\(\int \frac {1}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [137]

Optimal result

Integrand size = 35, antiderivative size = 223 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac {4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac {e^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^3}+\frac {3 c d e^2}{2 \left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (d+e x)}+\frac {10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac {10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6} \] Output:

-1/2*c^3*d^3/(-a*e^2+c*d^2)^4/(c*d*x+a*e)^2+4*c^3*d^3*e/(-a*e^2+c*d^2)^5/( 
c*d*x+a*e)+1/3*e^2/(-a*e^2+c*d^2)^3/(e*x+d)^3+3/2*c*d*e^2/(-a*e^2+c*d^2)^4 
/(e*x+d)^2+6*c^2*d^2*e^2/(-a*e^2+c*d^2)^5/(e*x+d)+10*c^3*d^3*e^2*ln(c*d*x+ 
a*e)/(-a*e^2+c*d^2)^6-10*c^3*d^3*e^2*ln(e*x+d)/(-a*e^2+c*d^2)^6
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {-\frac {3 c^3 d^3 \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {24 c^3 d^3 e \left (c d^2-a e^2\right )}{a e+c d x}-\frac {2 e^2 \left (-c d^2+a e^2\right )^3}{(d+e x)^3}+\frac {9 c d \left (c d^2 e-a e^3\right )^2}{(d+e x)^2}+\frac {36 c^2 d^2 e^2 \left (c d^2-a e^2\right )}{d+e x}+60 c^3 d^3 e^2 \log (a e+c d x)-60 c^3 d^3 e^2 \log (d+e x)}{6 \left (c d^2-a e^2\right )^6} \] Input:

Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3),x]
 

Output:

((-3*c^3*d^3*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (24*c^3*d^3*e*(c*d^2 - a 
*e^2))/(a*e + c*d*x) - (2*e^2*(-(c*d^2) + a*e^2)^3)/(d + e*x)^3 + (9*c*d*( 
c*d^2*e - a*e^3)^2)/(d + e*x)^2 + (36*c^2*d^2*e^2*(c*d^2 - a*e^2))/(d + e* 
x) + 60*c^3*d^3*e^2*Log[a*e + c*d*x] - 60*c^3*d^3*e^2*Log[d + e*x])/(6*(c* 
d^2 - a*e^2)^6)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 223, 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.

\(\displaystyle \int \frac {1}{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1121

\(\displaystyle \int \left (\frac {10 c^4 d^4 e^2}{\left (c d^2-a e^2\right )^6 (a e+c d x)}-\frac {4 c^4 d^4 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)^2}+\frac {c^4 d^4}{\left (c d^2-a e^2\right )^4 (a e+c d x)^3}-\frac {10 c^3 d^3 e^3}{(d+e x) \left (c d^2-a e^2\right )^6}-\frac {6 c^2 d^2 e^3}{(d+e x)^2 \left (c d^2-a e^2\right )^5}-\frac {3 c d e^3}{(d+e x)^3 \left (c d^2-a e^2\right )^4}-\frac {e^3}{(d+e x)^4 \left (c d^2-a e^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac {c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac {10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac {10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6}+\frac {6 c^2 d^2 e^2}{(d+e x) \left (c d^2-a e^2\right )^5}+\frac {3 c d e^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}+\frac {e^2}{3 (d+e x)^3 \left (c d^2-a e^2\right )^3}\)

Input:

Int[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3),x]
 

Output:

-1/2*(c^3*d^3)/((c*d^2 - a*e^2)^4*(a*e + c*d*x)^2) + (4*c^3*d^3*e)/((c*d^2 
 - a*e^2)^5*(a*e + c*d*x)) + e^2/(3*(c*d^2 - a*e^2)^3*(d + e*x)^3) + (3*c* 
d*e^2)/(2*(c*d^2 - a*e^2)^4*(d + e*x)^2) + (6*c^2*d^2*e^2)/((c*d^2 - a*e^2 
)^5*(d + e*x)) + (10*c^3*d^3*e^2*Log[a*e + c*d*x])/(c*d^2 - a*e^2)^6 - (10 
*c^3*d^3*e^2*Log[d + e*x])/(c*d^2 - a*e^2)^6
 

