\(\int \frac {1}{(d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [126]

Optimal result

Integrand size = 35, antiderivative size = 176 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c^3 d^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {e}{3 \left (c d^2-a e^2\right )^2 (d+e x)^3}-\frac {c d e}{\left (c d^2-a e^2\right )^3 (d+e x)^2}-\frac {3 c^2 d^2 e}{\left (c d^2-a e^2\right )^4 (d+e x)}-\frac {4 c^3 d^3 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c^3 d^3 e \log (d+e x)}{\left (c d^2-a e^2\right )^5} \] Output:

-c^3*d^3/(-a*e^2+c*d^2)^4/(c*d*x+a*e)-1/3*e/(-a*e^2+c*d^2)^2/(e*x+d)^3-c*d 
*e/(-a*e^2+c*d^2)^3/(e*x+d)^2-3*c^2*d^2*e/(-a*e^2+c*d^2)^4/(e*x+d)-4*c^3*d 
^3*e*ln(c*d*x+a*e)/(-a*e^2+c*d^2)^5+4*c^3*d^3*e*ln(e*x+d)/(-a*e^2+c*d^2)^5
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\frac {3 c^3 d^3 \left (c d^2-a e^2\right )}{a e+c d x}-\frac {e \left (-c d^2+a e^2\right )^3}{(d+e x)^3}+\frac {3 c d e \left (c d^2-a e^2\right )^2}{(d+e x)^2}+\frac {9 c^2 d^2 e \left (c d^2-a e^2\right )}{d+e x}+12 c^3 d^3 e \log (a e+c d x)-12 c^3 d^3 e \log (d+e x)}{3 \left (-c d^2+a e^2\right )^5} \] Input:

Integrate[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2),x]
 

Output:

((3*c^3*d^3*(c*d^2 - a*e^2))/(a*e + c*d*x) - (e*(-(c*d^2) + a*e^2)^3)/(d + 
 e*x)^3 + (3*c*d*e*(c*d^2 - a*e^2)^2)/(d + e*x)^2 + (9*c^2*d^2*e*(c*d^2 - 
a*e^2))/(d + e*x) + 12*c^3*d^3*e*Log[a*e + c*d*x] - 12*c^3*d^3*e*Log[d + e 
*x])/(3*(-(c*d^2) + a*e^2)^5)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 176, 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.

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2} \, dx\)

\(\Big \downarrow \) 1121

\(\displaystyle \int \left (-\frac {4 c^4 d^4 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {c^4 d^4}{\left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac {4 c^3 d^3 e^2}{(d+e x) \left (c d^2-a e^2\right )^5}+\frac {3 c^2 d^2 e^2}{(d+e x)^2 \left (c d^2-a e^2\right )^4}+\frac {2 c d e^2}{(d+e x)^3 \left (c d^2-a e^2\right )^3}+\frac {e^2}{(d+e x)^4 \left (c d^2-a e^2\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c^3 d^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac {4 c^3 d^3 e \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac {4 c^3 d^3 e \log (d+e x)}{\left (c d^2-a e^2\right )^5}-\frac {3 c^2 d^2 e}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac {c d e}{(d+e x)^2 \left (c d^2-a e^2\right )^3}-\frac {e}{3 (d+e x)^3 \left (c d^2-a e^2\right )^2}\)

Input:

Int[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2),x]
 

Output:

-((c^3*d^3)/((c*d^2 - a*e^2)^4*(a*e + c*d*x))) - e/(3*(c*d^2 - a*e^2)^2*(d 
 + e*x)^3) - (c*d*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)) - (4*c^3*d^3*e*Log[a*e + c*d*x])/(c*d^2 - a*e^2)^ 
5 + (4*c^3*d^3*e*Log[d + e*x])/(c*d^2 - a*e^2)^5
 

