3.1523 \(\int \frac{1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=155 \[ -\frac{e^5 \log (a+b x)}{(b d-a e)^6}+\frac{e^5 \log (d+e x)}{(b d-a e)^6}-\frac{e^4}{(a+b x) (b d-a e)^5}+\frac{e^3}{2 (a+b x)^2 (b d-a e)^4}-\frac{e^2}{3 (a+b x)^3 (b d-a e)^3}+\frac{e}{4 (a+b x)^4 (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.278365, antiderivative size = 155, 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 \[ -\frac{e^5 \log (a+b x)}{(b d-a e)^6}+\frac{e^5 \log (d+e x)}{(b d-a e)^6}-\frac{e^4}{(a+b x) (b d-a e)^5}+\frac{e^3}{2 (a+b x)^2 (b d-a e)^4}-\frac{e^2}{3 (a+b x)^3 (b d-a e)^3}+\frac{e}{4 (a+b x)^4 (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 108.882, size = 129, normalized size = 0.83 \[ - \frac{e^{5} \log{\left (a + b x \right )}}{\left (a e - b d\right )^{6}} + \frac{e^{5} \log{\left (d + e x \right )}}{\left (a e - b d\right )^{6}} + \frac{e^{4}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{e^{3}}{2 \left (a + b x\right )^{2} \left (a e - b d\right )^{4}} + \frac{e^{2}}{3 \left (a + b x\right )^{3} \left (a e - b d\right )^{3}} + \frac{e}{4 \left (a + b x\right )^{4} \left (a e - b d\right )^{2}} + \frac{1}{5 \left (a + b x\right )^{5} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-e**5*log(a + b*x)/(a*e - b*d)**6 + e**5*log(d + e*x)/(a*e - b*d)**6 + e**4/((a
+ b*x)*(a*e - b*d)**5) + e**3/(2*(a + b*x)**2*(a*e - b*d)**4) + e**2/(3*(a + b*x
)**3*(a*e - b*d)**3) + e/(4*(a + b*x)**4*(a*e - b*d)**2) + 1/(5*(a + b*x)**5*(a*
e - b*d))

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Mathematica [A]  time = 0.140264, size = 152, normalized size = 0.98 \[ \frac{60 e^5 (a+b x)^5 \log (d+e x)+60 e^4 (a+b x)^4 (a e-b d)+30 e^3 (a+b x)^3 (b d-a e)^2+20 e^2 (a+b x)^2 (a e-b d)^3+15 e (a+b x) (b d-a e)^4-12 (b d-a e)^5-60 e^5 (a+b x)^5 \log (a+b x)}{60 (a+b x)^5 (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-12*(b*d - a*e)^5 + 15*e*(b*d - a*e)^4*(a + b*x) + 20*e^2*(-(b*d) + a*e)^3*(a +
b*x)^2 + 30*e^3*(b*d - a*e)^2*(a + b*x)^3 + 60*e^4*(-(b*d) + a*e)*(a + b*x)^4 -
60*e^5*(a + b*x)^5*Log[a + b*x] + 60*e^5*(a + b*x)^5*Log[d + e*x])/(60*(b*d - a
*e)^6*(a + b*x)^5)

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Maple [A]  time = 0.019, size = 147, normalized size = 1. \[{\frac{1}{ \left ( 5\,ae-5\,bd \right ) \left ( bx+a \right ) ^{5}}}+{\frac{e}{4\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{4}}}+{\frac{{e}^{2}}{3\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{3}}}+{\frac{{e}^{3}}{2\, \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{2}}}+{\frac{{e}^{4}}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) }}-{\frac{{e}^{5}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{6}}}+{\frac{{e}^{5}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/5/(a*e-b*d)/(b*x+a)^5+1/4*e/(a*e-b*d)^2/(b*x+a)^4+1/3*e^2/(a*e-b*d)^3/(b*x+a)^
3+1/2*e^3/(a*e-b*d)^4/(b*x+a)^2+e^4/(a*e-b*d)^5/(b*x+a)-e^5/(a*e-b*d)^6*ln(b*x+a
)+e^5/(a*e-b*d)^6*ln(e*x+d)

