3.264 \(\int \frac{\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^5} \, dx\)

Optimal. Leaf size=317 \[ -\frac{x \left (-e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )-2 c d^2 e (3 a e+5 b d)+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )+2 c d^2 e (3 a e+5 b d)+35 c^2 d^4\right )}{128 d^{9/2} e^{9/2}}+\frac{x \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )-2 c d^2 e (59 b d-3 a e)+163 c^2 d^4\right )}{192 d^3 e^4 \left (d+e x^2\right )^2}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4}-\frac{x \left (-7 a e^2-9 b d e+25 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{48 d^2 e^4 \left (d+e x^2\right )^3} \]

[Out]

((c*d^2 - b*d*e + a*e^2)^2*x)/(8*d*e^4*(d + e*x^2)^4) - ((25*c*d^2 - 9*b*d*e - 7
*a*e^2)*(c*d^2 - b*d*e + a*e^2)*x)/(48*d^2*e^4*(d + e*x^2)^3) + ((163*c^2*d^4 -
2*c*d^2*e*(59*b*d - 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(192*
d^3*e^4*(d + e*x^2)^2) - ((93*c^2*d^4 - 2*c*d^2*e*(5*b*d + 3*a*e) - e^2*(3*b^2*d
^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(128*d^4*e^4*(d + e*x^2)) + ((35*c^2*d^4 + 2*c
*d^2*e*(5*b*d + 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*ArcTan[(Sqrt
[e]*x)/Sqrt[d]])/(128*d^(9/2)*e^(9/2))

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Rubi [A]  time = 1.23902, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{x \left (-e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )-2 c d^2 e (3 a e+5 b d)+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )+2 c d^2 e (3 a e+5 b d)+35 c^2 d^4\right )}{128 d^{9/2} e^{9/2}}+\frac{x \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )-2 c d^2 e (59 b d-3 a e)+163 c^2 d^4\right )}{192 d^3 e^4 \left (d+e x^2\right )^2}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4}-\frac{x \left (-7 a e^2-9 b d e+25 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{48 d^2 e^4 \left (d+e x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^2/(d + e*x^2)^5,x]

[Out]

((c*d^2 - b*d*e + a*e^2)^2*x)/(8*d*e^4*(d + e*x^2)^4) - ((25*c*d^2 - 9*b*d*e - 7
*a*e^2)*(c*d^2 - b*d*e + a*e^2)*x)/(48*d^2*e^4*(d + e*x^2)^3) + ((163*c^2*d^4 -
2*c*d^2*e*(59*b*d - 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(192*
d^3*e^4*(d + e*x^2)^2) - ((93*c^2*d^4 - 2*c*d^2*e*(5*b*d + 3*a*e) - e^2*(3*b^2*d
^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(128*d^4*e^4*(d + e*x^2)) + ((35*c^2*d^4 + 2*c
*d^2*e*(5*b*d + 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*ArcTan[(Sqrt
[e]*x)/Sqrt[d]])/(128*d^(9/2)*e^(9/2))

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Rubi in Sympy [A]  time = 173.066, size = 415, normalized size = 1.31 \[ - \frac{c^{2} x^{7}}{e \left (d + e x^{2}\right )^{4}} + \frac{x \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e - 7 c^{2} d^{4}\right )}{8 d e^{4} \left (d + e x^{2}\right )^{4}} + \frac{x \left (7 a^{2} e^{4} + 2 a b d e^{3} - 18 a c d^{2} e^{2} - 9 b^{2} d^{2} e^{2} + 34 b c d^{3} e + 119 c^{2} d^{4}\right )}{48 d^{2} e^{4} \left (d + e x^{2}\right )^{3}} + \frac{x \left (35 a^{2} e^{4} + 10 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} - 118 b c d^{3} e - 413 c^{2} d^{4}\right )}{192 d^{3} e^{4} \left (d + e x^{2}\right )^{2}} + \frac{x \left (35 a^{2} e^{4} + 10 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} + 10 b c d^{3} e + 35 c^{2} d^{4}\right )}{128 d^{4} e^{4} \left (d + e x^{2}\right )} + \frac{\left (35 a^{2} e^{4} + 10 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} + 10 b c d^{3} e + 35 c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{128 d^{\frac{9}{2}} e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**2/(e*x**2+d)**5,x)

