3.1514 \(\int \frac{(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=208 \[ \frac{4 e^7 (a+b x)^2 (b d-a e)}{b^9}+\frac{56 e^5 (b d-a e)^3 \log (a+b x)}{b^9}-\frac{70 e^4 (b d-a e)^4}{b^9 (a+b x)}-\frac{28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac{28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}-\frac{2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac{(b d-a e)^8}{5 b^9 (a+b x)^5}+\frac{e^8 (a+b x)^3}{3 b^9}+\frac{28 e^6 x (b d-a e)^2}{b^8} \]

[Out]

(28*e^6*(b*d - a*e)^2*x)/b^8 - (b*d - a*e)^8/(5*b^9*(a + b*x)^5) - (2*e*(b*d - a
*e)^7)/(b^9*(a + b*x)^4) - (28*e^2*(b*d - a*e)^6)/(3*b^9*(a + b*x)^3) - (28*e^3*
(b*d - a*e)^5)/(b^9*(a + b*x)^2) - (70*e^4*(b*d - a*e)^4)/(b^9*(a + b*x)) + (4*e
^7*(b*d - a*e)*(a + b*x)^2)/b^9 + (e^8*(a + b*x)^3)/(3*b^9) + (56*e^5*(b*d - a*e
)^3*Log[a + b*x])/b^9

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Rubi [A]  time = 0.667449, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{4 e^7 (a+b x)^2 (b d-a e)}{b^9}+\frac{56 e^5 (b d-a e)^3 \log (a+b x)}{b^9}-\frac{70 e^4 (b d-a e)^4}{b^9 (a+b x)}-\frac{28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac{28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}-\frac{2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac{(b d-a e)^8}{5 b^9 (a+b x)^5}+\frac{e^8 (a+b x)^3}{3 b^9}+\frac{28 e^6 x (b d-a e)^2}{b^8} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^8/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(28*e^6*(b*d - a*e)^2*x)/b^8 - (b*d - a*e)^8/(5*b^9*(a + b*x)^5) - (2*e*(b*d - a
*e)^7)/(b^9*(a + b*x)^4) - (28*e^2*(b*d - a*e)^6)/(3*b^9*(a + b*x)^3) - (28*e^3*
(b*d - a*e)^5)/(b^9*(a + b*x)^2) - (70*e^4*(b*d - a*e)^4)/(b^9*(a + b*x)) + (4*e
^7*(b*d - a*e)*(a + b*x)^2)/b^9 + (e^8*(a + b*x)^3)/(3*b^9) + (56*e^5*(b*d - a*e
)^3*Log[a + b*x])/b^9

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{64 e^{6} \left (a e - b d\right )^{2} \int \frac{7}{16}\, dx}{b^{8}} + \frac{e^{8} \left (a + b x\right )^{3}}{3 b^{9}} - \frac{4 e^{7} \left (a + b x\right )^{2} \left (a e - b d\right )}{b^{9}} - \frac{56 e^{5} \left (a e - b d\right )^{3} \log{\left (a + b x \right )}}{b^{9}} - \frac{70 e^{4} \left (a e - b d\right )^{4}}{b^{9} \left (a + b x\right )} + \frac{28 e^{3} \left (a e - b d\right )^{5}}{b^{9} \left (a + b x\right )^{2}} - \frac{28 e^{2} \left (a e - b d\right )^{6}}{3 b^{9} \left (a + b x\right )^{3}} + \frac{2 e \left (a e - b d\right )^{7}}{b^{9} \left (a + b x\right )^{4}} - \frac{\left (a e - b d\right )^{8}}{5 b^{9} \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**8/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

64*e**6*(a*e - b*d)**2*Integral(7/16, x)/b**8 + e**8*(a + b*x)**3/(3*b**9) - 4*e
**7*(a + b*x)**2*(a*e - b*d)/b**9 - 56*e**5*(a*e - b*d)**3*log(a + b*x)/b**9 - 7
0*e**4*(a*e - b*d)**4/(b**9*(a + b*x)) + 28*e**3*(a*e - b*d)**5/(b**9*(a + b*x)*
*2) - 28*e**2*(a*e - b*d)**6/(3*b**9*(a + b*x)**3) + 2*e*(a*e - b*d)**7/(b**9*(a
+ b*x)**4) - (a*e - b*d)**8/(5*b**9*(a + b*x)**5)

