c^{3} d^{3} e^{9}+24 x \,a^{2} c^{4} d^{5} e^{7}-24 x a \,c^{5} d^{7} e^{5}+12 \ln \left (e x +d \right ) x^{4} c^{6} d^{6} e^{6}-12 \ln \left (c d x +a e \right ) x^{4} c^{6} d^{6} e^{6}+24 \ln \left (e x +d \right ) x^{3} c^{6} d^{7} e^{5}-24 \ln \left (c d x +a e \right ) x^{3} c^{6} d^{7} e^{5}+12 \ln \left (e x +d \right ) x^{2} c^{6} d^{8} e^{4}+c^{6} d^{10} e^{2}-12 x^{3} c^{6} d^{7} e^{5}-18 x^{2} c^{6} d^{8} e^{4}-4 x \,c^{6} d^{9} e^{3}-12 \ln \left (c d x +a e \right ) x^{2} c^{6} d^{8} e^{4}+12 \ln \left (e x +d \right ) a^{2} c^{4} d^{6} e^{6}-12 \ln \left (c d x +a e \right ) a^{2} c^{4} d^{6} e^{6}}{2 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{2} c^{2} d^{2} e^{2}}\) | \(671\) |
1/2*c^2*d^2/(a*e^2-c*d^2)^3/(c*d*x+a*e)^2-6*c^2*d^2/(a*e^2-c*d^2)^5*e^2*ln (c*d*x+a*e)+3*c^2*d^2/(a*e^2-c*d^2)^4*e/(c*d*x+a*e)-1/2*e^2/(a*e^2-c*d^2)^ 3/(e*x+d)^2+6*c^2*d^2/(a*e^2-c*d^2)^5*e^2*ln(e*x+d)+3*e^2/(a*e^2-c*d^2)^4* c*d/(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (189) = 378\).
Time = 0.36 (sec) , antiderivative size = 828, normalized size of antiderivative = 4.34 \[ \int \frac {1}{\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} - 8 \, a c^{3} d^{6} e^{2} + 8 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} - 12 \, {\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} - 18 \, {\left (c^{4} d^{6} e^{2} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 4 \, {\left (c^{4} d^{7} e + 6 \, a c^{3} d^{5} e^{3} - 6 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x - 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a^{2} c^{2} d^{4} e^{4} + 2 \, {\left (c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + {\left (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} + 2 \, {\left (a c^{3} d^{5} e^{3} + a^{2} c^{2} d^{3} e^{5}\right )} x\right )} \log \left (c d x + a e\right ) + 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a^{2} c^{2} d^{4} e^{4} + 2 \, {\left (c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + {\left (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} + 2 \, {\left (a c^{3} d^{5} e^{3} + a^{2} c^{2} d^{3} e^{5}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} c^{5} d^{12} e^{2} - 5 \, a^{3} c^{4} d^{10} e^{4} + 10 \, a^{4} c^{3} d^{8} e^{6} - 10 \, a^{5} c^{2} d^{6} e^{8} + 5 \, a^{6} c d^{4} e^{10} - a^{7} d^{2} e^{12} + {\left (c^{7} d^{12} e^{2} - 5 \, a c^{6} d^{10} e^{4} + 10 \, a^{2} c^{5} d^{8} e^{6} - 10 \, a^{3} c^{4} d^{6} e^{8} + 5 \, a^{4} c^{3} d^{4} e^{10} - a^{5} c^{2} d^{2} e^{12}\right )} x^{4} + 2 \, {\left (c^{7} d^{13} e - 4 \, a c^{6} d^{11} e^{3} + 5 \, a^{2} c^{5} d^{9} e^{5} - 5 \, a^{4} c^{3} d^{5} e^{9} + 4 \, a^{5} c^{2} d^{3} e^{11} - a^{6} c d e^{13}\right )} x^{3} + {\left (c^{7} d^{14} - a c^{6} d^{12} e^{2} - 9 \, a^{2} c^{5} d^{10} e^{4} + 25 \, a^{3} c^{4} d^{8} e^{6} - 25 \, a^{4} c^{3} d^{6} e^{8} + 9 \, a^{5} c^{2} d^{4} e^{10} + a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} x^{2} + 2 \, {\left (a c^{6} d^{13} e - 4 \, a^{2} c^{5} d^{11} e^{3} + 5 \, a^{3} c^{4} d^{9} e^{5} - 5 \, a^{5} c^{2} d^{5} e^{9} + 4 \, a^{6} c d^{3} e^{11} - a^{7} d e^{13}\right )} x\right )}} \]