[Out]

-c**2*x**7/(e*(d + e*x**2)**4) + x*(a**2*e**4 - 2*a*b*d*e**3 + 2*a*c*d**2*e**2 +
b**2*d**2*e**2 - 2*b*c*d**3*e - 7*c**2*d**4)/(8*d*e**4*(d + e*x**2)**4) + x*(7*
a**2*e**4 + 2*a*b*d*e**3 - 18*a*c*d**2*e**2 - 9*b**2*d**2*e**2 + 34*b*c*d**3*e +
119*c**2*d**4)/(48*d**2*e**4*(d + e*x**2)**3) + x*(35*a**2*e**4 + 10*a*b*d*e**3
+ 6*a*c*d**2*e**2 + 3*b**2*d**2*e**2 - 118*b*c*d**3*e - 413*c**2*d**4)/(192*d**
3*e**4*(d + e*x**2)**2) + x*(35*a**2*e**4 + 10*a*b*d*e**3 + 6*a*c*d**2*e**2 + 3*
b**2*d**2*e**2 + 10*b*c*d**3*e + 35*c**2*d**4)/(128*d**4*e**4*(d + e*x**2)) + (3
5*a**2*e**4 + 10*a*b*d*e**3 + 6*a*c*d**2*e**2 + 3*b**2*d**2*e**2 + 10*b*c*d**3*e
+ 35*c**2*d**4)*atan(sqrt(e)*x/sqrt(d))/(128*d**(9/2)*e**(9/2))

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Mathematica [A]  time = 0.441256, size = 345, normalized size = 1.09 \[ \frac{-\frac{3 \sqrt{d} \sqrt{e} x \left (-e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )-2 c d^2 e (3 a e+5 b d)+93 c^2 d^4\right )}{d+e x^2}+3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )+2 c d^2 e (3 a e+5 b d)+35 c^2 d^4\right )-\frac{8 d^{5/2} \sqrt{e} x \left (e^2 \left (-7 a^2 e^2-2 a b d e+9 b^2 d^2\right )+2 c d^2 e (9 a e-17 b d)+25 c^2 d^4\right )}{\left (d+e x^2\right )^3}+\frac{2 d^{3/2} \sqrt{e} x \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )+2 c d^2 e (3 a e-59 b d)+163 c^2 d^4\right )}{\left (d+e x^2\right )^2}+\frac{48 d^{7/2} \sqrt{e} x \left (e (a e-b d)+c d^2\right )^2}{\left (d+e x^2\right )^4}}{384 d^{9/2} e^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x^2)^5,x]

[Out]

((48*d^(7/2)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^2*x)/(d + e*x^2)^4 - (8*d^(5/2)*
Sqrt[e]*(25*c^2*d^4 + 2*c*d^2*e*(-17*b*d + 9*a*e) + e^2*(9*b^2*d^2 - 2*a*b*d*e -
7*a^2*e^2))*x)/(d + e*x^2)^3 + (2*d^(3/2)*Sqrt[e]*(163*c^2*d^4 + 2*c*d^2*e*(-59
*b*d + 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(d + e*x^2)^2 - (3
*Sqrt[d]*Sqrt[e]*(93*c^2*d^4 - 2*c*d^2*e*(5*b*d + 3*a*e) - e^2*(3*b^2*d^2 + 10*a
*b*d*e + 35*a^2*e^2))*x)/(d + e*x^2) + 3*(35*c^2*d^4 + 2*c*d^2*e*(5*b*d + 3*a*e)
+ e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(384*
d^(9/2)*e^(9/2))