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Mathematica [A]  time = 0.312874, size = 195, normalized size = 0.94 \[ \frac{15 b e^6 x \left (21 a^2 e^2-48 a b d e+28 b^2 d^2\right )+15 b^2 e^7 x^2 (4 b d-3 a e)+840 e^5 (b d-a e)^3 \log (a+b x)-\frac{1050 e^4 (b d-a e)^4}{a+b x}+\frac{420 e^3 (a e-b d)^5}{(a+b x)^2}-\frac{140 e^2 (b d-a e)^6}{(a+b x)^3}+\frac{30 e (a e-b d)^7}{(a+b x)^4}-\frac{3 (b d-a e)^8}{(a+b x)^5}+5 b^3 e^8 x^3}{15 b^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^8/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(15*b*e^6*(28*b^2*d^2 - 48*a*b*d*e + 21*a^2*e^2)*x + 15*b^2*e^7*(4*b*d - 3*a*e)*
x^2 + 5*b^3*e^8*x^3 - (3*(b*d - a*e)^8)/(a + b*x)^5 + (30*e*(-(b*d) + a*e)^7)/(a
+ b*x)^4 - (140*e^2*(b*d - a*e)^6)/(a + b*x)^3 + (420*e^3*(-(b*d) + a*e)^5)/(a
+ b*x)^2 - (1050*e^4*(b*d - a*e)^4)/(a + b*x) + 840*e^5*(b*d - a*e)^3*Log[a + b*
x])/(15*b^9)

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Maple [B]  time = 0.023, size = 820, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^8/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/3*e^8/b^6*x^3-1/5/b/(b*x+a)^5*d^8+56/b^8*e^7/(b*x+a)^3*d*a^5-140/b^7*e^6/(b*x+
a)^3*d^2*a^4+560/3/b^6*e^5/(b*x+a)^3*d^3*a^3-140/b^5*e^4/(b*x+a)^3*d^4*a^2+56/b^
4*e^3/(b*x+a)^3*d^5*a+28/b^9*e^8/(b*x+a)^2*a^5-28/b^4*e^3/(b*x+a)^2*d^5-70/b^9*e
^8/(b*x+a)*a^4-70/b^5*e^4/(b*x+a)*d^4-28/3/b^9*e^8/(b*x+a)^3*a^6-28/3/b^3*e^2/(b
*x+a)^3*d^6-3*e^8/b^7*x^2*a+4*e^7/b^6*x^2*d+21*e^8/b^8*a^2*x+28*e^6/b^6*d^2*x+2/
b^9*e^8/(b*x+a)^4*a^7-2/b^2*e/(b*x+a)^4*d^7-1/5/b^9/(b*x+a)^5*a^8*e^8-56/b^9*e^8
*ln(b*x+a)*a^3+56/b^6*e^5*ln(b*x+a)*d^3-28/5/b^7/(b*x+a)^5*a^6*d^2*e^6+56/5/b^6/
(b*x+a)^5*a^5*d^3*e^5-70/b^6*e^5/(b*x+a)^4*a^4*d^3+70/b^5*e^4/(b*x+a)^4*a^3*d^4-
42/b^4*e^3/(b*x+a)^4*a^2*d^5+14/b^3*e^2/(b*x+a)^4*a*d^6+8/5/b^8/(b*x+a)^5*a^7*d*
e^7+168/b^8*e^7*ln(b*x+a)*a^2*d-168/b^7*e^6*ln(b*x+a)*a*d^2-140/b^8*e^7/(b*x+a)^
2*a^4*d+280/b^7*e^6/(b*x+a)^2*a^3*d^2-280/b^6*e^5/(b*x+a)^2*a^2*d^3+140/b^5*e^4/
(b*x+a)^2*a*d^4+280/b^8*e^7/(b*x+a)*a^3*d-420/b^7*e^6/(b*x+a)*d^2*a^2+280/b^6*e^
5/(b*x+a)*a*d^3-14/b^5/(b*x+a)^5*a^4*d^4*e^4+56/5/b^4/(b*x+a)^5*a^3*d^5*e^3-28/5
/b^3/(b*x+a)^5*a^2*d^6*e^2+8/5/b^2/(b*x+a)^5*a*d^7*e-48*e^7/b^7*d*a*x-14/b^8*e^7
/(b*x+a)^4*a^6*d+42/b^7*e^6/(b*x+a)^4*a^5*d^2