-1/2*(c^4*d^8 - 8*a*c^3*d^6*e^2 + 8*a^3*c*d^2*e^6 - a^4*e^8 - 12*(c^4*d^5* e^3 - a*c^3*d^3*e^5)*x^3 - 18*(c^4*d^6*e^2 - a^2*c^2*d^2*e^6)*x^2 - 4*(c^4 *d^7*e + 6*a*c^3*d^5*e^3 - 6*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x - 12*(c^4*d^ 4*e^4*x^4 + a^2*c^2*d^4*e^4 + 2*(c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + (c^4*d ^6*e^2 + 4*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 + a^2*c ^2*d^3*e^5)*x)*log(c*d*x + a*e) + 12*(c^4*d^4*e^4*x^4 + a^2*c^2*d^4*e^4 + 2*(c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + (c^4*d^6*e^2 + 4*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 + a^2*c^2*d^3*e^5)*x)*log(e*x + d))/( a^2*c^5*d^12*e^2 - 5*a^3*c^4*d^10*e^4 + 10*a^4*c^3*d^8*e^6 - 10*a^5*c^2*d^ 6*e^8 + 5*a^6*c*d^4*e^10 - a^7*d^2*e^12 + (c^7*d^12*e^2 - 5*a*c^6*d^10*e^4 + 10*a^2*c^5*d^8*e^6 - 10*a^3*c^4*d^6*e^8 + 5*a^4*c^3*d^4*e^10 - a^5*c^2* d^2*e^12)*x^4 + 2*(c^7*d^13*e - 4*a*c^6*d^11*e^3 + 5*a^2*c^5*d^9*e^5 - 5*a ^4*c^3*d^5*e^9 + 4*a^5*c^2*d^3*e^11 - a^6*c*d*e^13)*x^3 + (c^7*d^14 - a*c^ 6*d^12*e^2 - 9*a^2*c^5*d^10*e^4 + 25*a^3*c^4*d^8*e^6 - 25*a^4*c^3*d^6*e^8 + 9*a^5*c^2*d^4*e^10 + a^6*c*d^2*e^12 - a^7*e^14)*x^2 + 2*(a*c^6*d^13*e - 4*a^2*c^5*d^11*e^3 + 5*a^3*c^4*d^9*e^5 - 5*a^5*c^2*d^5*e^9 + 4*a^6*c*d^3*e ^11 - a^7*d*e^13)*x)
Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (187) = 374\).
Time = 2.35 (sec) , antiderivative size = 1001, normalized size of antiderivative = 5.24 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {6 c^{2} d^{2} e^{2} \log {\left (x + \frac {- \frac {6 a^{6} c^{2} d^{2} e^{14}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {36 a^{5} c^{3} d^{4} e^{12}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {90 a^{4} c^{4} d^{6} e^{10}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {120 a^{3} c^{5} d^{8} e^{8}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {90 a^{2} c^{6} d^{10} e^{6}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {36 a c^{7} d^{12} e^{4}}{\left (a e^{2} - c d^{2}\right )^{5}} + 6 a c^{2} d^{2} e^{4} - \frac {6 c^{8} d^{14} e^{2}}{\left (a e^{2} - c d^{2}\right )^{5}} + 6 c^{3} d^{4} e^{2}}{12 c^{3} d^{3} e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {6 c^{2} d^{2} e^{2} \log {\left (x + \frac {\frac {6 a^{6} c^{2} d^{2} e^{14}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {36 a^{5} c^{3} d^{4} e^{12}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {90 a^{4} c^{4} d^{6} e^{10}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {120 a^{3} c^{5} d^{8} e^{8}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {90 