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Maple [A]  time = 0.016, size = 412, normalized size = 1.3 \[{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{4}} \left ({\frac{ \left ( 35\,{a}^{2}{e}^{4}+10\,dab{e}^{3}+6\,ac{d}^{2}{e}^{2}+3\,{b}^{2}{d}^{2}{e}^{2}+10\,bc{d}^{3}e-93\,{c}^{2}{d}^{4} \right ){x}^{7}}{128\,{d}^{4}e}}+{\frac{ \left ( 385\,{a}^{2}{e}^{4}+110\,dab{e}^{3}+66\,ac{d}^{2}{e}^{2}+33\,{b}^{2}{d}^{2}{e}^{2}-146\,bc{d}^{3}e-511\,{c}^{2}{d}^{4} \right ){x}^{5}}{384\,{d}^{3}{e}^{2}}}+{\frac{ \left ( 511\,{a}^{2}{e}^{4}+146\,dab{e}^{3}-66\,ac{d}^{2}{e}^{2}-33\,{b}^{2}{d}^{2}{e}^{2}-110\,bc{d}^{3}e-385\,{c}^{2}{d}^{4} \right ){x}^{3}}{384\,{d}^{2}{e}^{3}}}+{\frac{ \left ( 93\,{a}^{2}{e}^{4}-10\,dab{e}^{3}-6\,ac{d}^{2}{e}^{2}-3\,{b}^{2}{d}^{2}{e}^{2}-10\,bc{d}^{3}e-35\,{c}^{2}{d}^{4} \right ) x}{128\,d{e}^{4}}} \right ) }+{\frac{35\,{a}^{2}}{128\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,ab}{64\,{d}^{3}e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,ac}{64\,{d}^{2}{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,{b}^{2}}{128\,{d}^{2}{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,bc}{64\,d{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{35\,{c}^{2}}{128\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^2/(e*x^2+d)^5,x)

[Out]

(1/128*(35*a^2*e^4+10*a*b*d*e^3+6*a*c*d^2*e^2+3*b^2*d^2*e^2+10*b*c*d^3*e-93*c^2*
d^4)/d^4/e*x^7+1/384*(385*a^2*e^4+110*a*b*d*e^3+66*a*c*d^2*e^2+33*b^2*d^2*e^2-14
6*b*c*d^3*e-511*c^2*d^4)/d^3/e^2*x^5+1/384*(511*a^2*e^4+146*a*b*d*e^3-66*a*c*d^2
*e^2-33*b^2*d^2*e^2-110*b*c*d^3*e-385*c^2*d^4)/d^2/e^3*x^3+1/128*(93*a^2*e^4-10*
a*b*d*e^3-6*a*c*d^2*e^2-3*b^2*d^2*e^2-10*b*c*d^3*e-35*c^2*d^4)/d/e^4*x)/(e*x^2+d
)^4+35/128/d^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a^2+5/64/d^3/e/(d*e)^(1/2)*ar
ctan(x*e/(d*e)^(1/2))*a*b+3/64/d^2/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a*c+3
/128/d^2/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b^2+5/64/d/e^3/(d*e)^(1/2)*arct
an(x*e/(d*e)^(1/2))*b*c+35/128/e^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*c^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((c*x^4 + b*x^2 + a)^2/(e*x^2 + d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280959, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/(e*x^2 + d)^5,x, algorithm="fricas")

[Out]