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 = 1.81 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {c^{3} d^{3}}{2 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right )^{2}}+\frac {10 c^{3} d^{3} e^{2} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{6}}-\frac {4 c^{3} d^{3} e}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (c d x +a e \right )}-\frac {e^{2}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (e x +d \right )^{3}}-\frac {10 c^{3} d^{3} e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{6}}-\frac {6 e^{2} c^{2} d^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (e x +d \right )}+\frac {3 e^{2} c d}{2 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (e x +d \right )^{2}}\)	\(218\)
risch	\(\frac {-\frac {10 c^{4} d^{4} e^{4} x^{4}}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}-\frac {5 c^{3} d^{3} e^{3} \left (3 a \,e^{2}+5 c \,d^{2}\right ) x^{3}}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}-\frac {5 \left (2 a^{2} e^{4}+23 a c \,d^{2} e^{2}+11 c^{2} d^{4}\right ) c^{2} d^{2} e^{2} x^{2}}{3 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}+\frac {5 \left (e^{6} a^{3}-11 d^{2} e^{4} a^{2} c -35 d^{4} e^{2} a \,c^{2}-3 d^{6} c^{3}\right ) c d e x}{6 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}-\frac {2 a^{4} e^{8}-13 a^{3} c \,d^{2} e^{6}+47 a^{2} c^{2} d^{4} e^{4}+27 a \,c^{3} d^{6} e^{2}-3 c^{4} d^{8}}{6 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}}{\left (e x +d \right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{2}}+\frac {10 e^{2} d^{3} c^{3} \ln \left (-c d x -a e \right )}{a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}}-\frac {10 e^{2} d^{3} c^{3} \ln \left (e x +d \right )}{a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}}\)	\(777\)
norman	\(\frac {-\frac {10 c^{4} d^{4} e^{4} x^{4}}{a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}}+\frac {\left (-15 a \,c^{5} d^{5} e^{7}-25 c^{6} d^{7} e^{5}\right ) x^{3}}{d^{2} c^{2} e^{2} \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}+\frac {-2 a^{4} c^{2} e^{9}+13 a^{3} c^{3} e^{7} d^{2}-47 a^{2} c^{4} e^{5} d^{4}-27 a \,c^{5} e^{3} d^{6}+3 c^{6} e \,d^{8}}{6 c^{2} e \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}+\frac {\left (5 a^{3} c^{3} e^{9} d -55 a^{2} c^{4} d^{3} e^{7}-175 a \,c^{5} d^{5} e^{5}-15 c^{6} d^{7} e^{3}\right ) x}{6 c^{2} e^{2} \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}+\frac {\left (-10 a^{2} c^{4} d^{3} e^{9}-115 a \,c^{5} d^{5} e^{7}-55 c^{6} d^{7} e^{5}\right ) x^{2}}{3 d \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) c^{2} e^{3}}}{\left (e x +d \right )^{3} \left (c d x +a e \right )^{2}}-\frac {10 e^{2} d^{3} c^{3} \ln \left (e x +d \right )}{a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}}+\frac {10 e^{2} d^{3} c^{3} \ln \left (c d x +a e \right )}{a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}}\)	\(805\)
