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

Optimal. Leaf size=295 \[ \frac{6 a^2 e^3+c d x \left (11 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac{8 a^3 e^5+c d x \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )}{16 a^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac{\sqrt{c} d \left (35 a^3 e^6+35 a^2 c d^2 e^4+21 a c^2 d^4 e^2+5 c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^4}+\frac{a e+c d x}{6 a \left (a+c x^2\right )^3 \left (a e^2+c d^2\right )}-\frac{e^7 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^4}+\frac{e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]

[Out]

(a*e + c*d*x)/(6*a*(c*d^2 + a*e^2)*(a + c*x^2)^3) + (6*a^2*e^3 + c*d*(5*c*d^2 +
11*a*e^2)*x)/(24*a^2*(c*d^2 + a*e^2)^2*(a + c*x^2)^2) + (8*a^3*e^5 + c*d*(5*c^2*
d^4 + 16*a*c*d^2*e^2 + 19*a^2*e^4)*x)/(16*a^3*(c*d^2 + a*e^2)^3*(a + c*x^2)) + (
Sqrt[c]*d*(5*c^3*d^6 + 21*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 35*a^3*e^6)*ArcTan[
(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*(c*d^2 + a*e^2)^4) + (e^7*Log[d + e*x])/(c*d^2
+ a*e^2)^4 - (e^7*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^4)

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Rubi [A]  time = 0.97338, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{6 a^2 e^3+c d x \left (11 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac{8 a^3 e^5+c d x \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )}{16 a^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac{\sqrt{c} d \left (35 a^3 e^6+35 a^2 c d^2 e^4+21 a c^2 d^4 e^2+5 c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^4}+\frac{a e+c d x}{6 a \left (a+c x^2\right )^3 \left (a e^2+c d^2\right )}-\frac{e^7 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^4}+\frac{e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

(a*e + c*d*x)/(6*a*(c*d^2 + a*e^2)*(a + c*x^2)^3) + (6*a^2*e^3 + c*d*(5*c*d^2 +
11*a*e^2)*x)/(24*a^2*(c*d^2 + a*e^2)^2*(a + c*x^2)^2) + (8*a^3*e^5 + c*d*(5*c^2*
d^4 + 16*a*c*d^2*e^2 + 19*a^2*e^4)*x)/(16*a^3*(c*d^2 + a*e^2)^3*(a + c*x^2)) + (
Sqrt[c]*d*(5*c^3*d^6 + 21*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 35*a^3*e^6)*ArcTan[
(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*(c*d^2 + a*e^2)^4) + (e^7*Log[d + e*x])/(c*d^2
+ a*e^2)^4 - (e^7*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^4)

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Rubi in Sympy [A]  time = 162.533, size = 280, normalized size = 0.95 \[ - \frac{e^{7} \log{\left (a + c x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{4}} + \frac{e^{7} \log{\left (d + e x \right )}}{\left (a e^{2} + c d^{2}\right )^{4}} + \frac{a e + c d x}{6 a \left (a + c x^{2}\right )^{3} \left (a e^{2} + c d^{2}\right )} + \frac{6 a^{2} e^{3} + c d x \left (11 a e^{2} + 5 c d^{2}\right )}{24 a^{2} \left (a + c x^{2}\right )^{2} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{24 a^{3} e^{5} + 3 c d x \left (19 a^{2} e^{4} + 16 a c d^{2} e^{2} + 5 c^{2} d^{4}\right )}{48 a^{3} \left (a + c x^{2}\right ) \left (a e^{2} + c d^{2}\right )^{3}} + \frac{\sqrt{c} d \left (35 a^{3} e^{6} + 35 a^{2} c d^{2} e^{4} + 21 a c^{2} d^{4} e^{2} + 5 c^{3} d^{6}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 a^{\frac{7}{2}} \left (a e^{2} + c d^{2}\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**7*log(a + c*x**2)/(2*(a*e**2 + c*d**2)**4) + e**7*log(d + e*x)/(a*e**2 + c*d
**2)**4 + (a*e + c*d*x)/(6*a*(a + c*x**2)**3*(a*e**2 + c*d**2)) + (6*a**2*e**3 +
c*d*x*(11*a*e**2 + 5*c*d**2))/(24*a**2*(a + c*x**2)**2*(a*e**2 + c*d**2)**2) +
(24*a**3*e**5 + 3*c*d*x*(19*a**2*e**4 + 16*a*c*d**2*e**2 + 5*c**2*d**4))/(48*a**
3*(a + c*x**2)*(a*e**2 + c*d**2)**3) + sqrt(c)*d*(35*a**3*e**6 + 35*a**2*c*d**2*
e**4 + 21*a*c**2*d**4*e**2 + 5*c**3*d**6)*atan(sqrt(c)*x/sqrt(a))/(16*a**(7/2)*(
a*e**2 + c*d**2)**4)