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Maxima [A]  time = 0.723719, size = 845, normalized size = 4.06 \[ -\frac{3 \, b^{8} d^{8} + 6 \, a b^{7} d^{7} e + 14 \, a^{2} b^{6} d^{6} e^{2} + 42 \, a^{3} b^{5} d^{5} e^{3} + 210 \, a^{4} b^{4} d^{4} e^{4} - 1918 \, a^{5} b^{3} d^{3} e^{5} + 3654 \, a^{6} b^{2} d^{2} e^{6} - 2754 \, a^{7} b d e^{7} + 743 \, a^{8} e^{8} + 1050 \,{\left (b^{8} d^{4} e^{4} - 4 \, a b^{7} d^{3} e^{5} + 6 \, a^{2} b^{6} d^{2} e^{6} - 4 \, a^{3} b^{5} d e^{7} + a^{4} b^{4} e^{8}\right )} x^{4} + 420 \,{\left (b^{8} d^{5} e^{3} + 5 \, a b^{7} d^{4} e^{4} - 30 \, a^{2} b^{6} d^{3} e^{5} + 50 \, a^{3} b^{5} d^{2} e^{6} - 35 \, a^{4} b^{4} d e^{7} + 9 \, a^{5} b^{3} e^{8}\right )} x^{3} + 140 \,{\left (b^{8} d^{6} e^{2} + 3 \, a b^{7} d^{5} e^{3} + 15 \, a^{2} b^{6} d^{4} e^{4} - 110 \, a^{3} b^{5} d^{3} e^{5} + 195 \, a^{4} b^{4} d^{2} e^{6} - 141 \, a^{5} b^{3} d e^{7} + 37 \, a^{6} b^{2} e^{8}\right )} x^{2} + 10 \,{\left (3 \, b^{8} d^{7} e + 7 \, a b^{7} d^{6} e^{2} + 21 \, a^{2} b^{6} d^{5} e^{3} + 105 \, a^{3} b^{5} d^{4} e^{4} - 875 \, a^{4} b^{4} d^{3} e^{5} + 1617 \, a^{5} b^{3} d^{2} e^{6} - 1197 \, a^{6} b^{2} d e^{7} + 319 \, a^{7} b e^{8}\right )} x}{15 \,{\left (b^{14} x^{5} + 5 \, a b^{13} x^{4} + 10 \, a^{2} b^{12} x^{3} + 10 \, a^{3} b^{11} x^{2} + 5 \, a^{4} b^{10} x + a^{5} b^{9}\right )}} + \frac{b^{2} e^{8} x^{3} + 3 \,{\left (4 \, b^{2} d e^{7} - 3 \, a b e^{8}\right )} x^{2} + 3 \,{\left (28 \, b^{2} d^{2} e^{6} - 48 \, a b d e^{7} + 21 \, a^{2} e^{8}\right )} x}{3 \, b^{8}} + \frac{56 \,{\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} \log \left (b x + a\right )}{b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^8/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

-1/15*(3*b^8*d^8 + 6*a*b^7*d^7*e + 14*a^2*b^6*d^6*e^2 + 42*a^3*b^5*d^5*e^3 + 210
*a^4*b^4*d^4*e^4 - 1918*a^5*b^3*d^3*e^5 + 3654*a^6*b^2*d^2*e^6 - 2754*a^7*b*d*e^
7 + 743*a^8*e^8 + 1050*(b^8*d^4*e^4 - 4*a*b^7*d^3*e^5 + 6*a^2*b^6*d^2*e^6 - 4*a^
3*b^5*d*e^7 + a^4*b^4*e^8)*x^4 + 420*(b^8*d^5*e^3 + 5*a*b^7*d^4*e^4 - 30*a^2*b^6
*d^3*e^5 + 50*a^3*b^5*d^2*e^6 - 35*a^4*b^4*d*e^7 + 9*a^5*b^3*e^8)*x^3 + 140*(b^8
*d^6*e^2 + 3*a*b^7*d^5*e^3 + 15*a^2*b^6*d^4*e^4 - 110*a^3*b^5*d^3*e^5 + 195*a^4*
b^4*d^2*e^6 - 141*a^5*b^3*d*e^7 + 37*a^6*b^2*e^8)*x^2 + 10*(3*b^8*d^7*e + 7*a*b^
7*d^6*e^2 + 21*a^2*b^6*d^5*e^3 + 105*a^3*b^5*d^4*e^4 - 875*a^4*b^4*d^3*e^5 + 161
7*a^5*b^3*d^2*e^6 - 1197*a^6*b^2*d*e^7 + 319*a^7*b*e^8)*x)/(b^14*x^5 + 5*a*b^13*
x^4 + 10*a^2*b^12*x^3 + 10*a^3*b^11*x^2 + 5*a^4*b^10*x + a^5*b^9) + 1/3*(b^2*e^8
*x^3 + 3*(4*b^2*d*e^7 - 3*a*b*e^8)*x^2 + 3*(28*b^2*d^2*e^6 - 48*a*b*d*e^7 + 21*a
^2*e^8)*x)/b^8 + 56*(b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8)*lo
g(b*x + a)/b^9