Defintions of rubi rules used

rule 1121

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]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 2.01 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.99

method	result	size
default	\(-\frac {c^{3} d^{3}}{\left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right )}+\frac {4 c^{3} d^{3} e \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{5}}-\frac {e}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{3}}-\frac {4 c^{3} d^{3} e \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{5}}-\frac {3 e \,c^{2} d^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{4} \left (e x +d \right )}+\frac {e c d}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \left (e x +d \right )^{2}}\)	\(174\)
risch	\(\frac {-\frac {4 c^{3} d^{3} e^{3} x^{3}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {2 \left (a \,e^{2}+5 c \,d^{2}\right ) e^{2} c^{2} d^{2} x^{2}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {2 \left (a^{2} e^{4}-8 a c \,d^{2} e^{2}-11 c^{2} d^{4}\right ) c d e x}{3 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}-\frac {e^{6} a^{3}-5 d^{2} e^{4} a^{2} c +13 d^{4} e^{2} a \,c^{2}+3 d^{6} c^{3}}{3 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}}{\left (e x +d \right )^{2} \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )}-\frac {4 c^{3} d^{3} e \ln \left (e x +d \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {4 c^{3} d^{3} e \ln \left (-c d x -a e \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}\)	\(546\)
norman	\(\frac {\frac {\left (-2 a \,c^{3} d^{2} e^{6}-10 c^{4} d^{4} e^{4}\right ) x^{2}}{e^{2} c \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}-\frac {4 c^{3} d^{3} e^{3} x^{3}}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {-a^{3} c \,e^{8}+5 a^{2} c^{2} d^{2} e^{6}-13 a \,c^{3} d^{4} e^{4}-3 c^{4} d^{6} e^{2}}{3 e^{2} c \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}+\frac {d \left (2 a^{2} c^{2} e^{8}-16 a \,c^{3} d^{2} e^{6}-22 c^{4} d^{4} e^{4}\right ) x}{3 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) c \,e^{3}}}{\left (c d x +a e \right ) \left (e x +d \right )^{3}}-\frac {4 c^{3} d^{3} e \ln \left (e x +d \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {4 c^{3} d^{3} e \ln \left (c d x +a e \right )}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}\)	\(562\)
parallelrisch	\(-\frac {-2 x \,a^{3} c^{2} d^{2} e^{10}+18 x \,a^{2} c^{3} d^{4} e^{8}+6 x a \,c^{4} d^{6} e^{6}+12 \ln \left (e x +d \right ) x^{4} c^{5} d^{5} e^{7}-12 \ln \left (c d x +a e \right ) x^{4} c^{5} d^{5} e^{7}+36 \ln \left (e x +d \right ) x^{3} c^{5} d^{6} e^{6}-36 \ln \left (c d x +a e \right ) x^{3} c^{5} d^{6} e^{6}+36 \ln \left (e x +d \right ) x^{2} c^{5} d^{7} e^{5}-36 \ln \left (c d x +a e \right ) x^{2} c^{5} d^{7} e^{5}+12 \ln \left (e x +d \right ) x \,c^{5} d^{8} e^{4}-12 \ln \left (c d x +a e \right ) x \,c^{5} d^{8} e^{4}+12 \ln \left (e x +d \right ) a \,c^{4} d^{7} e^{5}-12 \ln \left (c d x +a e \right ) a \,c^{4} d^{7} e^{5}+12 x^{3} a \,c^{4} d^{4} e^{8}+6 x^{2} a^{2} c^{3} d^{3} e^{9}+24 x^{2} a \,c^{4} d^{5} e^{7}+12 \ln \left (e x +d \right ) x^{3} a \,c^{4} d^{4} e^{8}-12 \ln \left (c d x +a e \right ) x^{3} a \,c^{4} d^{4} e^{8}+36 \ln \left (e x +d \right ) x^{2} a \,c^{4} d^{5} e^{7}-12 x^{3} c^{5} d^{6} e^{6}-30 x^{2} c^{5} d^{7} e^{5}-22 x \,c^{5} d^{8} e^{4}+a^{4} c d \,e^{11}-6 a^{3} c^{2} d^{3} e^{9}+18 a^{2} c^{3} d^{5} e^{7}-10 a \,c^{4} d^{7} e^{5}-36 \ln \left (c d x +a e \right ) x^{2} a \,c^{4} d^{5} e^{7}-3 c^{5} d^{9} e^{3}+36 \ln \left (e x +d \right ) x a \,c^{4} d^{6} e^{6}-36 \ln \left (c d x +a e \right ) x a \,c^{4} d^{6} e^{6}}{3 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right ) \left (e x +d \right )^{2} c d \,e^{3}}\)	\(644\)

Input:

int(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2,x,method=_RETURNVERBOS 
E)
 

Output:

-c^3*d^3/(a*e^2-c*d^2)^4/(c*d*x+a*e)+4*c^3*d^3/(a*e^2-c*d^2)^5*e*ln(c*d*x+ 
a*e)-1/3*e/(a*e^2-c*d^2)^2/(e*x+d)^3-4*c^3*d^3/(a*e^2-c*d^2)^5*e*ln(e*x+d) 
-3*e/(a*e^2-c*d^2)^4*c^2*d^2/(e*x+d)+e/(a*e^2-c*d^2)^3*c*d/(e*x+d)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (174) = 348\).