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Mathematica [A]  time = 0.4252, size = 265, normalized size = 0.9 \[ \frac{\frac{2 \left (a e^2+c d^2\right )^2 \left (6 a^2 e^3+11 a c d e^2 x+5 c^2 d^3 x\right )}{a^2 \left (a+c x^2\right )^2}+\frac{3 \left (a e^2+c d^2\right ) \left (8 a^3 e^5+19 a^2 c d e^4 x+16 a c^2 d^3 e^2 x+5 c^3 d^5 x\right )}{a^3 \left (a+c x^2\right )}+\frac{3 \sqrt{c} d \left (35 a^3 e^6+35 a^2 c d^2 e^4+21 a c^2 d^4 e^2+5 c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{7/2}}+\frac{8 \left (a e^2+c d^2\right )^3 (a e+c d x)}{a \left (a+c x^2\right )^3}-24 e^7 \log \left (a+c x^2\right )+48 e^7 \log (d+e x)}{48 \left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

((8*(c*d^2 + a*e^2)^3*(a*e + c*d*x))/(a*(a + c*x^2)^3) + (2*(c*d^2 + a*e^2)^2*(6
*a^2*e^3 + 5*c^2*d^3*x + 11*a*c*d*e^2*x))/(a^2*(a + c*x^2)^2) + (3*(c*d^2 + a*e^
2)*(8*a^3*e^5 + 5*c^3*d^5*x + 16*a*c^2*d^3*e^2*x + 19*a^2*c*d*e^4*x))/(a^3*(a +
c*x^2)) + (3*Sqrt[c]*d*(5*c^3*d^6 + 21*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 35*a^3
*e^6)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(7/2) + 48*e^7*Log[d + e*x] - 24*e^7*Log[a
+ c*x^2])/(48*(c*d^2 + a*e^2)^4)

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Maple [B]  time = 0.03, size = 945, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+a)^4,x)

[Out]

