3.1895 \(\int \frac{1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

Optimal. Leaf size=262 \[ -\frac{20 c^3 d^3 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^7}+\frac{20 c^3 d^3 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^7}-\frac{10 c^2 d^2 e^2 \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}+\frac{5 c d e \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2}-\frac{a e^2+c d^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.277692, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{20 c^3 d^3 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^7}+\frac{20 c^3 d^3 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^7}-\frac{10 c^2 d^2 e^2 \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^6 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}+\frac{5 c d e \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2}-\frac{a e^2+c d^2+2 c d e x}{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.0749, size = 246, normalized size = 0.94 \[ \frac{40 c^{3} d^{3} e^{3} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{a e^{2} - c d^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{7}} - \frac{10 c^{2} d^{2} e^{2} \left (a e^{2} + c d^{2} + 2 c d e x\right )}{\left (a e^{2} - c d^{2}\right )^{6} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )} + \frac{5 c d e \left (a e^{2} + c d^{2} + 2 c d e x\right )}{3 \left (a e^{2} - c d^{2}\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{2}} - \frac{a e^{2} + c d^{2} + 2 c d e x}{3 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

40*c**3*d**3*e**3*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(a*e**2 - c*d**2))/(a*e**2
- c*d**2)**7 - 10*c**2*d**2*e**2*(a*e**2 + c*d**2 + 2*c*d*e*x)/((a*e**2 - c*d**
2)**6*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 5*c*d*e*(a*e**2 + c*d**2 + 2
*c*d*e*x)/(3*(a*e**2 - c*d**2)**4*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**2)
- (a*e**2 + c*d**2 + 2*c*d*e*x)/(3*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x**2 + x
*(a*e**2 + c*d**2))**3)

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Mathematica [A]  time = 0.615838, size = 234, normalized size = 0.89 \[ \frac{60 c^3 d^3 e^3 \log (a e+c d x)+\frac{30 c^3 d^3 e^2 \left (c d^2-a e^2\right )}{a e+c d x}-\frac{6 c^3 d^3 e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac{c^3 d^3 \left (c d^2-a e^2\right )^3}{(a e+c d x)^3}+\frac{30 c^2 d^2 e^3 \left (c d^2-a e^2\right )}{d+e x}+\frac{\left (c d^2 e-a e^3\right )^3}{(d+e x)^3}+\frac{6 c d e^3 \left (c d^2-a e^2\right )^2}{(d+e x)^2}-60 c^3 d^3 e^3 \log (d+e x)}{3 \left (a e^2-c d^2\right )^7} \]

Antiderivative was successfully verified.

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

[Out]

((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*Lo
g[d + e*x])/(3*(-(c*d^2) + a*e^2)^7)

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Maple [A]  time = 0.025, size = 253, normalized size = 1. \[ -{\frac{{e}^{3}}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( ex+d \right ) ^{3}}}-20\,{\frac{{e}^{3}{c}^{3}{d}^{3}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{7}}}-10\,{\frac{{c}^{2}{d}^{2}{e}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{6} \left ( ex+d \right ) }}+2\,{\frac{d{e}^{3}c}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3}{d}^{3}}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdx+ae \right ) ^{3}}}+20\,{\frac{{e}^{3}{c}^{3}{d}^{3}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{7}}}-10\,{\frac{{c}^{3}{d}^{3}{e}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{6} \left ( cdx+ae \right ) }}-2\,{\frac{{c}^{3}{d}^{3}e}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( cdx+ae \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.777382, size = 1725, normalized size = 6.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-4),x, algorithm="maxima")

[Out]

-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 - 2
0*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*c^5*d^10*e^8 - 36*a^5*c^4*d^8*e^10 + 62*a^
6*c^3*d^6*e^12 - 30*a^7*c^2*d^4*e^14 + 3*a^8*c*d^2*e^16 + a^9*e^18)*x^3 + 3*(a*c
^8*d^17*e - 3*a^2*c^7*d^15*e^3 - 2*a^3*c^6*d^13*e^5 + 19*a^4*c^5*d^11*e^7 - 30*a
^5*c^4*d^9*e^9 + 19*a^6*c^3*d^7*e^11 - 2*a^7*c^2*d^5*e^13 - 3*a^8*c*d^3*e^15 + a
^9*d*e^17)*x^2 + 3*(a^2*c^7*d^16*e^2 - 5*a^3*c^6*d^14*e^4 + 9*a^4*c^5*d^12*e^6 -
5*a^5*c^4*d^10*e^8 - 5*a^6*c^3*d^8*e^10 + 9*a^7*c^2*d^6*e^12 - 5*a^8*c*d^4*e^14
+ a^9*d^2*e^16)*x)

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Fricas [A]  time = 0.312259, size = 2184, normalized size = 8.34 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-4),x, algorithm="fricas")

[Out]