[1/768*(3*(35*c^2*d^8 + 10*b*c*d^7*e + 10*a*b*d^5*e^3 + 35*a^2*d^4*e^4 + 3*(b^2
+ 2*a*c)*d^6*e^2 + (35*c^2*d^4*e^4 + 10*b*c*d^3*e^5 + 10*a*b*d*e^7 + 35*a^2*e^8
+ 3*(b^2 + 2*a*c)*d^2*e^6)*x^8 + 4*(35*c^2*d^5*e^3 + 10*b*c*d^4*e^4 + 10*a*b*d^2
*e^6 + 35*a^2*d*e^7 + 3*(b^2 + 2*a*c)*d^3*e^5)*x^6 + 6*(35*c^2*d^6*e^2 + 10*b*c*
d^5*e^3 + 10*a*b*d^3*e^5 + 35*a^2*d^2*e^6 + 3*(b^2 + 2*a*c)*d^4*e^4)*x^4 + 4*(35
*c^2*d^7*e + 10*b*c*d^6*e^2 + 10*a*b*d^4*e^4 + 35*a^2*d^3*e^5 + 3*(b^2 + 2*a*c)*
d^5*e^3)*x^2)*log((2*d*e*x + (e*x^2 - d)*sqrt(-d*e))/(e*x^2 + d)) - 2*(3*(93*c^2
*d^4*e^3 - 10*b*c*d^3*e^4 - 10*a*b*d*e^6 - 35*a^2*e^7 - 3*(b^2 + 2*a*c)*d^2*e^5)
*x^7 + (511*c^2*d^5*e^2 + 146*b*c*d^4*e^3 - 110*a*b*d^2*e^5 - 385*a^2*d*e^6 - 33
*(b^2 + 2*a*c)*d^3*e^4)*x^5 + (385*c^2*d^6*e + 110*b*c*d^5*e^2 - 146*a*b*d^3*e^4
- 511*a^2*d^2*e^5 + 33*(b^2 + 2*a*c)*d^4*e^3)*x^3 + 3*(35*c^2*d^7 + 10*b*c*d^6*
e + 10*a*b*d^4*e^3 - 93*a^2*d^3*e^4 + 3*(b^2 + 2*a*c)*d^5*e^2)*x)*sqrt(-d*e))/((
d^4*e^8*x^8 + 4*d^5*e^7*x^6 + 6*d^6*e^6*x^4 + 4*d^7*e^5*x^2 + d^8*e^4)*sqrt(-d*e
)), 1/384*(3*(35*c^2*d^8 + 10*b*c*d^7*e + 10*a*b*d^5*e^3 + 35*a^2*d^4*e^4 + 3*(b
^2 + 2*a*c)*d^6*e^2 + (35*c^2*d^4*e^4 + 10*b*c*d^3*e^5 + 10*a*b*d*e^7 + 35*a^2*e
^8 + 3*(b^2 + 2*a*c)*d^2*e^6)*x^8 + 4*(35*c^2*d^5*e^3 + 10*b*c*d^4*e^4 + 10*a*b*
d^2*e^6 + 35*a^2*d*e^7 + 3*(b^2 + 2*a*c)*d^3*e^5)*x^6 + 6*(35*c^2*d^6*e^2 + 10*b
*c*d^5*e^3 + 10*a*b*d^3*e^5 + 35*a^2*d^2*e^6 + 3*(b^2 + 2*a*c)*d^4*e^4)*x^4 + 4*
(35*c^2*d^7*e + 10*b*c*d^6*e^2 + 10*a*b*d^4*e^4 + 35*a^2*d^3*e^5 + 3*(b^2 + 2*a*
c)*d^5*e^3)*x^2)*arctan(sqrt(d*e)*x/d) - (3*(93*c^2*d^4*e^3 - 10*b*c*d^3*e^4 - 1
0*a*b*d*e^6 - 35*a^2*e^7 - 3*(b^2 + 2*a*c)*d^2*e^5)*x^7 + (511*c^2*d^5*e^2 + 146
*b*c*d^4*e^3 - 110*a*b*d^2*e^5 - 385*a^2*d*e^6 - 33*(b^2 + 2*a*c)*d^3*e^4)*x^5 +
(385*c^2*d^6*e + 110*b*c*d^5*e^2 - 146*a*b*d^3*e^4 - 511*a^2*d^2*e^5 + 33*(b^2
+ 2*a*c)*d^4*e^3)*x^3 + 3*(35*c^2*d^7 + 10*b*c*d^6*e + 10*a*b*d^4*e^3 - 93*a^2*d
^3*e^4 + 3*(b^2 + 2*a*c)*d^5*e^2)*x)*sqrt(d*e))/((d^4*e^8*x^8 + 4*d^5*e^7*x^6 +
6*d^6*e^6*x^4 + 4*d^7*e^5*x^2 + d^8*e^4)*sqrt(d*e))]