1/2*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^3*x^4*a*e^7+35/16*c/(a*e^2+c*d^2)^4/(a*c)^(1/2
)*arctan(c*x/(a*c)^(1/2))*d*e^6+5/16*c^4/(a*e^2+c*d^2)^4/a^3/(a*c)^(1/2)*arctan(
c*x/(a*c)^(1/2))*d^7+1/2*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^3*x^4*d^2*e^5+11/2*c^3/(a
*e^2+c*d^2)^4/(c*x^2+a)^3*d^3*x^3*e^4+5/6*c^5/(a*e^2+c*d^2)^4/(c*x^2+a)^3*d^7/a^
2*x^3+1/4*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^3*x^2*d^4*e^3+43/16*c^3/(a*e^2+c*d^2)^4/
(c*x^2+a)^3*d^5*x*e^2+11/16*c^4/(a*e^2+c*d^2)^4/(c*x^2+a)^3*d^7/a*x+3/4*c^2/(a*e
^2+c*d^2)^4/(c*x^2+a)^3*e^3*d^4*a+5/4*c/(a*e^2+c*d^2)^4/(c*x^2+a)^3*x^2*a^2*e^7+
3/2*c/(a*e^2+c*d^2)^4/(c*x^2+a)^3*a^2*d^2*e^5+e^7*ln(e*x+d)/(a*e^2+c*d^2)^4+21/1
6*c^3/(a*e^2+c*d^2)^4/a^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^5*e^2+3/2*c^2/(a
*e^2+c*d^2)^4/(c*x^2+a)^3*x^2*a*d^2*e^5+61/16*c^2/(a*e^2+c*d^2)^4/(c*x^2+a)^3*d^
3*a*x*e^4+29/16*c/(a*e^2+c*d^2)^4/(c*x^2+a)^3*d*a^2*x*e^6+19/16*c^3/(a*e^2+c*d^2
)^4/(c*x^2+a)^3*d*x^5*e^6+5/16*c^6/(a*e^2+c*d^2)^4/(c*x^2+a)^3*d^7/a^3*x^5+21/16
*c^5/(a*e^2+c*d^2)^4/(c*x^2+a)^3*d^5/a^2*x^5*e^2+17/6*c^2/(a*e^2+c*d^2)^4/(c*x^2
+a)^3*d*a*x^3*e^6+1/6*c^3/(a*e^2+c*d^2)^4/(c*x^2+a)^3*e*d^6+11/12/(a*e^2+c*d^2)^
4/(c*x^2+a)^3*e^7*a^3-1/2/(a*e^2+c*d^2)^4*e^7*ln(a^3*(c*x^2+a))+35/16*c^2/(a*e^2
+c*d^2)^4/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^3*e^4+35/16*c^4/(a*e^2+c*d^2)^
4/(c*x^2+a)^3*d^3/a*x^5*e^4+7/2*c^4/(a*e^2+c*d^2)^4/(c*x^2+a)^3*d^5/a*x^3*e^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 17.8034, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(16*a^3*c^3*d^6*e + 72*a^4*c^2*d^4*e^3 + 144*a^5*c*d^2*e^5 + 88*a^6*e^7 +
6*(5*c^6*d^7 + 21*a*c^5*d^5*e^2 + 35*a^2*c^4*d^3*e^4 + 19*a^3*c^3*d*e^6)*x^5 + 4
8*(a^3*c^3*d^2*e^5 + a^4*c^2*e^7)*x^4 + 16*(5*a*c^5*d^7 + 21*a^2*c^4*d^5*e^2 + 3
3*a^3*c^3*d^3*e^4 + 17*a^4*c^2*d*e^6)*x^3 + 24*(a^3*c^3*d^4*e^3 + 6*a^4*c^2*d^2*
e^5 + 5*a^5*c*e^7)*x^2 + 3*(5*a^3*c^3*d^7 + 21*a^4*c^2*d^5*e^2 + 35*a^5*c*d^3*e^
4 + 35*a^6*d*e^6 + (5*c^6*d^7 + 21*a*c^5*d^5*e^2 + 35*a^2*c^4*d^3*e^4 + 35*a^3*c
^3*d*e^6)*x^6 + 3*(5*a*c^5*d^7 + 21*a^2*c^4*d^5*e^2 + 35*a^3*c^3*d^3*e^4 + 35*a^
4*c^2*d*e^6)*x^4 + 3*(5*a^2*c^4*d^7 + 21*a^3*c^3*d^5*e^2 + 35*a^4*c^2*d^3*e^4 +