Time = 0.09 (sec) , antiderivative size = 807, normalized size of antiderivative = 4.59 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {3 \, c^{4} d^{8} + 10 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 6 \, 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} + 6 \, {\left (5 \, 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 (11 \, c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} - 9 \, 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 c^{3} d^{6} e^{2} + {\left (3 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (c^{4} d^{6} e^{2} + a c^{3} d^{4} e^{4}\right )} x^{2} + {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3}\right )} x\right )} \log \left (c d x + a e\right ) - 12 \, {\left (c^{4} d^{4} e^{4} x^{4} + a c^{3} d^{6} e^{2} + {\left (3 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (c^{4} d^{6} e^{2} + a c^{3} d^{4} e^{4}\right )} x^{2} + {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (a c^{5} d^{13} e - 5 \, a^{2} c^{4} d^{11} e^{3} + 10 \, a^{3} c^{3} d^{9} e^{5} - 10 \, a^{4} c^{2} d^{7} e^{7} + 5 \, a^{5} c d^{5} e^{9} - a^{6} d^{3} e^{11} + {\left (c^{6} d^{11} e^{3} - 5 \, a c^{5} d^{9} e^{5} + 10 \, a^{2} c^{4} d^{7} e^{7} - 10 \, a^{3} c^{3} d^{5} e^{9} + 5 \, a^{4} c^{2} d^{3} e^{11} - a^{5} c d e^{13}\right )} x^{4} + {\left (3 \, c^{6} d^{12} e^{2} - 14 \, a c^{5} d^{10} e^{4} + 25 \, a^{2} c^{4} d^{8} e^{6} - 20 \, a^{3} c^{3} d^{6} e^{8} + 5 \, a^{4} c^{2} d^{4} e^{10} + 2 \, a^{5} c d^{2} e^{12} - a^{6} e^{14}\right )} x^{3} + 3 \, {\left (c^{6} d^{13} e - 4 \, a c^{5} d^{11} e^{3} + 5 \, a^{2} c^{4} d^{9} e^{5} - 5 \, a^{4} c^{2} d^{5} e^{9} + 4 \, a^{5} c d^{3} e^{11} - a^{6} d e^{13}\right )} x^{2} + {\left (c^{6} d^{14} - 2 \, a c^{5} d^{12} e^{2} - 5 \, a^{2} c^{4} d^{10} e^{4} + 20 \, a^{3} c^{3} d^{8} e^{6} - 25 \, a^{4} c^{2} d^{6} e^{8} + 14 \, a^{5} c d^{4} e^{10} - 3 \, a^{6} d^{2} e^{12}\right )} x\right )}} \] Input:

integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fr 
icas")
 

Output:

-1/3*(3*c^4*d^8 + 10*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 6*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 + 6*(5*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*(11*c^4*d^7*e - 3*a*c^3*d^5*e^3 - 9 
*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*c^3*d^6*e^2 + 
(3*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + 3*(c^4*d^6*e^2 + a*c^3*d^4*e^4)*x^2 
+ (c^4*d^7*e + 3*a*c^3*d^5*e^3)*x)*log(c*d*x + a*e) - 12*(c^4*d^4*e^4*x^4 
+ a*c^3*d^6*e^2 + (3*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + 3*(c^4*d^6*e^2 + a 
*c^3*d^4*e^4)*x^2 + (c^4*d^7*e + 3*a*c^3*d^5*e^3)*x)*log(e*x + d))/(a*c^5* 
d^13*e - 5*a^2*c^4*d^11*e^3 + 10*a^3*c^3*d^9*e^5 - 10*a^4*c^2*d^7*e^7 + 5* 
a^5*c*d^5*e^9 - a^6*d^3*e^11 + (c^6*d^11*e^3 - 5*a*c^5*d^9*e^5 + 10*a^2*c^ 
4*d^7*e^7 - 10*a^3*c^3*d^5*e^9 + 5*a^4*c^2*d^3*e^11 - a^5*c*d*e^13)*x^4 + 
(3*c^6*d^12*e^2 - 14*a*c^5*d^10*e^4 + 25*a^2*c^4*d^8*e^6 - 20*a^3*c^3*d^6* 
e^8 + 5*a^4*c^2*d^4*e^10 + 2*a^5*c*d^2*e^12 - a^6*e^14)*x^3 + 3*(c^6*d^13* 
e - 4*a*c^5*d^11*e^3 + 5*a^2*c^4*d^9*e^5 - 5*a^4*c^2*d^5*e^9 + 4*a^5*c*d^3 
*e^11 - a^6*d*e^13)*x^2 + (c^6*d^14 - 2*a*c^5*d^12*e^2 - 5*a^2*c^4*d^10*e^ 
4 + 20*a^3*c^3*d^8*e^6 - 25*a^4*c^2*d^6*e^8 + 14*a^5*c*d^4*e^10 - 3*a^6*d^ 
2*e^12)*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 996 vs. \(2 (160) = 320\).

Time = 1.30 (sec) , antiderivative size = 996, normalized size of antiderivative = 5.66 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
 

Output:

-4*c**3*d**3*e*log(x + (-4*a**6*c**3*d**3*e**13/(a*e**2 - c*d**2)**5 + 24* 
a**5*c**4*d**5*e**11/(a*e**2 - c*d**2)**5 - 60*a**4*c**5*d**7*e**9/(a*e**2 
 - c*d**2)**5 + 80*a**3*c**6*d**9*e**7/(a*e**2 - c*d**2)**5 - 60*a**2*c**7 
*d**11*e**5/(a*e**2 - c*d**2)**5 + 24*a*c**8*d**13*e**3/(a*e**2 - c*d**2)* 
*5 + 4*a*c**3*d**3*e**3 - 4*c**9*d**15*e/(a*e**2 - c*d**2)**5 + 4*c**4*d** 
5*e)/(8*c**4*d**4*e**2))/(a*e**2 - c*d**2)**5 + 4*c**3*d**3*e*log(x + (4*a 
**6*c**3*d**3*e**13/(a*e**2 - c*d**2)**5 - 24*a**5*c**4*d**5*e**11/(a*e**2 
 - c*d**2)**5 + 60*a**4*c**5*d**7*e**9/(a*e**2 - c*d**2)**5 - 80*a**3*c**6 
*d**9*e**7/(a*e**2 - c*d**2)**5 + 60*a**2*c**7*d**11*e**5/(a*e**2 - c*d**2 
)**5 - 24*a*c**8*d**13*e**3/(a*e**2 - c*d**2)**5 + 4*a*c**3*d**3*e**3 + 4* 
c**9*d**15*e/(a*e**2 - c*d**2)**5 + 4*c**4*d**5*e)/(8*c**4*d**4*e**2))/(a* 
e**2 - c*d**2)**5 + (-a**3*e**6 + 5*a**2*c*d**2*e**4 - 13*a*c**2*d**4*e**2 
 - 3*c**3*d**6 - 12*c**3*d**3*e**3*x**3 + x**2*(-6*a*c**2*d**2*e**4 - 30*c 
**3*d**4*e**2) + x*(2*a**2*c*d*e**5 - 16*a*c**2*d**3*e**3 - 22*c**3*d**5*e 
))/(3*a**5*d**3*e**9 - 12*a**4*c*d**5*e**7 + 18*a**3*c**2*d**7*e**5 - 12*a 
**2*c**3*d**9*e**3 + 3*a*c**4*d**11*e + x**4*(3*a**4*c*d*e**11 - 12*a**3*c 
**2*d**3*e**9 + 18*a**2*c**3*d**5*e**7 - 12*a*c**4*d**7*e**5 + 3*c**5*d**9 
*e**3) + x**3*(3*a**5*e**12 - 3*a**4*c*d**2*e**10 - 18*a**3*c**2*d**4*e**8 
 + 42*a**2*c**3*d**6*e**6 - 33*a*c**4*d**8*e**4 + 9*c**5*d**10*e**2) + x** 
2*(9*a**5*d*e**11 - 27*a**4*c*d**3*e**9 + 18*a**3*c**2*d**5*e**7 + 18*a...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (174) = 348\).