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Maxima [A]  time = 0.730664, size = 1087, normalized size = 7.01 \[ -\frac{e^{5} \log \left (b x + a\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac{e^{5} \log \left (e x + d\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} - \frac{60 \, b^{4} e^{4} x^{4} + 12 \, b^{4} d^{4} - 63 \, a b^{3} d^{3} e + 137 \, a^{2} b^{2} d^{2} e^{2} - 163 \, a^{3} b d e^{3} + 137 \, a^{4} e^{4} - 30 \,{\left (b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 10 \,{\left (2 \, b^{4} d^{2} e^{2} - 13 \, a b^{3} d e^{3} + 47 \, a^{2} b^{2} e^{4}\right )} x^{2} - 5 \,{\left (3 \, b^{4} d^{3} e - 17 \, a b^{3} d^{2} e^{2} + 43 \, a^{2} b^{2} d e^{3} - 77 \, a^{3} b e^{4}\right )} x}{60 \,{\left (a^{5} b^{5} d^{5} - 5 \, a^{6} b^{4} d^{4} e + 10 \, a^{7} b^{3} d^{3} e^{2} - 10 \, a^{8} b^{2} d^{2} e^{3} + 5 \, a^{9} b d e^{4} - a^{10} e^{5} +{\left (b^{10} d^{5} - 5 \, a b^{9} d^{4} e + 10 \, a^{2} b^{8} d^{3} e^{2} - 10 \, a^{3} b^{7} d^{2} e^{3} + 5 \, a^{4} b^{6} d e^{4} - a^{5} b^{5} e^{5}\right )} x^{5} + 5 \,{\left (a b^{9} d^{5} - 5 \, a^{2} b^{8} d^{4} e + 10 \, a^{3} b^{7} d^{3} e^{2} - 10 \, a^{4} b^{6} d^{2} e^{3} + 5 \, a^{5} b^{5} d e^{4} - a^{6} b^{4} e^{5}\right )} x^{4} + 10 \,{\left (a^{2} b^{8} d^{5} - 5 \, a^{3} b^{7} d^{4} e + 10 \, a^{4} b^{6} d^{3} e^{2} - 10 \, a^{5} b^{5} d^{2} e^{3} + 5 \, a^{6} b^{4} d e^{4} - a^{7} b^{3} e^{5}\right )} x^{3} + 10 \,{\left (a^{3} b^{7} d^{5} - 5 \, a^{4} b^{6} d^{4} e + 10 \, a^{5} b^{5} d^{3} e^{2} - 10 \, a^{6} b^{4} d^{2} e^{3} + 5 \, a^{7} b^{3} d e^{4} - a^{8} b^{2} e^{5}\right )} x^{2} + 5 \,{\left (a^{4} b^{6} d^{5} - 5 \, a^{5} b^{5} d^{4} e + 10 \, a^{6} b^{4} d^{3} e^{2} - 10 \, a^{7} b^{3} d^{2} e^{3} + 5 \, a^{8} b^{2} d e^{4} - a^{9} b e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)),x, algorithm="maxima")

[Out]

-e^5*log(b*x + a)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3
*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6) + e^5*log(e*x + d)/(b^6*d^6
- 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4
- 6*a^5*b*d*e^5 + a^6*e^6) - 1/60*(60*b^4*e^4*x^4 + 12*b^4*d^4 - 63*a*b^3*d^3*e
+ 137*a^2*b^2*d^2*e^2 - 163*a^3*b*d*e^3 + 137*a^4*e^4 - 30*(b^4*d*e^3 - 9*a*b^3*
e^4)*x^3 + 10*(2*b^4*d^2*e^2 - 13*a*b^3*d*e^3 + 47*a^2*b^2*e^4)*x^2 - 5*(3*b^4*d
^3*e - 17*a*b^3*d^2*e^2 + 43*a^2*b^2*d*e^3 - 77*a^3*b*e^4)*x)/(a^5*b^5*d^5 - 5*a
^6*b^4*d^4*e + 10*a^7*b^3*d^3*e^2 - 10*a^8*b^2*d^2*e^3 + 5*a^9*b*d*e^4 - a^10*e^
5 + (b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*
b^6*d*e^4 - a^5*b^5*e^5)*x^5 + 5*(a*b^9*d^5 - 5*a^2*b^8*d^4*e + 10*a^3*b^7*d^3*e
^2 - 10*a^4*b^6*d^2*e^3 + 5*a^5*b^5*d*e^4 - a^6*b^4*e^5)*x^4 + 10*(a^2*b^8*d^5 -
5*a^3*b^7*d^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3 + 5*a^6*b^4*d*e^4 - a
^7*b^3*e^5)*x^3 + 10*(a^3*b^7*d^5 - 5*a^4*b^6*d^4*e + 10*a^5*b^5*d^3*e^2 - 10*a^
6*b^4*d^2*e^3 + 5*a^7*b^3*d*e^4 - a^8*b^2*e^5)*x^2 + 5*(a^4*b^6*d^5 - 5*a^5*b^5*
d^4*e + 10*a^6*b^4*d^3*e^2 - 10*a^7*b^3*d^2*e^3 + 5*a^8*b^2*d*e^4 - a^9*b*e^5)*x
)