35*a^5*c*d*e^6)*x^2)*sqrt(-c/a)*log((c*x^2 + 2*a*x*sqrt(-c/a) - a)/(c*x^2 + a))
+ 6*(11*a^2*c^4*d^7 + 43*a^3*c^3*d^5*e^2 + 61*a^4*c^2*d^3*e^4 + 29*a^5*c*d*e^6)*
x - 48*(a^3*c^3*e^7*x^6 + 3*a^4*c^2*e^7*x^4 + 3*a^5*c*e^7*x^2 + a^6*e^7)*log(c*x
^2 + a) + 96*(a^3*c^3*e^7*x^6 + 3*a^4*c^2*e^7*x^4 + 3*a^5*c*e^7*x^2 + a^6*e^7)*l
og(e*x + d))/(a^6*c^4*d^8 + 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4 + 4*a^9*c*d^2*
e^6 + a^10*e^8 + (a^3*c^7*d^8 + 4*a^4*c^6*d^6*e^2 + 6*a^5*c^5*d^4*e^4 + 4*a^6*c^
4*d^2*e^6 + a^7*c^3*e^8)*x^6 + 3*(a^4*c^6*d^8 + 4*a^5*c^5*d^6*e^2 + 6*a^6*c^4*d^
4*e^4 + 4*a^7*c^3*d^2*e^6 + a^8*c^2*e^8)*x^4 + 3*(a^5*c^5*d^8 + 4*a^6*c^4*d^6*e^
2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^6 + a^9*c*e^8)*x^2), 1/48*(8*a^3*c^3*d^6
*e + 36*a^4*c^2*d^4*e^3 + 72*a^5*c*d^2*e^5 + 44*a^6*e^7 + 3*(5*c^6*d^7 + 21*a*c^
5*d^5*e^2 + 35*a^2*c^4*d^3*e^4 + 19*a^3*c^3*d*e^6)*x^5 + 24*(a^3*c^3*d^2*e^5 + a
^4*c^2*e^7)*x^4 + 8*(5*a*c^5*d^7 + 21*a^2*c^4*d^5*e^2 + 33*a^3*c^3*d^3*e^4 + 17*
a^4*c^2*d*e^6)*x^3 + 12*(a^3*c^3*d^4*e^3 + 6*a^4*c^2*d^2*e^5 + 5*a^5*c*e^7)*x^2
+ 3*(5*a^3*c^3*d^7 + 21*a^4*c^2*d^5*e^2 + 35*a^5*c*d^3*e^4 + 35*a^6*d*e^6 + (5*c
^6*d^7 + 21*a*c^5*d^5*e^2 + 35*a^2*c^4*d^3*e^4 + 35*a^3*c^3*d*e^6)*x^6 + 3*(5*a*
c^5*d^7 + 21*a^2*c^4*d^5*e^2 + 35*a^3*c^3*d^3*e^4 + 35*a^4*c^2*d*e^6)*x^4 + 3*(5
*a^2*c^4*d^7 + 21*a^3*c^3*d^5*e^2 + 35*a^4*c^2*d^3*e^4 + 35*a^5*c*d*e^6)*x^2)*sq
rt(c/a)*arctan(c*x/(a*sqrt(c/a))) + 3*(11*a^2*c^4*d^7 + 43*a^3*c^3*d^5*e^2 + 61*
a^4*c^2*d^3*e^4 + 29*a^5*c*d*e^6)*x - 24*(a^3*c^3*e^7*x^6 + 3*a^4*c^2*e^7*x^4 +
3*a^5*c*e^7*x^2 + a^6*e^7)*log(c*x^2 + a) + 48*(a^3*c^3*e^7*x^6 + 3*a^4*c^2*e^7*
x^4 + 3*a^5*c*e^7*x^2 + a^6*e^7)*log(e*x + d))/(a^6*c^4*d^8 + 4*a^7*c^3*d^6*e^2
+ 6*a^8*c^2*d^4*e^4 + 4*a^9*c*d^2*e^6 + a^10*e^8 + (a^3*c^7*d^8 + 4*a^4*c^6*d^6*
e^2 + 6*a^5*c^5*d^4*e^4 + 4*a^6*c^4*d^2*e^6 + a^7*c^3*e^8)*x^6 + 3*(a^4*c^6*d^8
+ 4*a^5*c^5*d^6*e^2 + 6*a^6*c^4*d^4*e^4 + 4*a^7*c^3*d^2*e^6 + a^8*c^2*e^8)*x^4 +
3*(a^5*c^5*d^8 + 4*a^6*c^4*d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^6 + a^