Time = 0.05 (sec) , antiderivative size = 641, normalized size of antiderivative = 3.64 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {4 \, c^{3} d^{3} e \log \left (c d x + a e\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} + \frac {4 \, c^{3} d^{3} e \log \left (e x + d\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} - \frac {12 \, c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{6} + 13 \, a c^{2} d^{4} e^{2} - 5 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (11 \, c^{3} d^{5} e + 8 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{3 \, {\left (a c^{4} d^{11} e - 4 \, a^{2} c^{3} d^{9} e^{3} + 6 \, a^{3} c^{2} d^{7} e^{5} - 4 \, a^{4} c d^{5} e^{7} + a^{5} d^{3} e^{9} + {\left (c^{5} d^{9} e^{3} - 4 \, a c^{4} d^{7} e^{5} + 6 \, a^{2} c^{3} d^{5} e^{7} - 4 \, a^{3} c^{2} d^{3} e^{9} + a^{4} c d e^{11}\right )} x^{4} + {\left (3 \, c^{5} d^{10} e^{2} - 11 \, a c^{4} d^{8} e^{4} + 14 \, a^{2} c^{3} d^{6} e^{6} - 6 \, a^{3} c^{2} d^{4} e^{8} - a^{4} c d^{2} e^{10} + a^{5} e^{12}\right )} x^{3} + 3 \, {\left (c^{5} d^{11} e - 3 \, a c^{4} d^{9} e^{3} + 2 \, a^{2} c^{3} d^{7} e^{5} + 2 \, a^{3} c^{2} d^{5} e^{7} - 3 \, a^{4} c d^{3} e^{9} + a^{5} d e^{11}\right )} x^{2} + {\left (c^{5} d^{12} - a c^{4} d^{10} e^{2} - 6 \, a^{2} c^{3} d^{8} e^{4} + 14 \, a^{3} c^{2} d^{6} e^{6} - 11 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} d^{2} e^{10}\right )} x\right )}} \] Input:

integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="ma 
xima")
 

Output:

-4*c^3*d^3*e*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^3*d^3*e*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 + 3*c^3*d^6 + 
 13*a*c^2*d^4*e^2 - 5*a^2*c*d^2*e^4 + a^3*e^6 + 6*(5*c^3*d^4*e^2 + a*c^2*d 
^2*e^4)*x^2 + 2*(11*c^3*d^5*e + 8*a*c^2*d^3*e^3 - a^2*c*d*e^5)*x)/(a*c^4*d 
^11*e - 4*a^2*c^3*d^9*e^3 + 6*a^3*c^2*d^7*e^5 - 4*a^4*c*d^5*e^7 + a^5*d^3* 
e^9 + (c^5*d^9*e^3 - 4*a*c^4*d^7*e^5 + 6*a^2*c^3*d^5*e^7 - 4*a^3*c^2*d^3*e 
^9 + a^4*c*d*e^11)*x^4 + (3*c^5*d^10*e^2 - 11*a*c^4*d^8*e^4 + 14*a^2*c^3*d 
^6*e^6 - 6*a^3*c^2*d^4*e^8 - a^4*c*d^2*e^10 + a^5*e^12)*x^3 + 3*(c^5*d^11* 
e - 3*a*c^4*d^9*e^3 + 2*a^2*c^3*d^7*e^5 + 2*a^3*c^2*d^5*e^7 - 3*a^4*c*d^3* 
e^9 + a^5*d*e^11)*x^2 + (c^5*d^12 - a*c^4*d^10*e^2 - 6*a^2*c^3*d^8*e^4 + 1 
4*a^3*c^2*d^6*e^6 - 11*a^4*c*d^4*e^8 + 3*a^5*d^2*e^10)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (174) = 348\).