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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((c*x**4+b*x**2+a)**2/(e*x**2+d)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.267793, size = 491, normalized size = 1.55 \[ \frac{{\left (35 \, c^{2} d^{4} + 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} + 10 \, a b d e^{3} + 35 \, a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{128 \, d^{\frac{9}{2}}} - \frac{{\left (279 \, c^{2} d^{4} x^{7} e^{3} - 30 \, b c d^{3} x^{7} e^{4} + 511 \, c^{2} d^{5} x^{5} e^{2} - 9 \, b^{2} d^{2} x^{7} e^{5} - 18 \, a c d^{2} x^{7} e^{5} + 146 \, b c d^{4} x^{5} e^{3} + 385 \, c^{2} d^{6} x^{3} e - 30 \, a b d x^{7} e^{6} - 33 \, b^{2} d^{3} x^{5} e^{4} - 66 \, a c d^{3} x^{5} e^{4} + 110 \, b c d^{5} x^{3} e^{2} + 105 \, c^{2} d^{7} x - 105 \, a^{2} x^{7} e^{7} - 110 \, a b d^{2} x^{5} e^{5} + 33 \, b^{2} d^{4} x^{3} e^{3} + 66 \, a c d^{4} x^{3} e^{3} + 30 \, b c d^{6} x e - 385 \, a^{2} d x^{5} e^{6} - 146 \, a b d^{3} x^{3} e^{4} + 9 \, b^{2} d^{5} x e^{2} + 18 \, a c d^{5} x e^{2} - 511 \, a^{2} d^{2} x^{3} e^{5} + 30 \, a b d^{4} x e^{3} - 279 \, a^{2} d^{3} x e^{4}\right )} e^{\left (-4\right )}}{384 \,{\left (x^{2} e + d\right )}^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2/(e*x^2 + d)^5,x, algorithm="giac")

[Out]

1/128*(35*c^2*d^4 + 10*b*c*d^3*e + 3*b^2*d^2*e^2 + 6*a*c*d^2*e^2 + 10*a*b*d*e^3
+ 35*a^2*e^4)*arctan(x*e^(1/2)/sqrt(d))*e^(-9/2)/d^(9/2) - 1/384*(279*c^2*d^4*x^
7*e^3 - 30*b*c*d^3*x^7*e^4 + 511*c^2*d^5*x^5*e^2 - 9*b^2*d^2*x^7*e^5 - 18*a*c*d^
2*x^7*e^5 + 146*b*c*d^4*x^5*e^3 + 385*c^2*d^6*x^3*e - 30*a*b*d*x^7*e^6 - 33*b^2*
d^3*x^5*e^4 - 66*a*c*d^3*x^5*e^4 + 110*b*c*d^5*x^3*e^2 + 105*c^2*d^7*x - 105*a^2
*x^7*e^7 - 110*a*b*d^2*x^5*e^5 + 33*b^2*d^4*x^3*e^3 + 66*a*c*d^4*x^3*e^3 + 30*b*
c*d^6*x*e - 385*a^2*d*x^5*e^6 - 146*a*b*d^3*x^3*e^4 + 9*b^2*d^5*x*e^2 + 18*a*c*d
^5*x*e^2 - 511*a^2*d^2*x^3*e^5 + 30*a*b*d^4*x*e^3 - 279*a^2*d^3*x*e^4)*e^(-4)/((
x^2*e + d)^4*d^4)