9*c*e^8)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216269, size = 716, normalized size = 2.43 \[ -\frac{e^{7}{\rm ln}\left (c x^{2} + a\right )}{2 \,{\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} + \frac{e^{8}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e + 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} + 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} + \frac{{\left (5 \, c^{4} d^{7} + 21 \, a c^{3} d^{5} e^{2} + 35 \, a^{2} c^{2} d^{3} e^{4} + 35 \, a^{3} c d e^{6}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \,{\left (a^{3} c^{4} d^{8} + 4 \, a^{4} c^{3} d^{6} e^{2} + 6 \, a^{5} c^{2} d^{4} e^{4} + 4 \, a^{6} c d^{2} e^{6} + a^{7} e^{8}\right )} \sqrt{a c}} + \frac{8 \, a^{3} c^{3} d^{6} e + 36 \, a^{4} c^{2} d^{4} e^{3} + 72 \, a^{5} c d^{2} e^{5} + 44 \, a^{6} e^{7} + 3 \,{\left (5 \, c^{6} d^{7} + 21 \, a c^{5} d^{5} e^{2} + 35 \, a^{2} c^{4} d^{3} e^{4} + 19 \, a^{3} c^{3} d e^{6}\right )} x^{5} + 24 \,{\left (a^{3} c^{3} d^{2} e^{5} + a^{4} c^{2} e^{7}\right )} x^{4} + 8 \,{\left (5 \, a c^{5} d^{7} + 21 \, a^{2} c^{4} d^{5} e^{2} + 33 \, a^{3} c^{3} d^{3} e^{4} + 17 \, a^{4} c^{2} d e^{6}\right )} x^{3} + 12 \,{\left (a^{3} c^{3} d^{4} e^{3} + 6 \, a^{4} c^{2} d^{2} e^{5} + 5 \, a^{5} c e^{7}\right )} x^{2} + 3 \,{\left (11 \, a^{2} c^{4} d^{7} + 43 \, a^{3} c^{3} d^{5} e^{2} + 61 \, a^{4} c^{2} d^{3} e^{4} + 29 \, a^{5} c d e^{6}\right )} x}{48 \,{\left (c d^{2} + a e^{2}\right )}^{4}{\left (c x^{2} + a\right )}^{3} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*e^7*ln(c*x^2 + a)/(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) + e^8*ln(abs(x*e + d))/(c^4*d^8*e + 4*a*c^3*d^6*e^3 + 6*a^2*c
^2*d^4*e^5 + 4*a^3*c*d^2*e^7 + a^4*e^9) + 1/16*(5*c^4*d^7 + 21*a*c^3*d^5*e^2 + 3
5*a^2*c^2*d^3*e^4 + 35*a^3*c*d*e^6)*arctan(c*x/sqrt(a*c))/((a^3*c^4*d^8 + 4*a^4*
c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4 + 4*a^6*c*d^2*e^6 + a^7*e^8)*sqrt(a*c)) + 1/48*(
8*a^3*c^3*d^6*e + 36*a^4*c^2*d^4*e^3 + 72*a^5*c*d^2*e^5 + 44*a^6*e^7 + 3*(5*c^6*
d^7 + 21*a*c^5*d^5*e^2 + 35*a^2*c^4*d^3*e^4 + 19*a^3*c^3*d*e^6)*x^5 + 24*(a^3*c^
3*d^2*e^5 + a^4*c^2*e^7)*x^4 + 8*(5*a*c^5*d^7 + 21*a^2*c^4*d^5*e^2 + 33*a^3*c^3*
d^3*e^4 + 17*a^4*c^2*d*e^6)*x^3 + 12*(a^3*c^3*d^4*e^3 + 6*a^4*c^2*d^2*e^5 + 5*a^
5*c*e^7)*x^2 + 3*(11*a^2*c^4*d^7 + 43*a^3*c^3*d^5*e^2 + 61*a^4*c^2*d^3*e^4 + 29*
a^5*c*d*e^6)*x)/((c*d^2 + a*e^2)^4*(c*x^2 + a)^3*a^3)