parallelrisch	\(-\frac {-60 x^{4} c^{7} d^{8} e^{7}-150 x^{3} c^{7} d^{9} e^{6}-110 x^{2} c^{7} d^{10} e^{5}-15 x \,c^{7} d^{11} e^{4}+3 e^{3} d^{12} c^{7}+60 \ln \left (e x +d \right ) x^{3} a^{2} c^{5} d^{5} e^{10}+360 \ln \left (e x +d \right ) x^{3} a \,c^{6} d^{7} e^{8}-60 \ln \left (c d x +a e \right ) x^{3} a^{2} c^{5} d^{5} e^{10}-360 \ln \left (c d x +a e \right ) x^{3} a \,c^{6} d^{7} e^{8}+180 \ln \left (e x +d \right ) x^{2} a^{2} c^{5} d^{6} e^{9}+360 \ln \left (e x +d \right ) x^{2} a \,c^{6} d^{8} e^{7}-180 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{5} d^{6} e^{9}-360 \ln \left (c d x +a e \right ) x^{2} a \,c^{6} d^{8} e^{7}+180 \ln \left (e x +d \right ) x \,a^{2} c^{5} d^{7} e^{8}+120 \ln \left (e x +d \right ) x a \,c^{6} d^{9} e^{6}-180 \ln \left (c d x +a e \right ) x \,a^{2} c^{5} d^{7} e^{8}-120 \ln \left (c d x +a e \right ) x a \,c^{6} d^{9} e^{6}-160 x a \,c^{6} d^{9} e^{6}+60 \ln \left (e x +d \right ) x^{5} c^{7} d^{7} e^{8}-60 \ln \left (c d x +a e \right ) x^{5} c^{7} d^{7} e^{8}+180 \ln \left (e x +d \right ) x^{4} c^{7} d^{8} e^{7}-180 \ln \left (c d x +a e \right ) x^{4} c^{7} d^{8} e^{7}+180 \ln \left (e x +d \right ) x^{3} c^{7} d^{9} e^{6}-180 \ln \left (c d x +a e \right ) x^{3} c^{7} d^{9} e^{6}+60 \ln \left (e x +d \right ) x^{2} c^{7} d^{10} e^{5}-60 \ln \left (c d x +a e \right ) x^{2} c^{7} d^{10} e^{5}+60 \ln \left (e x +d \right ) a^{2} c^{5} d^{8} e^{7}+120 \ln \left (e x +d \right ) x^{4} a \,c^{6} d^{6} e^{9}-120 \ln \left (c d x +a e \right ) x^{4} a \,c^{6} d^{6} e^{9}-5 x \,a^{4} c^{3} d^{3} e^{12}+60 x \,a^{3} c^{4} d^{5} e^{10}-60 \ln \left (c d x +a e \right ) a^{2} c^{5} d^{8} e^{7}+2 a^{5} c^{2} d^{2} e^{13}-15 a^{4} c^{3} d^{4} e^{11}+60 a^{3} c^{4} d^{6} e^{9}-20 a^{2} c^{5} d^{8} e^{7}-30 a \,c^{6} d^{10} e^{5}+60 x^{4} a \,c^{6} d^{6} e^{9}+90 x^{3} a^{2} c^{5} d^{5} e^{10}+60 x^{3} a \,c^{6} d^{7} e^{8}+20 x^{2} a^{3} c^{4} d^{4} e^{11}+210 x^{2} a^{2} c^{5} d^{6} e^{9}-120 x^{2} a \,c^{6} d^{8} e^{7}+120 x \,a^{2} c^{5} d^{7} e^{8}}{6 \left (a^{6} e^{12}-6 a^{5} d^{2} e^{10} c +15 a^{4} d^{4} e^{8} c^{2}-20 a^{3} d^{6} e^{6} c^{3}+15 a^{2} d^{8} e^{4} c^{4}-6 a \,d^{10} e^{2} c^{5}+d^{12} c^{6}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{2} \left (e x +d \right ) c^{2} d^{2} e^{3}}\)	\(950\)

