\(\int \frac {1}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [148]

Optimal result

Integrand size = 27, antiderivative size = 256 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {c^3 d^3}{3 \left (c d^2-a e^2\right )^4 (a e+c d x)^3}+\frac {2 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)^2}-\frac {10 c^3 d^3 e^2}{\left (c d^2-a e^2\right )^6 (a e+c d x)}-\frac {e^3}{3 \left (c d^2-a e^2\right )^4 (d+e x)^3}-\frac {2 c d e^3}{\left (c d^2-a e^2\right )^5 (d+e x)^2}-\frac {10 c^2 d^2 e^3}{\left (c d^2-a e^2\right )^6 (d+e x)}-\frac {20 c^3 d^3 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^7}+\frac {20 c^3 d^3 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^7} \] Output:

-1/3*c^3*d^3/(-a*e^2+c*d^2)^4/(c*d*x+a*e)^3+2*c^3*d^3*e/(-a*e^2+c*d^2)^5/( 
c*d*x+a*e)^2-10*c^3*d^3*e^2/(-a*e^2+c*d^2)^6/(c*d*x+a*e)-1/3*e^3/(-a*e^2+c 
*d^2)^4/(e*x+d)^3-2*c*d*e^3/(-a*e^2+c*d^2)^5/(e*x+d)^2-10*c^2*d^2*e^3/(-a* 
e^2+c*d^2)^6/(e*x+d)-20*c^3*d^3*e^3*ln(c*d*x+a*e)/(-a*e^2+c*d^2)^7+20*c^3* 
d^3*e^3*ln(e*x+d)/(-a*e^2+c*d^2)^7
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {\frac {c^3 d^3 \left (c d^2-a e^2\right )^3}{(a e+c d x)^3}-\frac {6 c^3 d^3 e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac {30 c^3 d^3 e^2 \left (c d^2-a e^2\right )}{a e+c d x}+\frac {\left (c d^2 e-a e^3\right )^3}{(d+e x)^3}+\frac {6 c d e^3 \left (c d^2-a e^2\right )^2}{(d+e x)^2}+\frac {30 c^2 d^2 e^3 \left (c d^2-a e^2\right )}{d+e x}+60 c^3 d^3 e^3 \log (a e+c d x)-60 c^3 d^3 e^3 \log (d+e x)}{3 \left (-c d^2+a e^2\right )^7} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-4),x]
 

Output:

((c^3*d^3*(c*d^2 - a*e^2)^3)/(a*e + c*d*x)^3 - (6*c^3*d^3*e*(c*d^2 - a*e^2 
)^2)/(a*e + c*d*x)^2 + (30*c^3*d^3*e^2*(c*d^2 - a*e^2))/(a*e + c*d*x) + (c 
*d^2*e - a*e^3)^3/(d + e*x)^3 + (6*c*d*e^3*(c*d^2 - a*e^2)^2)/(d + e*x)^2 
+ (30*c^2*d^2*e^3*(c*d^2 - a*e^2))/(d + e*x) + 60*c^3*d^3*e^3*Log[a*e + c* 
d*x] - 60*c^3*d^3*e^3*Log[d + e*x])/(3*(-(c*d^2) + a*e^2)^7)
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.10, 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.

\(\displaystyle \int \frac {1}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^4} \, dx\)

