3.2325 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=281 \[ -\frac{x \left (A c e (3 c d-2 b e)-B \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^5}-\frac{\log (d+e x) \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^6}+\frac{\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{e^6 (d+e x)}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{2 e^6 (d+e x)^2}-\frac{c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}+\frac{B c^2 x^3}{3 e^3} \]

[Out]

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

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Rubi [A]  time = 1.13981, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{x \left (A c e (3 c d-2 b e)-B \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^5}-\frac{\log (d+e x) \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^6}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^6 (d+e x)}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{2 e^6 (d+e x)^2}-\frac{c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}+\frac{B c^2 x^3}{3 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.321006, size = 262, normalized size = 0.93 \[ \frac{6 e x \left (B \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+A c e (2 b e-3 c d)\right )+6 \log (d+e x) \left (A e \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+B \left (6 c d e (2 b d-a e)+b e^2 (2 a e-3 b d)-10 c^2 d^3\right )\right )-\frac{6 \left (e (a e-b d)+c d^2\right ) \left (B e (a e-3 b d)+2 A e (b e-2 c d)+5 B c d^2\right )}{d+e x}+\frac{3 (B d-A e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}+3 c e^2 x^2 (A c e+2 b B e-3 B c d)+2 B c^2 e^3 x^3}{6 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(6*e*(A*c*e*(-3*c*d + 2*b*e) + B*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e)))*x
+ 3*c*e^2*(-3*B*c*d + 2*b*B*e + A*c*e)*x^2 + 2*B*c^2*e^3*x^3 + (3*(B*d - A*e)*(
c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 - (6*(c*d^2 + e*(-(b*d) + a*e))*(5*B*c*
d^2 + B*e*(-3*b*d + a*e) + 2*A*e*(-2*c*d + b*e)))/(d + e*x) + 6*(A*e*(6*c^2*d^2
+ b^2*e^2 + 2*c*e*(-3*b*d + a*e)) + B*(-10*c^2*d^3 + 6*c*d*e*(2*b*d - a*e) + b*e
^2*(-3*b*d + 2*a*e)))*Log[d + e*x])/(6*e^6)

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Maple [B]  time = 0.016, size = 654, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*c^2*x^3/e^3+b^2*B*x/e^3-1/2/e/(e*x+d)^2*A*a^2+1/2/e^3*A*x^2*c^2+1/e^3*ln(e
*x+d)*A*b^2-1/e^2/(e*x+d)*B*a^2-6/e^4*B*b*c*d*x-6/e^4*ln(e*x+d)*A*b*c*d-6/e^4*ln
(e*x+d)*B*a*c*d+12/e^5*ln(e*x+d)*B*b*c*d^2+4/e^3/(e*x+d)*A*a*c*d-6/e^4/(e*x+d)*A
*b*c*d^2+4/e^3/(e*x+d)*B*a*b*d-6/e^4/(e*x+d)*B*a*c*d^2+8/e^5/(e*x+d)*B*b*c*d^3+1
/e^2/(e*x+d)^2*A*d*a*b-1/e^3/(e*x+d)^2*A*d^2*a*c+1/e^4/(e*x+d)^2*A*b*c*d^3-1/e^3
/(e*x+d)^2*B*d^2*a*b+1/e^4/(e*x+d)^2*B*a*c*d^3-1/e^5/(e*x+d)^2*B*b*c*d^4+1/e^3*B
*x^2*b*c-3/2/e^4*B*x^2*c^2*d-3/e^4*A*c^2*d*x-3/e^4/(e*x+d)*B*b^2*d^2-5/e^6/(e*x+
d)*B*c^2*d^4-1/2/e^3/(e*x+d)^2*A*d^2*b^2-1/2/e^5/(e*x+d)^2*A*c^2*d^4+1/2/e^2/(e*
x+d)^2*B*d*a^2+1/2/e^4/(e*x+d)^2*B*b^2*d^3+1/2/e^6/(e*x+d)^2*B*c^2*d^5+2/e^3*B*a
*c*x+6/e^5*B*c^2*d^2*x+2/e^3*ln(e*x+d)*A*a*c+6/e^5*ln(e*x+d)*A*c^2*d^2+2/e^3*ln(
e*x+d)*B*a*b-3/e^4*ln(e*x+d)*B*b^2*d-10/e^6*ln(e*x+d)*B*c^2*d^3-2/e^2/(e*x+d)*A*
a*b+2/e^3/(e*x+d)*A*b^2*d+4/e^5/(e*x+d)*A*c^2*d^3+2/e^3*A*b*c*x