Input:

int(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*c^3*d^3/(a*e^2-c*d^2)^4/(c*d*x+a*e)^2+10*c^3*d^3/(a*e^2-c*d^2)^6*e^2* 
ln(c*d*x+a*e)-4*c^3*d^3/(a*e^2-c*d^2)^5*e/(c*d*x+a*e)-1/3*e^2/(a*e^2-c*d^2 
)^3/(e*x+d)^3-10*c^3*d^3/(a*e^2-c*d^2)^6*e^2*ln(e*x+d)-6*e^2/(a*e^2-c*d^2) 
^5*c^2*d^2/(e*x+d)+3/2*e^2/(a*e^2-c*d^2)^4*c*d/(e*x+d)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1222 vs. \(2 (217) = 434\).

Time = 0.11 (sec) , antiderivative size = 1222, normalized size of antiderivative = 5.48 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fric 
as")
 

Output:

-1/6*(3*c^5*d^10 - 30*a*c^4*d^8*e^2 - 20*a^2*c^3*d^6*e^4 + 60*a^3*c^2*d^4* 
e^6 - 15*a^4*c*d^2*e^8 + 2*a^5*e^10 - 60*(c^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 
 - 30*(5*c^5*d^7*e^3 - 2*a*c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*x^3 - 10*(11*c 
^5*d^8*e^2 + 12*a*c^4*d^6*e^4 - 21*a^2*c^3*d^4*e^6 - 2*a^3*c^2*d^2*e^8)*x^ 
2 - 5*(3*c^5*d^9*e + 32*a*c^4*d^7*e^3 - 24*a^2*c^3*d^5*e^5 - 12*a^3*c^2*d^ 
3*e^7 + a^4*c*d*e^9)*x - 60*(c^5*d^5*e^5*x^5 + a^2*c^3*d^6*e^4 + (3*c^5*d^ 
6*e^4 + 2*a*c^4*d^4*e^6)*x^4 + (3*c^5*d^7*e^3 + 6*a*c^4*d^5*e^5 + a^2*c^3* 
d^3*e^7)*x^3 + (c^5*d^8*e^2 + 6*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6)*x^2 + ( 
2*a*c^4*d^7*e^3 + 3*a^2*c^3*d^5*e^5)*x)*log(c*d*x + a*e) + 60*(c^5*d^5*e^5 
*x^5 + a^2*c^3*d^6*e^4 + (3*c^5*d^6*e^4 + 2*a*c^4*d^4*e^6)*x^4 + (3*c^5*d^ 
7*e^3 + 6*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^3 + (c^5*d^8*e^2 + 6*a*c^4*d^ 
6*e^4 + 3*a^2*c^3*d^4*e^6)*x^2 + (2*a*c^4*d^7*e^3 + 3*a^2*c^3*d^5*e^5)*x)* 
log(e*x + d))/(a^2*c^6*d^15*e^2 - 6*a^3*c^5*d^13*e^4 + 15*a^4*c^4*d^11*e^6 
 - 20*a^5*c^3*d^9*e^8 + 15*a^6*c^2*d^7*e^10 - 6*a^7*c*d^5*e^12 + a^8*d^3*e 
^14 + (c^8*d^14*e^3 - 6*a*c^7*d^12*e^5 + 15*a^2*c^6*d^10*e^7 - 20*a^3*c^5* 
d^8*e^9 + 15*a^4*c^4*d^6*e^11 - 6*a^5*c^3*d^4*e^13 + a^6*c^2*d^2*e^15)*x^5 
 + (3*c^8*d^15*e^2 - 16*a*c^7*d^13*e^4 + 33*a^2*c^6*d^11*e^6 - 30*a^3*c^5* 
d^9*e^8 + 5*a^4*c^4*d^7*e^10 + 12*a^5*c^3*d^5*e^12 - 9*a^6*c^2*d^3*e^14 + 
2*a^7*c*d*e^16)*x^4 + (3*c^8*d^16*e - 12*a*c^7*d^14*e^3 + 10*a^2*c^6*d^12* 
e^5 + 24*a^3*c^5*d^10*e^7 - 60*a^4*c^4*d^8*e^9 + 52*a^5*c^3*d^6*e^11 - ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1357 vs. \(2 (206) = 412\).

Time = 2.28 (sec) , antiderivative size = 1357, normalized size of antiderivative = 6.09 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
 

Output:

-10*c**3*d**3*e**2*log(x + (-10*a**7*c**3*d**3*e**16/(a*e**2 - c*d**2)**6 
+ 70*a**6*c**4*d**5*e**14/(a*e**2 - c*d**2)**6 - 210*a**5*c**5*d**7*e**12/ 
(a*e**2 - c*d**2)**6 + 350*a**4*c**6*d**9*e**10/(a*e**2 - c*d**2)**6 - 350 
*a**3*c**7*d**11*e**8/(a*e**2 - c*d**2)**6 + 210*a**2*c**8*d**13*e**6/(a*e 
**2 - c*d**2)**6 - 70*a*c**9*d**15*e**4/(a*e**2 - c*d**2)**6 + 10*a*c**3*d 
**3*e**4 + 10*c**10*d**17*e**2/(a*e**2 - c*d**2)**6 + 10*c**4*d**5*e**2)/( 
20*c**4*d**4*e**3))/(a*e**2 - c*d**2)**6 + 10*c**3*d**3*e**2*log(x + (10*a 
**7*c**3*d**3*e**16/(a*e**2 - c*d**2)**6 - 70*a**6*c**4*d**5*e**14/(a*e**2 
 - c*d**2)**6 + 210*a**5*c**5*d**7*e**12/(a*e**2 - c*d**2)**6 - 350*a**4*c 
**6*d**9*e**10/(a*e**2 - c*d**2)**6 + 350*a**3*c**7*d**11*e**8/(a*e**2 - c 
*d**2)**6 - 210*a**2*c**8*d**13*e**6/(a*e**2 - c*d**2)**6 + 70*a*c**9*d**1 
5*e**4/(a*e**2 - c*d**2)**6 + 10*a*c**3*d**3*e**4 - 10*c**10*d**17*e**2/(a 
*e**2 - c*d**2)**6 + 10*c**4*d**5*e**2)/(20*c**4*d**4*e**3))/(a*e**2 - c*d 
**2)**6 + (-2*a**4*e**8 + 13*a**3*c*d**2*e**6 - 47*a**2*c**2*d**4*e**4 - 2 
7*a*c**3*d**6*e**2 + 3*c**4*d**8 - 60*c**4*d**4*e**4*x**4 + x**3*(-90*a*c* 
*3*d**3*e**5 - 150*c**4*d**5*e**3) + x**2*(-20*a**2*c**2*d**2*e**6 - 230*a 
*c**3*d**4*e**4 - 110*c**4*d**6*e**2) + x*(5*a**3*c*d*e**7 - 55*a**2*c**2* 
d**3*e**5 - 175*a*c**3*d**5*e**3 - 15*c**4*d**7*e))/(6*a**7*d**3*e**12 - 3 
0*a**6*c*d**5*e**10 + 60*a**5*c**2*d**7*e**8 - 60*a**4*c**3*d**9*e**6 + 30 
*a**3*c**4*d**11*e**4 - 6*a**2*c**5*d**13*e**2 + x**5*(6*a**5*c**2*d**2...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 947 vs. \(2 (217) = 434\).

Time = 0.08 (sec) , antiderivative size = 947, normalized size of antiderivative = 4.25 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxi 
ma")
 

Output:

10*c^3*d^3*e^2*log(c*d*x + a*e)/(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) - 10*c^3*d^3*e^2*log(e*x + d)/(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) + 1/6*(60*c^4*d^4*e^4*x^4 - 3*c^4*d^8 + 27*a*c^3*d^6*e^2 + 47* 
a^2*c^2*d^4*e^4 - 13*a^3*c*d^2*e^6 + 2*a^4*e^8 + 30*(5*c^4*d^5*e^3 + 3*a*c 
^3*d^3*e^5)*x^3 + 10*(11*c^4*d^6*e^2 + 23*a*c^3*d^4*e^4 + 2*a^2*c^2*d^2*e^ 
6)*x^2 + 5*(3*c^4*d^7*e + 35*a*c^3*d^5*e^3 + 11*a^2*c^2*d^3*e^5 - a^3*c*d* 
e^7)*x)/(a^2*c^5*d^13*e^2 - 5*a^3*c^4*d^11*e^4 + 10*a^4*c^3*d^9*e^6 - 10*a 
^5*c^2*d^7*e^8 + 5*a^6*c*d^5*e^10 - a^7*d^3*e^12 + (c^7*d^12*e^3 - 5*a*c^6 
*d^10*e^5 + 10*a^2*c^5*d^8*e^7 - 10*a^3*c^4*d^6*e^9 + 5*a^4*c^3*d^4*e^11 - 
 a^5*c^2*d^2*e^13)*x^5 + (3*c^7*d^13*e^2 - 13*a*c^6*d^11*e^4 + 20*a^2*c^5* 
d^9*e^6 - 10*a^3*c^4*d^7*e^8 - 5*a^4*c^3*d^5*e^10 + 7*a^5*c^2*d^3*e^12 - 2 
*a^6*c*d*e^14)*x^4 + (3*c^7*d^14*e - 9*a*c^6*d^12*e^3 + a^2*c^5*d^10*e^5 + 
 25*a^3*c^4*d^8*e^7 - 35*a^4*c^3*d^6*e^9 + 17*a^5*c^2*d^4*e^11 - a^6*c*d^2 
*e^13 - a^7*e^15)*x^3 + (c^7*d^15 + a*c^6*d^13*e^2 - 17*a^2*c^5*d^11*e^4 + 
 35*a^3*c^4*d^9*e^6 - 25*a^4*c^3*d^7*e^8 - a^5*c^2*d^5*e^10 + 9*a^6*c*d^3* 
e^12 - 3*a^7*d*e^14)*x^2 + (2*a*c^6*d^14*e - 7*a^2*c^5*d^12*e^3 + 5*a^3*c^ 
4*d^10*e^5 + 10*a^4*c^3*d^8*e^7 - 20*a^5*c^2*d^6*e^9 + 13*a^6*c*d^4*e^11 - 
 3*a^7*d^2*e^13)*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (217) = 434\).