a^{2} c^{6} d^{10} e^{6}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac {36 a c^{7} d^{12} e^{4}}{\left (a e^{2} - c d^{2}\right )^{5}} + 6 a c^{2} d^{2} e^{4} + \frac {6 c^{8} d^{14} e^{2}}{\left (a e^{2} - c d^{2}\right )^{5}} + 6 c^{3} d^{4} e^{2}}{12 c^{3} d^{3} e^{3}} \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac {- a^{3} e^{6} + 7 a^{2} c d^{2} e^{4} + 7 a c^{2} d^{4} e^{2} - c^{3} d^{6} + 12 c^{3} d^{3} e^{3} x^{3} + x^{2} \cdot \left (18 a c^{2} d^{2} e^{4} + 18 c^{3} d^{4} e^{2}\right ) + x \left (4 a^{2} c d e^{5} + 28 a c^{2} d^{3} e^{3} + 4 c^{3} d^{5} e\right )}{2 a^{6} d^{2} e^{10} - 8 a^{5} c d^{4} e^{8} + 12 a^{4} c^{2} d^{6} e^{6} - 8 a^{3} c^{3} d^{8} e^{4} + 2 a^{2} c^{4} d^{10} e^{2} + x^{4} \cdot \left (2 a^{4} c^{2} d^{2} e^{10} - 8 a^{3} c^{3} d^{4} e^{8} + 12 a^{2} c^{4} d^{6} e^{6} - 8 a c^{5} d^{8} e^{4} + 2 c^{6} d^{10} e^{2}\right ) + x^{3} \cdot \left (4 a^{5} c d e^{11} - 12 a^{4} c^{2} d^{3} e^{9} + 8 a^{3} c^{3} d^{5} e^{7} + 8 a^{2} c^{4} d^{7} e^{5} - 12 a c^{5} d^{9} e^{3} + 4 c^{6} d^{11} e\right ) + x^{2} \cdot \left (2 a^{6} e^{12} - 18 a^{4} c^{2} d^{4} e^{8} + 32 a^{3} c^{3} d^{6} e^{6} - 18 a^{2} c^{4} d^{8} e^{4} + 2 c^{6} d^{12}\right ) + x \left (4 a^{6} d e^{11} - 12 a^{5} c d^{3} e^{9} + 8 a^{4} c^{2} d^{5} e^{7} + 8 a^{3} c^{3} d^{7} e^{5} - 12 a^{2} c^{4} d^{9} e^{3} + 4 a c^{5} d^{11} e\right )} \]
6*c**2*d**2*e**2*log(x + (-6*a**6*c**2*d**2*e**14/(a*e**2 - c*d**2)**5 + 3 6*a**5*c**3*d**4*e**12/(a*e**2 - c*d**2)**5 - 90*a**4*c**4*d**6*e**10/(a*e **2 - c*d**2)**5 + 120*a**3*c**5*d**8*e**8/(a*e**2 - c*d**2)**5 - 90*a**2* c**6*d**10*e**6/(a*e**2 - c*d**2)**5 + 36*a*c**7*d**12*e**4/(a*e**2 - c*d* *2)**5 + 6*a*c**2*d**2*e**4 - 6*c**8*d**14*e**2/(a*e**2 - c*d**2)**5 + 6*c **3*d**4*e**2)/(12*c**3*d**3*e**3))/(a*e**2 - c*d**2)**5 - 6*c**2*d**2*e** 2*log(x + (6*a**6*c**2*d**2*e**14/(a*e**2 - c*d**2)**5 - 36*a**5*c**3*d**4 *e**12/(a*e**2 - c*d**2)**5 + 90*a**4*c**4*d**6*e**10/(a*e**2 - c*d**2)**5 - 120*a**3*c**5*d**8*e**8/(a*e**2 - c*d**2)**5 + 90*a**2*c**6*d**10*e**6/ (a*e**2 - c*d**2)**5 - 36*a*c**7*d**12*e**4/(a*e**2 - c*d**2)**5 + 6*a*c** 2*d**2*e**4 + 6*c**8*d**14*e**2/(a*e**2 - c*d**2)**5 + 6*c**3*d**4*e**2)/( 12*c**3*d**3*e**3))/(a*e**2 - c*d**2)**5 + (-a**3*e**6 + 7*a**2*c*d**2*e** 4 + 7*a*c**2*d**4*e**2 - c**3*d**6 + 12*c**3*d**3*e**3*x**3 + x**2*(18*a*c **2*d**2*e**4 + 18*c**3*d**4*e**2) + x*(4*a**2*c*d*e**5 + 28*a*c**2*d**3*e **3 + 4*c**3*d**5*e))/(2*a**6*d**2*e**10 - 8*a**5*c*d**4*e**8 + 12*a**4*c* *2*d**6*e**6 - 8*a**3*c**3*d**8*e**4 + 2*a**2*c**4*d**10*e**2 + x**4*(2*a* *4*c**2*d**2*e**10 - 8*a**3*c**3*d**4*e**8 + 12*a**2*c**4*d**6*e**6 - 8*a* c**5*d**8*e**4 + 2*c**6*d**10*e**2) + x**3*(4*a**5*c*d*e**11 - 12*a**4*c** 2*d**3*e**9 + 8*a**3*c**3*d**5*e**7 + 8*a**2*c**4*d**7*e**5 - 12*a*c**5*d* *9*e**3 + 4*c**6*d**11*e) + x**2*(2*a**6*e**12 - 18*a**4*c**2*d**4*e**8...
Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (189) = 378\).
Time = 0.19 (sec) , antiderivative size = 642, normalized size of antiderivative = 3.36 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {6 \, c^{2} d^{2} e^{2} \log \left (c d x + a e\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} - \frac {6 \, c^{2} d^{2} e^{2} \log \left (e x + d\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} + \frac {12 \, c^{3} d^{3} e^{3} x^{3} - c^{3} d^{6} + 7 \, a c^{2} d^{4} e^{2} + 7 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 18 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (c^{3} d^{5} e + 7 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (a^{2} c^{4} d^{10} e^{2} - 4 \, a^{3} c^{3} d^{8} e^{4} + 6 \, a^{4} c^{2} d^{6} e^{6} - 4 \, a^{5} c d^{4} e^{8} + a^{6} d^{2} e^{10} + {\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{4} + 2 \, {\left (c^{6} d^{11} e - 3 \, a c^{5} d^{9} e^{3} + 2 \, a^{2} c^{4} d^{7} e^{5} + 2 \, a^{3} c^{3} d^{5} e^{7} - 3 \, a^{4} c^{2} d^{3} e^{9} + a^{5} c d e^{11}\right )} x^{3} + {\left (c^{6} d^{12} - 9 \, a^{2} c^{4} d^{8} e^{4} + 16 \, a^{3} c^{3} d^{6} e^{6} - 9 \, a^{4} c^{2} d^{4} e^{8} + a^{6} e^{12}\right )} x^{2} + 2 \, {\left (a c^{5} d^{11} e - 3 \, a^{2} c^{4} d^{9} e^{3} + 2 \, a^{3} c^{3} d^{7} e^{5} + 2 \, a^{4} c^{2} d^{5} e^{7} - 3 \, a^{5} c d^{3} e^{9} + a^{6} d e^{11}\right )} x\right )}} \]
6*c^2*d^2*e^2*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) - 6*c^2*d^2*e^2*l og(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) + 1/2*(12*c^3*d^3*e^3*x^3 - c^3*d^6 + 7*a*c^2*d^4*e^2 + 7*a^2*c*d^2*e^4 - a^3*e^6 + 18*(c^3*d^4*e^2 + a*c^2*d^ 2*e^4)*x^2 + 4*(c^3*d^5*e + 7*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(a^2*c^4*d^1 0*e^2 - 4*a^3*c^3*d^8*e^4 + 6*a^4*c^2*d^6*e^6 - 4*a^5*c*d^4*e^8 + a^6*d^2* e^10 + (c^6*d^10*e^2 - 4*a*c^5*d^8*e^4 + 6*a^2*c^4*d^6*e^6 - 4*a^3*c^3*d^4 *e^8 + a^4*c^2*d^2*e^10)*x^4 + 2*(c^6*d^11*e - 3*a*c^5*d^9*e^3 + 2*a^2*c^4 *d^7*e^5 + 2*a^3*c^3*d^5*e^7 - 3*a^4*c^2*d^3*e^9 + a^5*c*d*e^11)*x^3 + (c^ 6*d^12 - 9*a^2*c^4*d^8*e^4 + 16*a^3*c^3*d^6*e^6 - 9*a^4*c^2*d^4*e^8 + a^6* e^12)*x^2 + 2*(a*c^5*d^11*e - 3*a^2*c^4*d^9*e^3 + 2*a^3*c^3*d^7*e^5 + 2*a^ 4*c^2*d^5*e^7 - 3*a^5*c*d^3*e^9 + a^6*d*e^11)*x)
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (189) = 378\).