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Fricas [A]  time = 0.204958, size = 1273, normalized size = 6.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^8/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

1/15*(5*b^8*e^8*x^8 - 3*b^8*d^8 - 6*a*b^7*d^7*e - 14*a^2*b^6*d^6*e^2 - 42*a^3*b^
5*d^5*e^3 - 210*a^4*b^4*d^4*e^4 + 1918*a^5*b^3*d^3*e^5 - 3654*a^6*b^2*d^2*e^6 +
2754*a^7*b*d*e^7 - 743*a^8*e^8 + 20*(3*b^8*d*e^7 - a*b^7*e^8)*x^7 + 140*(3*b^8*d
^2*e^6 - 3*a*b^7*d*e^7 + a^2*b^6*e^8)*x^6 + 25*(84*a*b^7*d^2*e^6 - 120*a^2*b^6*d
*e^7 + 47*a^3*b^5*e^8)*x^5 - 25*(42*b^8*d^4*e^4 - 168*a*b^7*d^3*e^5 + 84*a^2*b^6
*d^2*e^6 + 96*a^3*b^5*d*e^7 - 67*a^4*b^4*e^8)*x^4 - 10*(42*b^8*d^5*e^3 + 210*a*b
^7*d^4*e^4 - 1260*a^2*b^6*d^3*e^5 + 1680*a^3*b^5*d^2*e^6 - 780*a^4*b^4*d*e^7 + 8
5*a^5*b^3*e^8)*x^3 - 10*(14*b^8*d^6*e^2 + 42*a*b^7*d^5*e^3 + 210*a^2*b^6*d^4*e^4
- 1540*a^3*b^5*d^3*e^5 + 2520*a^4*b^4*d^2*e^6 - 1620*a^5*b^3*d*e^7 + 365*a^6*b^
2*e^8)*x^2 - 5*(6*b^8*d^7*e + 14*a*b^7*d^6*e^2 + 42*a^2*b^6*d^5*e^3 + 210*a^3*b^
5*d^4*e^4 - 1750*a^4*b^4*d^3*e^5 + 3150*a^5*b^3*d^2*e^6 - 2250*a^6*b^2*d*e^7 + 5
75*a^7*b*e^8)*x + 840*(a^5*b^3*d^3*e^5 - 3*a^6*b^2*d^2*e^6 + 3*a^7*b*d*e^7 - a^8
*e^8 + (b^8*d^3*e^5 - 3*a*b^7*d^2*e^6 + 3*a^2*b^6*d*e^7 - a^3*b^5*e^8)*x^5 + 5*(
a*b^7*d^3*e^5 - 3*a^2*b^6*d^2*e^6 + 3*a^3*b^5*d*e^7 - a^4*b^4*e^8)*x^4 + 10*(a^2
*b^6*d^3*e^5 - 3*a^3*b^5*d^2*e^6 + 3*a^4*b^4*d*e^7 - a^5*b^3*e^8)*x^3 + 10*(a^3*
b^5*d^3*e^5 - 3*a^4*b^4*d^2*e^6 + 3*a^5*b^3*d*e^7 - a^6*b^2*e^8)*x^2 + 5*(a^4*b^
4*d^3*e^5 - 3*a^5*b^3*d^2*e^6 + 3*a^6*b^2*d*e^7 - a^7*b*e^8)*x)*log(b*x + a))/(b
^14*x^5 + 5*a*b^13*x^4 + 10*a^2*b^12*x^3 + 10*a^3*b^11*x^2 + 5*a^4*b^10*x + a^5*
b^9)