Time = 0.12 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.73 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {4 \, c^{3} d^{3} e^{2} \log \left ({\left | c d - \frac {c d^{2}}{e x + d} + \frac {a e^{2}}{e x + d} \right |}\right )}{c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}} - \frac {c^{4} d^{4} e}{{\left (c d^{2} - a e^{2}\right )}^{5} {\left (c d - \frac {c d^{2}}{e x + d} + \frac {a e^{2}}{e x + d}\right )}} - \frac {\frac {9 \, c^{4} d^{6} e^{7}}{e x + d} + \frac {3 \, c^{4} d^{7} e^{7}}{{\left (e x + d\right )}^{2}} + \frac {c^{4} d^{8} e^{7}}{{\left (e x + d\right )}^{3}} - \frac {18 \, a c^{3} d^{4} e^{9}}{e x + d} - \frac {9 \, a c^{3} d^{5} e^{9}}{{\left (e x + d\right )}^{2}} - \frac {4 \, a c^{3} d^{6} e^{9}}{{\left (e x + d\right )}^{3}} + \frac {9 \, a^{2} c^{2} d^{2} e^{11}}{e x + d} + \frac {9 \, a^{2} c^{2} d^{3} e^{11}}{{\left (e x + d\right )}^{2}} + \frac {6 \, a^{2} c^{2} d^{4} e^{11}}{{\left (e x + d\right )}^{3}} - \frac {3 \, a^{3} c d e^{13}}{{\left (e x + d\right )}^{2}} - \frac {4 \, a^{3} c d^{2} e^{13}}{{\left (e x + d\right )}^{3}} + \frac {a^{4} e^{15}}{{\left (e x + d\right )}^{3}}}{3 \, {\left (c^{6} d^{12} e^{6} - 6 \, a c^{5} d^{10} e^{8} + 15 \, a^{2} c^{4} d^{8} e^{10} - 20 \, a^{3} c^{3} d^{6} e^{12} + 15 \, a^{4} c^{2} d^{4} e^{14} - 6 \, a^{5} c d^{2} e^{16} + a^{6} e^{18}\right )}} \] Input:

integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="gi 
ac")
 

Output:

-4*c^3*d^3*e^2*log(abs(c*d - c*d^2/(e*x + d) + a*e^2/(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) - c^4*d^4*e/((c*d^2 - a*e^2)^5*(c*d - c*d^2/(e*x + d) + 
 a*e^2/(e*x + d))) - 1/3*(9*c^4*d^6*e^7/(e*x + d) + 3*c^4*d^7*e^7/(e*x + d 
)^2 + c^4*d^8*e^7/(e*x + d)^3 - 18*a*c^3*d^4*e^9/(e*x + d) - 9*a*c^3*d^5*e 
^9/(e*x + d)^2 - 4*a*c^3*d^6*e^9/(e*x + d)^3 + 9*a^2*c^2*d^2*e^11/(e*x + d 
) + 9*a^2*c^2*d^3*e^11/(e*x + d)^2 + 6*a^2*c^2*d^4*e^11/(e*x + d)^3 - 3*a^ 
3*c*d*e^13/(e*x + d)^2 - 4*a^3*c*d^2*e^13/(e*x + d)^3 + a^4*e^15/(e*x + d) 
^3)/(c^6*d^12*e^6 - 6*a*c^5*d^10*e^8 + 15*a^2*c^4*d^8*e^10 - 20*a^3*c^3*d^ 
6*e^12 + 15*a^4*c^2*d^4*e^14 - 6*a^5*c*d^2*e^16 + a^6*e^18)
 

Mupad [B] (verification not implemented)

Time = 5.82 (sec) , antiderivative size = 595, normalized size of antiderivative = 3.38 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {8\,c^3\,d^3\,e\,\mathrm {atanh}\left (\frac {a^5\,e^{10}-3\,a^4\,c\,d^2\,e^8+2\,a^3\,c^2\,d^4\,e^6+2\,a^2\,c^3\,d^6\,e^4-3\,a\,c^4\,d^8\,e^2+c^5\,d^{10}}{{\left (a\,e^2-c\,d^2\right )}^5}+\frac {2\,c\,d\,e\,x\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{{\left (a\,e^2-c\,d^2\right )}^5}\right )}{{\left (a\,e^2-c\,d^2\right )}^5}-\frac {\frac {a^3\,e^6-5\,a^2\,c\,d^2\,e^4+13\,a\,c^2\,d^4\,e^2+3\,c^3\,d^6}{3\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {4\,c^3\,d^3\,e^3\,x^3}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}+\frac {2\,c\,d\,x\,\left (-a^2\,e^5+8\,a\,c\,d^2\,e^3+11\,c^2\,d^4\,e\right )}{3\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}+\frac {2\,c^2\,d^2\,x^2\,\left (5\,c\,d^2\,e^2+a\,e^4\right )}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}}{x\,\left (c\,d^4+3\,a\,d^2\,e^2\right )+x^3\,\left (3\,c\,d^2\,e^2+a\,e^4\right )+x^2\,\left (3\,c\,d^3\,e+3\,a\,d\,e^3\right )+a\,d^3\,e+c\,d\,e^3\,x^4} \] Input:

int(1/((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2),x)
 