Time = 0.14 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.26 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {10 \, c^{4} d^{4} e^{2} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{13} - 6 \, a c^{6} d^{11} e^{2} + 15 \, a^{2} c^{5} d^{9} e^{4} - 20 \, a^{3} c^{4} d^{7} e^{6} + 15 \, a^{4} c^{3} d^{5} e^{8} - 6 \, a^{5} c^{2} d^{3} e^{10} + a^{6} c d e^{12}} - \frac {10 \, c^{3} d^{3} e^{3} \log \left ({\left | e x + d \right |}\right )}{c^{6} d^{12} e - 6 \, a c^{5} d^{10} e^{3} + 15 \, a^{2} c^{4} d^{8} e^{5} - 20 \, a^{3} c^{3} d^{6} e^{7} + 15 \, a^{4} c^{2} d^{4} e^{9} - 6 \, a^{5} c d^{2} e^{11} + a^{6} e^{13}} - \frac {3 \, c^{5} d^{10} - 30 \, a c^{4} d^{8} e^{2} - 20 \, a^{2} c^{3} d^{6} e^{4} + 60 \, a^{3} c^{2} d^{4} e^{6} - 15 \, a^{4} c d^{2} e^{8} + 2 \, a^{5} e^{10} - 60 \, {\left (c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} - 30 \, {\left (5 \, c^{5} d^{7} e^{3} - 2 \, a c^{4} d^{5} e^{5} - 3 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{3} - 10 \, {\left (11 \, c^{5} d^{8} e^{2} + 12 \, a c^{4} d^{6} e^{4} - 21 \, a^{2} c^{3} d^{4} e^{6} - 2 \, a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 5 \, {\left (3 \, c^{5} d^{9} e + 32 \, a c^{4} d^{7} e^{3} - 24 \, a^{2} c^{3} d^{5} e^{5} - 12 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x}{6 \, {\left (c d^{2} - a e^{2}\right )}^{6} {\left (c d x + a e\right )}^{2} {\left (e x + d\right )}^{3}} \] Input:

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac 
")
 

Output:

10*c^4*d^4*e^2*log(abs(c*d*x + a*e))/(c^7*d^13 - 6*a*c^6*d^11*e^2 + 15*a^2 
*c^5*d^9*e^4 - 20*a^3*c^4*d^7*e^6 + 15*a^4*c^3*d^5*e^8 - 6*a^5*c^2*d^3*e^1 
0 + a^6*c*d*e^12) - 10*c^3*d^3*e^3*log(abs(e*x + d))/(c^6*d^12*e - 6*a*c^5 
*d^10*e^3 + 15*a^2*c^4*d^8*e^5 - 20*a^3*c^3*d^6*e^7 + 15*a^4*c^2*d^4*e^9 - 
 6*a^5*c*d^2*e^11 + a^6*e^13) - 1/6*(3*c^5*d^10 - 30*a*c^4*d^8*e^2 - 20*a^ 
2*c^3*d^6*e^4 + 60*a^3*c^2*d^4*e^6 - 15*a^4*c*d^2*e^8 + 2*a^5*e^10 - 60*(c 
^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 - 30*(5*c^5*d^7*e^3 - 2*a*c^4*d^5*e^5 - 3* 
a^2*c^3*d^3*e^7)*x^3 - 10*(11*c^5*d^8*e^2 + 12*a*c^4*d^6*e^4 - 21*a^2*c^3* 
d^4*e^6 - 2*a^3*c^2*d^2*e^8)*x^2 - 5*(3*c^5*d^9*e + 32*a*c^4*d^7*e^3 - 24* 
a^2*c^3*d^5*e^5 - 12*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)*x)/((c*d^2 - a*e^2)^6* 
(c*d*x + a*e)^2*(e*x + d)^3)
 

Mupad [B] (verification not implemented)

Time = 5.68 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.94 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {20\,c^3\,d^3\,e^2\,\mathrm {atanh}\left (\frac {a^6\,e^{12}-4\,a^5\,c\,d^2\,e^{10}+5\,a^4\,c^2\,d^4\,e^8-5\,a^2\,c^4\,d^8\,e^4+4\,a\,c^5\,d^{10}\,e^2-c^6\,d^{12}}{{\left (a\,e^2-c\,d^2\right )}^6}+\frac {2\,c\,d\,e\,x\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}{{\left (a\,e^2-c\,d^2\right )}^6}\right )}{{\left (a\,e^2-c\,d^2\right )}^6}-\frac {\frac {2\,a^4\,e^8-13\,a^3\,c\,d^2\,e^6+47\,a^2\,c^2\,d^4\,e^4+27\,a\,c^3\,d^6\,e^2-3\,c^4\,d^8}{6\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}+\frac {5\,c^2\,d\,x^3\,\left (5\,c^2\,d^4\,e^3+3\,a\,c\,d^2\,e^5\right )}{a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}+\frac {5\,c^2\,d^2\,x^2\,\left (2\,a^2\,e^6+23\,a\,c\,d^2\,e^4+11\,c^2\,d^4\,e^2\right )}{3\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}+\frac {10\,c^4\,d^4\,e^4\,x^4}{a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}+\frac {5\,c\,d\,e\,x\,\left (-a^3\,e^6+11\,a^2\,c\,d^2\,e^4+35\,a\,c^2\,d^4\,e^2+3\,c^3\,d^6\right )}{6\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}}{x\,\left (3\,a^2\,d^2\,e^3+2\,c\,a\,d^4\,e\right )+x^2\,\left (3\,a^2\,d\,e^4+6\,a\,c\,d^3\,e^2+c^2\,d^5\right )+x^3\,\left (a^2\,e^5+6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )+x^4\,\left (3\,c^2\,d^3\,e^2+2\,a\,c\,d\,e^4\right )+a^2\,d^3\,e^2+c^2\,d^2\,e^3\,x^5} \] Input:

int(1/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3),x)
 

Output:

(20*c^3*d^3*e^2*atanh((a^6*e^12 - c^6*d^12 + 4*a*c^5*d^10*e^2 - 4*a^5*c*d^ 
2*e^10 - 5*a^2*c^4*d^8*e^4 + 5*a^4*c^2*d^4*e^8)/(a*e^2 - c*d^2)^6 + (2*c*d 
*e*x*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3 
*d^6*e^4 + 10*a^3*c^2*d^4*e^6))/(a*e^2 - c*d^2)^6))/(a*e^2 - c*d^2)^6 - (( 
2*a^4*e^8 - 3*c^4*d^8 + 27*a*c^3*d^6*e^2 - 13*a^3*c*d^2*e^6 + 47*a^2*c^2*d 
^4*e^4)/(6*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a 
^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6)) + (5*c^2*d*x^3*(5*c^2*d^4*e^3 + 3*a* 
c*d^2*e^5))/(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10* 
a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6) + (5*c^2*d^2*x^2*(2*a^2*e^6 + 11*c^2 
*d^4*e^2 + 23*a*c*d^2*e^4))/(3*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5* 
a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6)) + (10*c^4*d^4*e^ 
4*x^4)/(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 - 5*a^4*c*d^2*e^8 - 10*a^2*c 
^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6) + (5*c*d*e*x*(3*c^3*d^6 - a^3*e^6 + 35*a* 
c^2*d^4*e^2 + 11*a^2*c*d^2*e^4))/(6*(a^5*e^10 - c^5*d^10 + 5*a*c^4*d^8*e^2 
 - 5*a^4*c*d^2*e^8 - 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6)))/(x*(3*a^2* 
d^2*e^3 + 2*a*c*d^4*e) + x^2*(c^2*d^5 + 3*a^2*d*e^4 + 6*a*c*d^3*e^2) + x^3 
*(a^2*e^5 + 3*c^2*d^4*e + 6*a*c*d^2*e^3) + x^4*(3*c^2*d^3*e^2 + 2*a*c*d*e^ 
4) + a^2*d^3*e^2 + c^2*d^2*e^3*x^5)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1970, normalized size of antiderivative = 8.83 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
                                                                                    
                                                                                    
 

Output:

(120*log(a*e + c*d*x)*a**3*c**3*d**6*e**6 + 360*log(a*e + c*d*x)*a**3*c**3 
*d**5*e**7*x + 360*log(a*e + c*d*x)*a**3*c**3*d**4*e**8*x**2 + 120*log(a*e 
 + c*d*x)*a**3*c**3*d**3*e**9*x**3 + 180*log(a*e + c*d*x)*a**2*c**4*d**8*e 
**4 + 780*log(a*e + c*d*x)*a**2*c**4*d**7*e**5*x + 1260*log(a*e + c*d*x)*a 
**2*c**4*d**6*e**6*x**2 + 900*log(a*e + c*d*x)*a**2*c**4*d**5*e**7*x**3 + 
240*log(a*e + c*d*x)*a**2*c**4*d**4*e**8*x**4 + 360*log(a*e + c*d*x)*a*c** 
5*d**9*e**3*x + 1200*log(a*e + c*d*x)*a*c**5*d**8*e**4*x**2 + 1440*log(a*e 
 + c*d*x)*a*c**5*d**7*e**5*x**3 + 720*log(a*e + c*d*x)*a*c**5*d**6*e**6*x* 
*4 + 120*log(a*e + c*d*x)*a*c**5*d**5*e**7*x**5 + 180*log(a*e + c*d*x)*c** 
6*d**10*e**2*x**2 + 540*log(a*e + c*d*x)*c**6*d**9*e**3*x**3 + 540*log(a*e 
 + c*d*x)*c**6*d**8*e**4*x**4 + 180*log(a*e + c*d*x)*c**6*d**7*e**5*x**5 - 
 120*log(d + e*x)*a**3*c**3*d**6*e**6 - 360*log(d + e*x)*a**3*c**3*d**5*e* 
*7*x - 360*log(d + e*x)*a**3*c**3*d**4*e**8*x**2 - 120*log(d + e*x)*a**3*c 
**3*d**3*e**9*x**3 - 180*log(d + e*x)*a**2*c**4*d**8*e**4 - 780*log(d + e* 
x)*a**2*c**4*d**7*e**5*x - 1260*log(d + e*x)*a**2*c**4*d**6*e**6*x**2 - 90 
0*log(d + e*x)*a**2*c**4*d**5*e**7*x**3 - 240*log(d + e*x)*a**2*c**4*d**4* 
e**8*x**4 - 360*log(d + e*x)*a*c**5*d**9*e**3*x - 1200*log(d + e*x)*a*c**5 
*d**8*e**4*x**2 - 1440*log(d + e*x)*a*c**5*d**7*e**5*x**3 - 720*log(d + e* 
x)*a*c**5*d**6*e**6*x**4 - 120*log(d + e*x)*a*c**5*d**5*e**7*x**5 - 180*lo 
g(d + e*x)*c**6*d**10*e**2*x**2 - 540*log(d + e*x)*c**6*d**9*e**3*x**3 ...