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Fricas [A]  time = 0.21964, size = 1246, normalized size = 8.04 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)),x, algorithm="fricas")

[Out]

-1/60*(12*b^5*d^5 - 75*a*b^4*d^4*e + 200*a^2*b^3*d^3*e^2 - 300*a^3*b^2*d^2*e^3 +
300*a^4*b*d*e^4 - 137*a^5*e^5 + 60*(b^5*d*e^4 - a*b^4*e^5)*x^4 - 30*(b^5*d^2*e^
3 - 10*a*b^4*d*e^4 + 9*a^2*b^3*e^5)*x^3 + 10*(2*b^5*d^3*e^2 - 15*a*b^4*d^2*e^3 +
60*a^2*b^3*d*e^4 - 47*a^3*b^2*e^5)*x^2 - 5*(3*b^5*d^4*e - 20*a*b^4*d^3*e^2 + 60
*a^2*b^3*d^2*e^3 - 120*a^3*b^2*d*e^4 + 77*a^4*b*e^5)*x + 60*(b^5*e^5*x^5 + 5*a*b
^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*
log(b*x + a) - 60*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b
^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*log(e*x + d))/(a^5*b^6*d^6 - 6*a^6*b^5*d^5
*e + 15*a^7*b^4*d^4*e^2 - 20*a^8*b^3*d^3*e^3 + 15*a^9*b^2*d^2*e^4 - 6*a^10*b*d*e
^5 + a^11*e^6 + (b^11*d^6 - 6*a*b^10*d^5*e + 15*a^2*b^9*d^4*e^2 - 20*a^3*b^8*d^3
*e^3 + 15*a^4*b^7*d^2*e^4 - 6*a^5*b^6*d*e^5 + a^6*b^5*e^6)*x^5 + 5*(a*b^10*d^6 -
6*a^2*b^9*d^5*e + 15*a^3*b^8*d^4*e^2 - 20*a^4*b^7*d^3*e^3 + 15*a^5*b^6*d^2*e^4
- 6*a^6*b^5*d*e^5 + a^7*b^4*e^6)*x^4 + 10*(a^2*b^9*d^6 - 6*a^3*b^8*d^5*e + 15*a^
4*b^7*d^4*e^2 - 20*a^5*b^6*d^3*e^3 + 15*a^6*b^5*d^2*e^4 - 6*a^7*b^4*d*e^5 + a^8*
b^3*e^6)*x^3 + 10*(a^3*b^8*d^6 - 6*a^4*b^7*d^5*e + 15*a^5*b^6*d^4*e^2 - 20*a^6*b
^5*d^3*e^3 + 15*a^7*b^4*d^2*e^4 - 6*a^8*b^3*d*e^5 + a^9*b^2*e^6)*x^2 + 5*(a^4*b^
7*d^6 - 6*a^5*b^6*d^5*e + 15*a^6*b^5*d^4*e^2 - 20*a^7*b^4*d^3*e^3 + 15*a^8*b^3*d
^2*e^4 - 6*a^9*b^2*d*e^5 + a^10*b*e^6)*x)

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Sympy [A]  time = 15.7374, size = 1081, normalized size = 6.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