Output:

(8*c^3*d^3*e*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 - ((a^3*e^6 + 3*c^3*d^6 + 13*a*c^ 
2*d^4*e^2 - 5*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)) + (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) + (2*c*d*x*(11* 
c^2*d^4*e - a^2*e^5 + 8*a*c*d^2*e^3))/(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^2*d^2*x^2*(a*e^4 + 5*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*(c*d^4 + 3*a*d^2*e^2) + x^3*(a*e^4 + 3*c*d^2*e^2) + x^2*(3 
*a*d*e^3 + 3*c*d^3*e) + a*d^3*e + c*d*e^3*x^4)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1331, normalized size of antiderivative = 7.56 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
                                                                                    
                                                                                    
 

Output:

(12*log(a*e + c*d*x)*a**2*c**3*d**6*e**4 + 36*log(a*e + c*d*x)*a**2*c**3*d 
**5*e**5*x + 36*log(a*e + c*d*x)*a**2*c**3*d**4*e**6*x**2 + 12*log(a*e + c 
*d*x)*a**2*c**3*d**3*e**7*x**3 + 36*log(a*e + c*d*x)*a*c**4*d**8*e**2 + 12 
0*log(a*e + c*d*x)*a*c**4*d**7*e**3*x + 144*log(a*e + c*d*x)*a*c**4*d**6*e 
**4*x**2 + 72*log(a*e + c*d*x)*a*c**4*d**5*e**5*x**3 + 12*log(a*e + c*d*x) 
*a*c**4*d**4*e**6*x**4 + 36*log(a*e + c*d*x)*c**5*d**9*e*x + 108*log(a*e + 
 c*d*x)*c**5*d**8*e**2*x**2 + 108*log(a*e + c*d*x)*c**5*d**7*e**3*x**3 + 3 
6*log(a*e + c*d*x)*c**5*d**6*e**4*x**4 - 12*log(d + e*x)*a**2*c**3*d**6*e* 
*4 - 36*log(d + e*x)*a**2*c**3*d**5*e**5*x - 36*log(d + e*x)*a**2*c**3*d** 
4*e**6*x**2 - 12*log(d + e*x)*a**2*c**3*d**3*e**7*x**3 - 36*log(d + e*x)*a 
*c**4*d**8*e**2 - 120*log(d + e*x)*a*c**4*d**7*e**3*x - 144*log(d + e*x)*a 
*c**4*d**6*e**4*x**2 - 72*log(d + e*x)*a*c**4*d**5*e**5*x**3 - 12*log(d + 
e*x)*a*c**4*d**4*e**6*x**4 - 36*log(d + e*x)*c**5*d**9*e*x - 108*log(d + e 
*x)*c**5*d**8*e**2*x**2 - 108*log(d + e*x)*c**5*d**7*e**3*x**3 - 36*log(d 
+ e*x)*c**5*d**6*e**4*x**4 - a**5*e**10 + 3*a**4*c*d**2*e**8 + 2*a**4*c*d* 
e**9*x - 12*a**3*c**2*d**3*e**7*x - 6*a**3*c**2*d**2*e**8*x**2 - 32*a**2*c 
**3*d**6*e**4 - 24*a**2*c**3*d**5*e**5*x - 6*a**2*c**3*d**4*e**6*x**2 + 21 
*a*c**4*d**8*e**2 - 20*a*c**4*d**7*e**3*x - 42*a*c**4*d**6*e**4*x**2 + 12* 
a*c**4*d**4*e**6*x**4 + 9*c**5*d**10 + 54*c**5*d**9*e*x + 54*c**5*d**8*e** 
2*x**2 - 12*c**5*d**6*e**4*x**4)/(3*(a**7*d**3*e**13 + 3*a**7*d**2*e**1...