-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 + 150*(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 + 18*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^10*d^18*e^2 - 6*a*c^9*d^16*e^4 + 14*a^2*c^8*
d^14*e^6 - 14*a^3*c^7*d^12*e^8 + 14*a^5*c^5*d^8*e^12 - 14*a^6*c^4*d^6*e^14 + 6*a
^7*c^3*d^4*e^16 - a^8*c^2*d^2*e^18)*x^5 + 3*(c^10*d^19*e - 4*a*c^9*d^17*e^3 + a^
2*c^8*d^15*e^5 + 21*a^3*c^7*d^13*e^7 - 49*a^4*c^6*d^11*e^9 + 49*a^5*c^5*d^9*e^11
- 21*a^6*c^4*d^7*e^13 - a^7*c^3*d^5*e^15 + 4*a^8*c^2*d^3*e^17 - a^9*c*d*e^19)*x
^4 + (c^10*d^20 + 2*a*c^9*d^18*e^2 - 33*a^2*c^8*d^16*e^4 + 92*a^3*c^7*d^14*e^6 -
98*a^4*c^6*d^12*e^8 + 98*a^6*c^4*d^8*e^12 - 92*a^7*c^3*d^6*e^14 + 33*a^8*c^2*d^
4*e^16 - 2*a^9*c*d^2*e^18 - a^10*e^20)*x^3 + 3*(a*c^9*d^19*e - 4*a^2*c^8*d^17*e^
3 + a^3*c^7*d^15*e^5 + 21*a^4*c^6*d^13*e^7 - 49*a^5*c^5*d^11*e^9 + 49*a^6*c^4*d^
9*e^11 - 21*a^7*c^3*d^7*e^13 - a^8*c^2*d^5*e^15 + 4*a^9*c*d^3*e^17 - a^10*d*e^19
)*x^2 + 3*(a^2*c^8*d^18*e^2 - 6*a^3*c^7*d^16*e^4 + 14*a^4*c^6*d^14*e^6 - 14*a^5*
c^5*d^12*e^8 + 14*a^7*c^3*d^8*e^12 - 14*a^8*c^2*d^6*e^14 + 6*a^9*c*d^4*e^16 - a^
10*d^2*e^18)*x)

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Sympy [A]  time = 31.5122, size = 1742, normalized size = 6.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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 - 1400*a**4*c**7*d**1
1*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 + 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 + 2
0*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 + 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 + 1400*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 - 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)/(4
0*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 + 150*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 + 42*a**3*c**2*d**3*e**7 + 222*a**2*c**3*d**5
*e**5 + 42*a*c**4*d**7*e**3 - 3*c**5*d**9*e))/(3*a**9*d**3*e**15 - 18*a**8*c*d**
5*e**13 + 45*a**7*c**2*d**7*e**11 - 60*a**6*c**3*d**9*e**9 + 45*a**5*c**4*d**11*
e**7 - 18*a**4*c**5*d**13*e**5 + 3*a**3*c**6*d**15*e**3 + x**6*(3*a**6*c**3*d**3
*e**15 - 18*a**5*c**4*d**5*e**13 + 45*a**4*c**5*d**7*e**11 - 60*a**3*c**6*d**9*e
**9 + 45*a**2*c**7*d**11*e**7 - 18*a*c**8*d**13*e**5 + 3*c**9*d**15*e**3) + x**5
*(9*a**7*c**2*d**2*e**16 - 45*a**6*c**3*d**4*e**14 + 81*a**5*c**4*d**6*e**12 - 4
5*a**4*c**5*d**8*e**10 - 45*a**3*c**6*d**10*e**8 + 81*a**2*c**7*d**12*e**6 - 45*
a*c**8*d**14*e**4 + 9*c**9*d**16*e**2) + x**4*(9*a**8*c*d*e**17 - 27*a**7*c**2*d
**3*e**15 - 18*a**6*c**3*d**5*e**13 + 171*a**5*c**4*d**7*e**11 - 270*a**4*c**5*d
**9*e**9 + 171*a**3*c**6*d**11*e**7 - 18*a**2*c**7*d**13*e**5 - 27*a*c**8*d**15*
e**3 + 9*c**9*d**17*e) + x**3*(3*a**9*e**18 + 9*a**8*c*d**2*e**16 - 90*a**7*c**2
*d**4*e**14 + 186*a**6*c**3*d**6*e**12 - 108*a**5*c**4*d**8*e**10 - 108*a**4*c**
5*d**10*e**8 + 186*a**3*c**6*d**12*e**6 - 90*a**2*c**7*d**14*e**4 + 9*a*c**8*d**
16*e**2 + 3*c**9*d**18) + x**2*(9*a**9*d*e**17 - 27*a**8*c*d**3*e**15 - 18*a**7*
c**2*d**5*e**13 + 171*a**6*c**3*d**7*e**11 - 270*a**5*c**4*d**9*e**9 + 171*a**4*
c**5*d**11*e**7 - 18*a**3*c**6*d**13*e**5 - 27*a**2*c**7*d**15*e**3 + 9*a*c**8*d
**17*e) + x*(9*a**9*d**2*e**16 - 45*a**8*c*d**4*e**14 + 81*a**7*c**2*d**6*e**12
- 45*a**6*c**3*d**8*e**10 - 45*a**5*c**4*d**10*e**8 + 81*a**4*c**5*d**12*e**6 -
45*a**3*c**6*d**14*e**4 + 9*a**2*c**7*d**16*e**2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218238, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-4),x, algorithm="giac")

[Out]

Done