\(\Big \downarrow \) 1084

\(\displaystyle c^4 d^4 e^4 \int \left (\frac {20}{c d \left (c d^2-a e^2\right )^7 (d+e x)}+\frac {10}{c^2 d^2 \left (c d^2-a e^2\right )^6 (d+e x)^2}+\frac {4}{c^3 d^3 \left (c d^2-a e^2\right )^5 (d+e x)^3}+\frac {1}{c^4 d^4 \left (c d^2-a e^2\right )^4 (d+e x)^4}-\frac {20}{e \left (c d^2-a e^2\right )^7 (a e+c d x)}+\frac {10}{e^2 \left (c d^2-a e^2\right )^6 (a e+c d x)^2}-\frac {4}{e^3 \left (c d^2-a e^2\right )^5 (a e+c d x)^3}+\frac {1}{e^4 \left (c d^2-a e^2\right )^4 (a e+c d x)^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle c^4 d^4 e^4 \left (-\frac {1}{3 c^4 d^4 e (d+e x)^3 \left (c d^2-a e^2\right )^4}-\frac {2}{c^3 d^3 e (d+e x)^2 \left (c d^2-a e^2\right )^5}-\frac {10}{c^2 d^2 e (d+e x) \left (c d^2-a e^2\right )^6}-\frac {10}{c d e^2 \left (c d^2-a e^2\right )^6 (a e+c d x)}-\frac {20 \log (a e+c d x)}{c d e \left (c d^2-a e^2\right )^7}+\frac {20 \log (d+e x)}{c d e \left (c d^2-a e^2\right )^7}-\frac {1}{3 c d e^4 \left (c d^2-a e^2\right )^4 (a e+c d x)^3}+\frac {2}{c d e^3 \left (c d^2-a e^2\right )^5 (a e+c d x)^2}\right )\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-4),x]
 

Output:

c^4*d^4*e^4*(-1/3*1/(c*d*e^4*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^3) + 2/(c*d*e 
^3*(c*d^2 - a*e^2)^5*(a*e + c*d*x)^2) - 10/(c*d*e^2*(c*d^2 - a*e^2)^6*(a*e 
 + c*d*x)) - 1/(3*c^4*d^4*e*(c*d^2 - a*e^2)^4*(d + e*x)^3) - 2/(c^3*d^3*e* 
(c*d^2 - a*e^2)^5*(d + e*x)^2) - 10/(c^2*d^2*e*(c*d^2 - a*e^2)^6*(d + e*x) 
) - (20*Log[a*e + c*d*x])/(c*d*e*(c*d^2 - a*e^2)^7) + (20*Log[d + e*x])/(c 
*d*e*(c*d^2 - a*e^2)^7))
 

Defintions of rubi rules used

rule 1084

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]
 

rule 2009

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

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.99

method	result	size
default	\(-\frac {c^{3} d^{3}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (c d x +a e \right )^{3}}+\frac {20 c^{3} d^{3} e^{3} \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{7}}-\frac {10 c^{3} d^{3} e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{6} \left (c d x +a e \right )}-\frac {2 c^{3} d^{3} e}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (c d x +a e \right )^{2}}-\frac {e^{3}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{4} \left (e x +d \right )^{3}}-\frac {20 c^{3} d^{3} e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{7}}-\frac {10 e^{3} c^{2} d^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{6} \left (e x +d \right )}+\frac {2 e^{3} c d}{\left (a \,e^{2}-c \,d^{2}\right )^{5} \left (e x +d \right )^{2}}\)	\(253\)
risch	\(\text {Expression too large to display}\)	\(1023\)
norman	\(\text {Expression too large to display}\)	\(1074\)
parallelrisch	\(\text {Expression too large to display}\)	\(1216\)

Input:

int(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/3*c^3*d^3/(a*e^2-c*d^2)^4/(c*d*x+a*e)^3+20*c^3*d^3/(a*e^2-c*d^2)^7*e^3* 
ln(c*d*x+a*e)-10*c^3*d^3/(a*e^2-c*d^2)^6*e^2/(c*d*x+a*e)-2*c^3*d^3/(a*e^2- 
c*d^2)^5*e/(c*d*x+a*e)^2-1/3*e^3/(a*e^2-c*d^2)^4/(e*x+d)^3-20*c^3*d^3/(a*e 
^2-c*d^2)^7*e^3*ln(e*x+d)-10*e^3/(a*e^2-c*d^2)^6*c^2*d^2/(e*x+d)+2*e^3/(a* 
e^2-c*d^2)^5*c*d/(e*x+d)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1618 vs. \(2 (252) = 504\).