Time = 0.27 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.02 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {6 \, c^{3} d^{3} e^{2} \log \left ({\left | c d x + a e \right |}\right )}{c^{6} d^{11} - 5 \, a c^{5} d^{9} e^{2} + 10 \, a^{2} c^{4} d^{7} e^{4} - 10 \, a^{3} c^{3} d^{5} e^{6} + 5 \, a^{4} c^{2} d^{3} e^{8} - a^{5} c d e^{10}} - \frac {6 \, c^{2} d^{2} e^{3} \log \left ({\left | e x + d \right |}\right )}{c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}} + \frac {12 \, c^{3} d^{3} e^{3} x^{3} + 18 \, c^{3} d^{4} e^{2} x^{2} + 18 \, a c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d^{5} e x + 28 \, a c^{2} d^{3} e^{3} x + 4 \, a^{2} c d e^{5} x - c^{3} d^{6} + 7 \, a c^{2} d^{4} e^{2} + 7 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}}{2 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left (c d e x^{2} + c d^{2} x + a e^{2} x + a d e\right )}^{2}} \]
6*c^3*d^3*e^2*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) - 6*c^ 2*d^2*e^3*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) + 1/2*(12*c^3*d^3* e^3*x^3 + 18*c^3*d^4*e^2*x^2 + 18*a*c^2*d^2*e^4*x^2 + 4*c^3*d^5*e*x + 28*a *c^2*d^3*e^3*x + 4*a^2*c*d*e^5*x - c^3*d^6 + 7*a*c^2*d^4*e^2 + 7*a^2*c*d^2 *e^4 - a^3*e^6)/((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*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)^2)
Time = 10.21 (sec) , antiderivative size = 616, normalized size of antiderivative = 3.23 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {\frac {9\,c\,x^2\,\left (c^2\,d^4\,e^2+a\,c\,d^2\,e^4\right )}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}-\frac {a^3\,e^6-7\,a^2\,c\,d^2\,e^4-7\,a\,c^2\,d^4\,e^2+c^3\,d^6}{2\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {6\,c^3\,d^3\,e^3\,x^3}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}+\frac {2\,c\,d\,e\,x\,\left (a^2\,e^4+7\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}}{x^2\,\left (a^2\,e^4+4\,a\,c\,d^2\,e^2+c^2\,d^4\right )+x^3\,\left (2\,c^2\,d^3\,e+2\,a\,c\,d\,e^3\right )+x\,\left (2\,a^2\,d\,e^3+2\,c\,a\,d^3\,e\right )+a^2\,d^2\,e^2+c^2\,d^2\,e^2\,x^4}-\frac {12\,c^2\,d^2\,e^2\,\mathrm {atanh}\left (\frac {a^5\,e^{10}-3\,a^4\,c\,d^2\,e^8+2\,a^3\,c^2\,d^4\,e^6+2\,a^2\,c^3\,d^6\,e^4-3\,a\,c^4\,d^8\,e^2+c^5\,d^{10}}{{\left (a\,e^2-c\,d^2\right )}^5}+\frac {2\,c\,d\,e\,x\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{{\left (a\,e^2-c\,d^2\right )}^5}\right )}{{\left (a\,e^2-c\,d^2\right )}^5} \]
((9*c*x^2*(c^2*d^4*e^2 + a*c*d^2*e^4))/(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^3*e^6 + c^3*d^6 - 7*a*c^2*d^ 4*e^2 - 7*a^2*c*d^2*e^4)/(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)) + (6*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) + (2*c*d*e*x*(a^2*e ^4 + c^2*d^4 + 7*a*c*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))/(x^2*(a^2*e^4 + c^2*d^4 + 4*a*c*d^2*e^2) + x^3*(2*c^2*d^3*e + 2*a*c*d*e^3) + x*(2*a^2*d*e^3 + 2*a*c*d^3*e) + a^2*d ^2*e^2 + c^2*d^2*e^2*x^4) - (12*c^2*d^2*e^2*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