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Maxima [A]  time = 0.69575, size = 536, normalized size = 1.91 \[ -\frac{9 \, B c^{2} d^{5} + A a^{2} e^{5} - 7 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 5 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 3 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} +{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 2 \,{\left (5 \, B c^{2} d^{4} e - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} +{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{2 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac{2 \, B c^{2} e^{2} x^{3} - 3 \,{\left (3 \, B c^{2} d e -{\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{2} + 6 \,{\left (6 \, B c^{2} d^{2} - 3 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac{{\left (10 \, B c^{2} d^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{2} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(9*B*c^2*d^5 + A*a^2*e^5 - 7*(2*B*b*c + A*c^2)*d^4*e + 5*(B*b^2 + 2*(B*a +
A*b)*c)*d^3*e^2 - 3*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + (B*a^2 + 2*A*a*b)*d*e^
4 + 2*(5*B*c^2*d^4*e - 4*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*(B*a + A*b)*c)
*d^2*e^3 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + (B*a^2 + 2*A*a*b)*e^5)*x)/(e^8*
x^2 + 2*d*e^7*x + d^2*e^6) + 1/6*(2*B*c^2*e^2*x^3 - 3*(3*B*c^2*d*e - (2*B*b*c +
A*c^2)*e^2)*x^2 + 6*(6*B*c^2*d^2 - 3*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*(B*a + A
*b)*c)*e^2)*x)/e^5 - (10*B*c^2*d^3 - 6*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*(B
*a + A*b)*c)*d*e^2 - (2*B*a*b + A*b^2 + 2*A*a*c)*e^3)*log(e*x + d)/e^6

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Fricas [A]  time = 0.264626, size = 837, normalized size = 2.98 \[ \frac{2 \, B c^{2} e^{5} x^{5} - 27 \, B c^{2} d^{5} - 3 \, A a^{2} e^{5} + 21 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 15 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} + 9 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} -{\left (5 \, B c^{2} d e^{4} - 3 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \,{\left (10 \, B c^{2} d^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 3 \,{\left (21 \, B c^{2} d^{3} e^{2} - 11 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 4 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4}\right )} x^{2} + 6 \,{\left (B c^{2} d^{4} e +{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 2 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} + 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} -{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x - 6 \,{\left (10 \, B c^{2} d^{5} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} +{\left (10 \, B c^{2} d^{3} e^{2} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 2 \,{\left (10 \, B c^{2} d^{4} e - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*B*c^2*e^5*x^5 - 27*B*c^2*d^5 - 3*A*a^2*e^5 + 21*(2*B*b*c + A*c^2)*d^4*e -
15*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + 9*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 -
3*(B*a^2 + 2*A*a*b)*d*e^4 - (5*B*c^2*d*e^4 - 3*(2*B*b*c + A*c^2)*e^5)*x^4 + 2*(1
0*B*c^2*d^2*e^3 - 6*(2*B*b*c + A*c^2)*d*e^4 + 3*(B*b^2 + 2*(B*a + A*b)*c)*e^5)*x
^3 + 3*(21*B*c^2*d^3*e^2 - 11*(2*B*b*c + A*c^2)*d^2*e^3 + 4*(B*b^2 + 2*(B*a + A*
b)*c)*d*e^4)*x^2 + 6*(B*c^2*d^4*e + (2*B*b*c + A*c^2)*d^3*e^2 - 2*(B*b^2 + 2*(B*
a + A*b)*c)*d^2*e^3 + 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 - (B*a^2 + 2*A*a*b)*e^
5)*x - 6*(10*B*c^2*d^5 - 6*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*(B*a + A*b)*c)
*d^3*e^2 - (2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + (10*B*c^2*d^3*e^2 - 6*(2*B*b*c
+ A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4 - (2*B*a*b + A*b^2 + 2*A*a*
c)*e^5)*x^2 + 2*(10*B*c^2*d^4*e - 6*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*(B*
a + A*b)*c)*d^2*e^3 - (2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4)*x)*log(e*x + d))/(e^8*x
^2 + 2*d*e^7*x + d^2*e^6)