Time = 0.13 (sec) , antiderivative size = 1618, normalized size of antiderivative = 6.32 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")
 

Output:

-1/3*(c^6*d^12 - 9*a*c^5*d^10*e^2 + 45*a^2*c^4*d^8*e^4 - 45*a^4*c^2*d^4*e^ 
8 + 9*a^5*c*d^2*e^10 - a^6*e^12 + 60*(c^6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 1 
50*(c^6*d^8*e^4 - a^2*c^4*d^4*e^8)*x^4 + 10*(11*c^6*d^9*e^3 + 27*a*c^5*d^7 
*e^5 - 27*a^2*c^4*d^5*e^7 - 11*a^3*c^3*d^3*e^9)*x^3 + 15*(c^6*d^10*e^2 + 1 
8*a*c^5*d^8*e^4 - 18*a^3*c^3*d^4*e^8 - a^4*c^2*d^2*e^10)*x^2 - 3*(c^6*d^11 
*e - 15*a*c^5*d^9*e^3 - 60*a^2*c^4*d^7*e^5 + 60*a^3*c^3*d^5*e^7 + 15*a^4*c 
^2*d^3*e^9 - a^5*c*d*e^11)*x + 60*(c^6*d^6*e^6*x^6 + a^3*c^3*d^6*e^6 + 3*( 
c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^5 + 3*(c^6*d^8*e^4 + 3*a*c^5*d^6*e^6 + a^2* 
c^4*d^4*e^8)*x^4 + (c^6*d^9*e^3 + 9*a*c^5*d^7*e^5 + 9*a^2*c^4*d^5*e^7 + a^ 
3*c^3*d^3*e^9)*x^3 + 3*(a*c^5*d^8*e^4 + 3*a^2*c^4*d^6*e^6 + a^3*c^3*d^4*e^ 
8)*x^2 + 3*(a^2*c^4*d^7*e^5 + a^3*c^3*d^5*e^7)*x)*log(c*d*x + a*e) - 60*(c 
^6*d^6*e^6*x^6 + a^3*c^3*d^6*e^6 + 3*(c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^5 + 3 
*(c^6*d^8*e^4 + 3*a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + (c^6*d^9*e^3 + 9* 
a*c^5*d^7*e^5 + 9*a^2*c^4*d^5*e^7 + a^3*c^3*d^3*e^9)*x^3 + 3*(a*c^5*d^8*e^ 
4 + 3*a^2*c^4*d^6*e^6 + a^3*c^3*d^4*e^8)*x^2 + 3*(a^2*c^4*d^7*e^5 + a^3*c^ 
3*d^5*e^7)*x)*log(e*x + d))/(a^3*c^7*d^17*e^3 - 7*a^4*c^6*d^15*e^5 + 21*a^ 
5*c^5*d^13*e^7 - 35*a^6*c^4*d^11*e^9 + 35*a^7*c^3*d^9*e^11 - 21*a^8*c^2*d^ 
7*e^13 + 7*a^9*c*d^5*e^15 - a^10*d^3*e^17 + (c^10*d^17*e^3 - 7*a*c^9*d^15* 
e^5 + 21*a^2*c^8*d^13*e^7 - 35*a^3*c^7*d^11*e^9 + 35*a^4*c^6*d^9*e^11 - 21 
*a^5*c^5*d^7*e^13 + 7*a^6*c^4*d^5*e^15 - a^7*c^3*d^3*e^17)*x^6 + 3*(c^1...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1748 vs. \(2 (238) = 476\).