e**5*log(x + (-a**7*e**12/(a*e - b*d)**6 + 7*a**6*b*d*e**11/(a*e - b*d)**6 - 21*
a**5*b**2*d**2*e**10/(a*e - b*d)**6 + 35*a**4*b**3*d**3*e**9/(a*e - b*d)**6 - 35
*a**3*b**4*d**4*e**8/(a*e - b*d)**6 + 21*a**2*b**5*d**5*e**7/(a*e - b*d)**6 - 7*
a*b**6*d**6*e**6/(a*e - b*d)**6 + a*e**6 + b**7*d**7*e**5/(a*e - b*d)**6 + b*d*e
**5)/(2*b*e**6))/(a*e - b*d)**6 - e**5*log(x + (a**7*e**12/(a*e - b*d)**6 - 7*a*
*6*b*d*e**11/(a*e - b*d)**6 + 21*a**5*b**2*d**2*e**10/(a*e - b*d)**6 - 35*a**4*b
**3*d**3*e**9/(a*e - b*d)**6 + 35*a**3*b**4*d**4*e**8/(a*e - b*d)**6 - 21*a**2*b
**5*d**5*e**7/(a*e - b*d)**6 + 7*a*b**6*d**6*e**6/(a*e - b*d)**6 + a*e**6 - b**7
*d**7*e**5/(a*e - b*d)**6 + b*d*e**5)/(2*b*e**6))/(a*e - b*d)**6 + (137*a**4*e**
4 - 163*a**3*b*d*e**3 + 137*a**2*b**2*d**2*e**2 - 63*a*b**3*d**3*e + 12*b**4*d**
4 + 60*b**4*e**4*x**4 + x**3*(270*a*b**3*e**4 - 30*b**4*d*e**3) + x**2*(470*a**2
*b**2*e**4 - 130*a*b**3*d*e**3 + 20*b**4*d**2*e**2) + x*(385*a**3*b*e**4 - 215*a
**2*b**2*d*e**3 + 85*a*b**3*d**2*e**2 - 15*b**4*d**3*e))/(60*a**10*e**5 - 300*a*
*9*b*d*e**4 + 600*a**8*b**2*d**2*e**3 - 600*a**7*b**3*d**3*e**2 + 300*a**6*b**4*
d**4*e - 60*a**5*b**5*d**5 + x**5*(60*a**5*b**5*e**5 - 300*a**4*b**6*d*e**4 + 60
0*a**3*b**7*d**2*e**3 - 600*a**2*b**8*d**3*e**2 + 300*a*b**9*d**4*e - 60*b**10*d
**5) + x**4*(300*a**6*b**4*e**5 - 1500*a**5*b**5*d*e**4 + 3000*a**4*b**6*d**2*e*
*3 - 3000*a**3*b**7*d**3*e**2 + 1500*a**2*b**8*d**4*e - 300*a*b**9*d**5) + x**3*
(600*a**7*b**3*e**5 - 3000*a**6*b**4*d*e**4 + 6000*a**5*b**5*d**2*e**3 - 6000*a*
*4*b**6*d**3*e**2 + 3000*a**3*b**7*d**4*e - 600*a**2*b**8*d**5) + x**2*(600*a**8
*b**2*e**5 - 3000*a**7*b**3*d*e**4 + 6000*a**6*b**4*d**2*e**3 - 6000*a**5*b**5*d
**3*e**2 + 3000*a**4*b**6*d**4*e - 600*a**3*b**7*d**5) + x*(300*a**9*b*e**5 - 15
00*a**8*b**2*d*e**4 + 3000*a**7*b**3*d**2*e**3 - 3000*a**6*b**4*d**3*e**2 + 1500
*a**5*b**5*d**4*e - 300*a**4*b**6*d**5))

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GIAC/XCAS [A]  time = 0.213661, size = 568, normalized size = 3.66 \[ -\frac{b e^{5}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac{e^{6}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} - \frac{12 \, b^{5} d^{5} - 75 \, a b^{4} d^{4} e + 200 \, a^{2} b^{3} d^{3} e^{2} - 300 \, a^{3} b^{2} d^{2} e^{3} + 300 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 60 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \,{\left (b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 9 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \,{\left (2 \, b^{5} d^{3} e^{2} - 15 \, a b^{4} d^{2} e^{3} + 60 \, a^{2} b^{3} d e^{4} - 47 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \,{\left (3 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 60 \, a^{2} b^{3} d^{2} e^{3} - 120 \, a^{3} b^{2} d e^{4} + 77 \, a^{4} b e^{5}\right )} x}{60 \,{\left (b d - a e\right )}^{6}{\left (b x + a\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)),x, algorithm="giac")

[Out]

-b*e^5*ln(abs(b*x + a))/(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b
^4*d^3*e^3 + 15*a^4*b^3*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6) + e^6*ln(abs(x*e
+ d))/(b^6*d^6*e - 6*a*b^5*d^5*e^2 + 15*a^2*b^4*d^4*e^3 - 20*a^3*b^3*d^3*e^4 + 1
5*a^4*b^2*d^2*e^5 - 6*a^5*b*d*e^6 + a^6*e^7) - 1/60*(12*b^5*d^5 - 75*a*b^4*d^4*e
+ 200*a^2*b^3*d^3*e^2 - 300*a^3*b^2*d^2*e^3 + 300*a^4*b*d*e^4 - 137*a^5*e^5 + 6
0*(b^5*d*e^4 - a*b^4*e^5)*x^4 - 30*(b^5*d^2*e^3 - 10*a*b^4*d*e^4 + 9*a^2*b^3*e^5
)*x^3 + 10*(2*b^5*d^3*e^2 - 15*a*b^4*d^2*e^3 + 60*a^2*b^3*d*e^4 - 47*a^3*b^2*e^5
)*x^2 - 5*(3*b^5*d^4*e - 20*a*b^4*d^3*e^2 + 60*a^2*b^3*d^2*e^3 - 120*a^3*b^2*d*e
^4 + 77*a^4*b*e^5)*x)/((b*d - a*e)^6*(b*x + a)^5)