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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((e*x+d)**8/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214036, size = 737, normalized size = 3.54 \[ \frac{56 \,{\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{9}} - \frac{3 \, b^{8} d^{8} + 6 \, a b^{7} d^{7} e + 14 \, a^{2} b^{6} d^{6} e^{2} + 42 \, a^{3} b^{5} d^{5} e^{3} + 210 \, a^{4} b^{4} d^{4} e^{4} - 1918 \, a^{5} b^{3} d^{3} e^{5} + 3654 \, a^{6} b^{2} d^{2} e^{6} - 2754 \, a^{7} b d e^{7} + 743 \, a^{8} e^{8} + 1050 \,{\left (b^{8} d^{4} e^{4} - 4 \, a b^{7} d^{3} e^{5} + 6 \, a^{2} b^{6} d^{2} e^{6} - 4 \, a^{3} b^{5} d e^{7} + a^{4} b^{4} e^{8}\right )} x^{4} + 420 \,{\left (b^{8} d^{5} e^{3} + 5 \, a b^{7} d^{4} e^{4} - 30 \, a^{2} b^{6} d^{3} e^{5} + 50 \, a^{3} b^{5} d^{2} e^{6} - 35 \, a^{4} b^{4} d e^{7} + 9 \, a^{5} b^{3} e^{8}\right )} x^{3} + 140 \,{\left (b^{8} d^{6} e^{2} + 3 \, a b^{7} d^{5} e^{3} + 15 \, a^{2} b^{6} d^{4} e^{4} - 110 \, a^{3} b^{5} d^{3} e^{5} + 195 \, a^{4} b^{4} d^{2} e^{6} - 141 \, a^{5} b^{3} d e^{7} + 37 \, a^{6} b^{2} e^{8}\right )} x^{2} + 10 \,{\left (3 \, b^{8} d^{7} e + 7 \, a b^{7} d^{6} e^{2} + 21 \, a^{2} b^{6} d^{5} e^{3} + 105 \, a^{3} b^{5} d^{4} e^{4} - 875 \, a^{4} b^{4} d^{3} e^{5} + 1617 \, a^{5} b^{3} d^{2} e^{6} - 1197 \, a^{6} b^{2} d e^{7} + 319 \, a^{7} b e^{8}\right )} x}{15 \,{\left (b x + a\right )}^{5} b^{9}} + \frac{b^{12} x^{3} e^{8} + 12 \, b^{12} d x^{2} e^{7} + 84 \, b^{12} d^{2} x e^{6} - 9 \, a b^{11} x^{2} e^{8} - 144 \, a b^{11} d x e^{7} + 63 \, a^{2} b^{10} x e^{8}}{3 \, b^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^8/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

56*(b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8)*ln(abs(b*x + a))/b^
9 - 1/15*(3*b^8*d^8 + 6*a*b^7*d^7*e + 14*a^2*b^6*d^6*e^2 + 42*a^3*b^5*d^5*e^3 +
210*a^4*b^4*d^4*e^4 - 1918*a^5*b^3*d^3*e^5 + 3654*a^6*b^2*d^2*e^6 - 2754*a^7*b*d
*e^7 + 743*a^8*e^8 + 1050*(b^8*d^4*e^4 - 4*a*b^7*d^3*e^5 + 6*a^2*b^6*d^2*e^6 - 4
*a^3*b^5*d*e^7 + a^4*b^4*e^8)*x^4 + 420*(b^8*d^5*e^3 + 5*a*b^7*d^4*e^4 - 30*a^2*
b^6*d^3*e^5 + 50*a^3*b^5*d^2*e^6 - 35*a^4*b^4*d*e^7 + 9*a^5*b^3*e^8)*x^3 + 140*(
b^8*d^6*e^2 + 3*a*b^7*d^5*e^3 + 15*a^2*b^6*d^4*e^4 - 110*a^3*b^5*d^3*e^5 + 195*a
^4*b^4*d^2*e^6 - 141*a^5*b^3*d*e^7 + 37*a^6*b^2*e^8)*x^2 + 10*(3*b^8*d^7*e + 7*a
*b^7*d^6*e^2 + 21*a^2*b^6*d^5*e^3 + 105*a^3*b^5*d^4*e^4 - 875*a^4*b^4*d^3*e^5 +
1617*a^5*b^3*d^2*e^6 - 1197*a^6*b^2*d*e^7 + 319*a^7*b*e^8)*x)/((b*x + a)^5*b^9)
+ 1/3*(b^12*x^3*e^8 + 12*b^12*d*x^2*e^7 + 84*b^12*d^2*x*e^6 - 9*a*b^11*x^2*e^8 -
144*a*b^11*d*x*e^7 + 63*a^2*b^10*x*e^8)/b^18