Time = 3.61 (sec) , antiderivative size = 1748, normalized size of antiderivative = 6.83 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)
 

Output:

-20*c**3*d**3*e**3*log(x + (-20*a**8*c**3*d**3*e**19/(a*e**2 - c*d**2)**7 
+ 160*a**7*c**4*d**5*e**17/(a*e**2 - c*d**2)**7 - 560*a**6*c**5*d**7*e**15 
/(a*e**2 - c*d**2)**7 + 1120*a**5*c**6*d**9*e**13/(a*e**2 - c*d**2)**7 - 1 
400*a**4*c**7*d**11*e**11/(a*e**2 - c*d**2)**7 + 1120*a**3*c**8*d**13*e**9 
/(a*e**2 - c*d**2)**7 - 560*a**2*c**9*d**15*e**7/(a*e**2 - c*d**2)**7 + 16 
0*a*c**10*d**17*e**5/(a*e**2 - c*d**2)**7 + 20*a*c**3*d**3*e**5 - 20*c**11 
*d**19*e**3/(a*e**2 - c*d**2)**7 + 20*c**4*d**5*e**3)/(40*c**4*d**4*e**4)) 
/(a*e**2 - c*d**2)**7 + 20*c**3*d**3*e**3*log(x + (20*a**8*c**3*d**3*e**19 
/(a*e**2 - c*d**2)**7 - 160*a**7*c**4*d**5*e**17/(a*e**2 - c*d**2)**7 + 56 
0*a**6*c**5*d**7*e**15/(a*e**2 - c*d**2)**7 - 1120*a**5*c**6*d**9*e**13/(a 
*e**2 - c*d**2)**7 + 1400*a**4*c**7*d**11*e**11/(a*e**2 - c*d**2)**7 - 112 
0*a**3*c**8*d**13*e**9/(a*e**2 - c*d**2)**7 + 560*a**2*c**9*d**15*e**7/(a* 
e**2 - c*d**2)**7 - 160*a*c**10*d**17*e**5/(a*e**2 - c*d**2)**7 + 20*a*c** 
3*d**3*e**5 + 20*c**11*d**19*e**3/(a*e**2 - c*d**2)**7 + 20*c**4*d**5*e**3 
)/(40*c**4*d**4*e**4))/(a*e**2 - c*d**2)**7 + (-a**5*e**10 + 8*a**4*c*d**2 
*e**8 - 37*a**3*c**2*d**4*e**6 - 37*a**2*c**3*d**6*e**4 + 8*a*c**4*d**8*e* 
*2 - c**5*d**10 - 60*c**5*d**5*e**5*x**5 + x**4*(-150*a*c**4*d**4*e**6 - 1 
50*c**5*d**6*e**4) + x**3*(-110*a**2*c**3*d**3*e**7 - 380*a*c**4*d**5*e**5 
 - 110*c**5*d**7*e**3) + x**2*(-15*a**3*c**2*d**2*e**8 - 285*a**2*c**3*d** 
4*e**6 - 285*a*c**4*d**6*e**4 - 15*c**5*d**8*e**2) + x*(3*a**4*c*d*e**9...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1278 vs. \(2 (252) = 504\).

Time = 0.07 (sec) , antiderivative size = 1278, normalized size of antiderivative = 4.99 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")
 

Output:

-20*c^3*d^3*e^3*log(c*d*x + a*e)/(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5 
*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 
+ 7*a^6*c*d^2*e^12 - a^7*e^14) + 20*c^3*d^3*e^3*log(e*x + d)/(c^7*d^14 - 7 
*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^ 
6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14) - 1/3*(60*c^5*d 
^5*e^5*x^5 + c^5*d^10 - 8*a*c^4*d^8*e^2 + 37*a^2*c^3*d^6*e^4 + 37*a^3*c^2* 
d^4*e^6 - 8*a^4*c*d^2*e^8 + a^5*e^10 + 150*(c^5*d^6*e^4 + a*c^4*d^4*e^6)*x 
^4 + 10*(11*c^5*d^7*e^3 + 38*a*c^4*d^5*e^5 + 11*a^2*c^3*d^3*e^7)*x^3 + 15* 
(c^5*d^8*e^2 + 19*a*c^4*d^6*e^4 + 19*a^2*c^3*d^4*e^6 + a^3*c^2*d^2*e^8)*x^ 
2 - 3*(c^5*d^9*e - 14*a*c^4*d^7*e^3 - 74*a^2*c^3*d^5*e^5 - 14*a^3*c^2*d^3* 
e^7 + a^4*c*d*e^9)*x)/(a^3*c^6*d^15*e^3 - 6*a^4*c^5*d^13*e^5 + 15*a^5*c^4* 
d^11*e^7 - 20*a^6*c^3*d^9*e^9 + 15*a^7*c^2*d^7*e^11 - 6*a^8*c*d^5*e^13 + a 
^9*d^3*e^15 + (c^9*d^15*e^3 - 6*a*c^8*d^13*e^5 + 15*a^2*c^7*d^11*e^7 - 20* 
a^3*c^6*d^9*e^9 + 15*a^4*c^5*d^7*e^11 - 6*a^5*c^4*d^5*e^13 + a^6*c^3*d^3*e 
^15)*x^6 + 3*(c^9*d^16*e^2 - 5*a*c^8*d^14*e^4 + 9*a^2*c^7*d^12*e^6 - 5*a^3 
*c^6*d^10*e^8 - 5*a^4*c^5*d^8*e^10 + 9*a^5*c^4*d^6*e^12 - 5*a^6*c^3*d^4*e^ 
14 + a^7*c^2*d^2*e^16)*x^5 + 3*(c^9*d^17*e - 3*a*c^8*d^15*e^3 - 2*a^2*c^7* 
d^13*e^5 + 19*a^3*c^6*d^11*e^7 - 30*a^4*c^5*d^9*e^9 + 19*a^5*c^4*d^7*e^11 
- 2*a^6*c^3*d^5*e^13 - 3*a^7*c^2*d^3*e^15 + a^8*c*d*e^17)*x^4 + (c^9*d^18 
+ 3*a*c^8*d^16*e^2 - 30*a^2*c^7*d^14*e^4 + 62*a^3*c^6*d^12*e^6 - 36*a^4...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (252) = 504\).

Time = 0.16 (sec) , antiderivative size = 635, normalized size of antiderivative = 2.48 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {20 \, c^{4} d^{4} e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{8} d^{15} - 7 \, a c^{7} d^{13} e^{2} + 21 \, a^{2} c^{6} d^{11} e^{4} - 35 \, a^{3} c^{5} d^{9} e^{6} + 35 \, a^{4} c^{4} d^{7} e^{8} - 21 \, a^{5} c^{3} d^{5} e^{10} + 7 \, a^{6} c^{2} d^{3} e^{12} - a^{7} c d e^{14}} + \frac {20 \, c^{3} d^{3} e^{4} \log \left ({\left | e x + d \right |}\right )}{c^{7} d^{14} e - 7 \, a c^{6} d^{12} e^{3} + 21 \, a^{2} c^{5} d^{10} e^{5} - 35 \, a^{3} c^{4} d^{8} e^{7} + 35 \, a^{4} c^{3} d^{6} e^{9} - 21 \, a^{5} c^{2} d^{4} e^{11} + 7 \, a^{6} c d^{2} e^{13} - a^{7} e^{15}} - \frac {60 \, c^{5} d^{5} e^{5} x^{5} + 150 \, c^{5} d^{6} e^{4} x^{4} + 150 \, a c^{4} d^{4} e^{6} x^{4} + 110 \, c^{5} d^{7} e^{3} x^{3} + 380 \, a c^{4} d^{5} e^{5} x^{3} + 110 \, a^{2} c^{3} d^{3} e^{7} x^{3} + 15 \, c^{5} d^{8} e^{2} x^{2} + 285 \, a c^{4} d^{6} e^{4} x^{2} + 285 \, a^{2} c^{3} d^{4} e^{6} x^{2} + 15 \, a^{3} c^{2} d^{2} e^{8} x^{2} - 3 \, c^{5} d^{9} e x + 42 \, a c^{4} d^{7} e^{3} x + 222 \, a^{2} c^{3} d^{5} e^{5} x + 42 \, a^{3} c^{2} d^{3} e^{7} x - 3 \, a^{4} c d e^{9} x + c^{5} d^{10} - 8 \, a c^{4} d^{8} e^{2} + 37 \, a^{2} c^{3} d^{6} e^{4} + 37 \, a^{3} c^{2} d^{4} e^{6} - 8 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}}{3 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} {\left (c d e x^{2} + c d^{2} x + a e^{2} x + a d e\right )}^{3}} \] Input:

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")
 

Output:

-20*c^4*d^4*e^3*log(abs(c*d*x + a*e))/(c^8*d^15 - 7*a*c^7*d^13*e^2 + 21*a^ 
2*c^6*d^11*e^4 - 35*a^3*c^5*d^9*e^6 + 35*a^4*c^4*d^7*e^8 - 21*a^5*c^3*d^5* 
e^10 + 7*a^6*c^2*d^3*e^12 - a^7*c*d*e^14) + 20*c^3*d^3*e^4*log(abs(e*x + d 
))/(c^7*d^14*e - 7*a*c^6*d^12*e^3 + 21*a^2*c^5*d^10*e^5 - 35*a^3*c^4*d^8*e 
^7 + 35*a^4*c^3*d^6*e^9 - 21*a^5*c^2*d^4*e^11 + 7*a^6*c*d^2*e^13 - a^7*e^1 
5) - 1/3*(60*c^5*d^5*e^5*x^5 + 150*c^5*d^6*e^4*x^4 + 150*a*c^4*d^4*e^6*x^4 
 + 110*c^5*d^7*e^3*x^3 + 380*a*c^4*d^5*e^5*x^3 + 110*a^2*c^3*d^3*e^7*x^3 + 
 15*c^5*d^8*e^2*x^2 + 285*a*c^4*d^6*e^4*x^2 + 285*a^2*c^3*d^4*e^6*x^2 + 15 
*a^3*c^2*d^2*e^8*x^2 - 3*c^5*d^9*e*x + 42*a*c^4*d^7*e^3*x + 222*a^2*c^3*d^ 
5*e^5*x + 42*a^3*c^2*d^3*e^7*x - 3*a^4*c*d*e^9*x + c^5*d^10 - 8*a*c^4*d^8* 
e^2 + 37*a^2*c^3*d^6*e^4 + 37*a^3*c^2*d^4*e^6 - 8*a^4*c*d^2*e^8 + a^5*e^10 
)/((c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 
+ 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*(c*d*e*x^2 + c*d^2*x + 
 a*e^2*x + a*d*e)^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx =\text {Too large to display} \] Input:

int(1/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)
 

Output:

piecewise(0 < (a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2, - (20*c^3*d^3*e^3*log(((a 
*e^2)/2 + (c*d^2)/2 - ((a*e^2 + c*d^2)^2/4 - a*c*d^2*e^2)^(1/2) + c*d*e*x) 
/((a*e^2)/2 + (c*d^2)/2 + ((a*e^2 + c*d^2)^2/4 - a*c*d^2*e^2)^(1/2) + c*d* 
e*x)))/((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)^(7/2) - (20*((a*e^2)/2 + (c*d^2 
)/2 + c*d*e*x)*((c*d*e)/(30*((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)*(x*(a*e^2 
+ c*d^2) + a*d*e + c*d*e*x^2)^3) - (c^2*d^2*e^2)/(6*((a*e^2 + c*d^2)^2 - 4 
*a*c*d^2*e^2)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2) + (c^3*d^3*e^3) 
/(((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e 
*x^2))))/(c*d*e), (a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2 < 0, - (20*((a*e^2)/2 
+ (c*d^2)/2 + c*d*e*x)*((c*d*e)/(30*((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)*(x 
*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3) - (c^2*d^2*e^2)/(6*((a*e^2 + c*d^ 
2)^2 - 4*a*c*d^2*e^2)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2) + (c^3* 
d^3*e^3)/(((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)^3*(x*(a*e^2 + c*d^2) + a*d*e 
 + c*d*e*x^2))))/(c*d*e) - (20*c^3*d^3*e^3*atan(((a*e^2)/2 + (c*d^2)/2 + c 
*d*e*x)/(- (a*e^2 + c*d^2)^2/4 + a*c*d^2*e^2)^(1/2)))/((- (a*e^2 + c*d^2)^ 
2/4 + a*c*d^2*e^2)^(1/2)*((a*e^2 + c*d^2)^2 - 4*a*c*d^2*e^2)^3), ~in((a*e^ 
2 + c*d^2)^2 - 4*a*c*d^2*e^2, 'real') | (a*e^2 + c*d^2)^2 == 4*a*c*d^2*e^2 
, int(1/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4, x))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 2480, normalized size of antiderivative = 9.69 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx =\text {Too large to display} \] Input:

int(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)
 

Output:

(60*log(a*e + c*d*x)*a**4*c**3*d**6*e**8 + 180*log(a*e + c*d*x)*a**4*c**3* 
d**5*e**9*x + 180*log(a*e + c*d*x)*a**4*c**3*d**4*e**10*x**2 + 60*log(a*e 
+ c*d*x)*a**4*c**3*d**3*e**11*x**3 + 60*log(a*e + c*d*x)*a**3*c**4*d**8*e* 
*6 + 360*log(a*e + c*d*x)*a**3*c**4*d**7*e**7*x + 720*log(a*e + c*d*x)*a** 
3*c**4*d**6*e**8*x**2 + 600*log(a*e + c*d*x)*a**3*c**4*d**5*e**9*x**3 + 18 
0*log(a*e + c*d*x)*a**3*c**4*d**4*e**10*x**4 + 180*log(a*e + c*d*x)*a**2*c 
**5*d**9*e**5*x + 720*log(a*e + c*d*x)*a**2*c**5*d**8*e**6*x**2 + 1080*log 
(a*e + c*d*x)*a**2*c**5*d**7*e**7*x**3 + 720*log(a*e + c*d*x)*a**2*c**5*d* 
*6*e**8*x**4 + 180*log(a*e + c*d*x)*a**2*c**5*d**5*e**9*x**5 + 180*log(a*e 
 + c*d*x)*a*c**6*d**10*e**4*x**2 + 600*log(a*e + c*d*x)*a*c**6*d**9*e**5*x 
**3 + 720*log(a*e + c*d*x)*a*c**6*d**8*e**6*x**4 + 360*log(a*e + c*d*x)*a* 
c**6*d**7*e**7*x**5 + 60*log(a*e + c*d*x)*a*c**6*d**6*e**8*x**6 + 60*log(a 
*e + c*d*x)*c**7*d**11*e**3*x**3 + 180*log(a*e + c*d*x)*c**7*d**10*e**4*x* 
*4 + 180*log(a*e + c*d*x)*c**7*d**9*e**5*x**5 + 60*log(a*e + c*d*x)*c**7*d 
**8*e**6*x**6 - 60*log(d + e*x)*a**4*c**3*d**6*e**8 - 180*log(d + e*x)*a** 
4*c**3*d**5*e**9*x - 180*log(d + e*x)*a**4*c**3*d**4*e**10*x**2 - 60*log(d 
 + e*x)*a**4*c**3*d**3*e**11*x**3 - 60*log(d + e*x)*a**3*c**4*d**8*e**6 - 
360*log(d + e*x)*a**3*c**4*d**7*e**7*x - 720*log(d + e*x)*a**3*c**4*d**6*e 
**8*x**2 - 600*log(d + e*x)*a**3*c**4*d**5*e**9*x**3 - 180*log(d + e*x)*a* 
*3*c**4*d**4*e**10*x**4 - 180*log(d + e*x)*a**2*c**5*d**9*e**5*x - 720*...