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Sympy [A]  time = 112.614, size = 527, normalized size = 1.88 \[ \frac{B c^{2} x^{3}}{3 e^{3}} - \frac{A a^{2} e^{5} + 2 A a b d e^{4} - 6 A a c d^{2} e^{3} - 3 A b^{2} d^{2} e^{3} + 10 A b c d^{3} e^{2} - 7 A c^{2} d^{4} e + B a^{2} d e^{4} - 6 B a b d^{2} e^{3} + 10 B a c d^{3} e^{2} + 5 B b^{2} d^{3} e^{2} - 14 B b c d^{4} e + 9 B c^{2} d^{5} + x \left (4 A a b e^{5} - 8 A a c d e^{4} - 4 A b^{2} d e^{4} + 12 A b c d^{2} e^{3} - 8 A c^{2} d^{3} e^{2} + 2 B a^{2} e^{5} - 8 B a b d e^{4} + 12 B a c d^{2} e^{3} + 6 B b^{2} d^{2} e^{3} - 16 B b c d^{3} e^{2} + 10 B c^{2} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} + \frac{x^{2} \left (A c^{2} e + 2 B b c e - 3 B c^{2} d\right )}{2 e^{4}} + \frac{x \left (2 A b c e^{2} - 3 A c^{2} d e + 2 B a c e^{2} + B b^{2} e^{2} - 6 B b c d e + 6 B c^{2} d^{2}\right )}{e^{5}} + \frac{\left (2 A a c e^{3} + A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e + 2 B a b e^{3} - 6 B a c d e^{2} - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right ) \log{\left (d + e x \right )}}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*c**2*x**3/(3*e**3) - (A*a**2*e**5 + 2*A*a*b*d*e**4 - 6*A*a*c*d**2*e**3 - 3*A*b
**2*d**2*e**3 + 10*A*b*c*d**3*e**2 - 7*A*c**2*d**4*e + B*a**2*d*e**4 - 6*B*a*b*d
**2*e**3 + 10*B*a*c*d**3*e**2 + 5*B*b**2*d**3*e**2 - 14*B*b*c*d**4*e + 9*B*c**2*
d**5 + x*(4*A*a*b*e**5 - 8*A*a*c*d*e**4 - 4*A*b**2*d*e**4 + 12*A*b*c*d**2*e**3 -
8*A*c**2*d**3*e**2 + 2*B*a**2*e**5 - 8*B*a*b*d*e**4 + 12*B*a*c*d**2*e**3 + 6*B*
b**2*d**2*e**3 - 16*B*b*c*d**3*e**2 + 10*B*c**2*d**4*e))/(2*d**2*e**6 + 4*d*e**7
*x + 2*e**8*x**2) + x**2*(A*c**2*e + 2*B*b*c*e - 3*B*c**2*d)/(2*e**4) + x*(2*A*b
*c*e**2 - 3*A*c**2*d*e + 2*B*a*c*e**2 + B*b**2*e**2 - 6*B*b*c*d*e + 6*B*c**2*d**
2)/e**5 + (2*A*a*c*e**3 + A*b**2*e**3 - 6*A*b*c*d*e**2 + 6*A*c**2*d**2*e + 2*B*a
*b*e**3 - 6*B*a*c*d*e**2 - 3*B*b**2*d*e**2 + 12*B*b*c*d**2*e - 10*B*c**2*d**3)*l
og(d + e*x)/e**6

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GIAC/XCAS [A]  time = 0.285525, size = 581, normalized size = 2.07 \[ -{\left (10 \, B c^{2} d^{3} - 12 \, B b c d^{2} e - 6 \, A c^{2} d^{2} e + 3 \, B b^{2} d e^{2} + 6 \, B a c d e^{2} + 6 \, A b c d e^{2} - 2 \, B a b e^{3} - A b^{2} e^{3} - 2 \, A a c e^{3}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B c^{2} x^{3} e^{6} - 9 \, B c^{2} d x^{2} e^{5} + 36 \, B c^{2} d^{2} x e^{4} + 6 \, B b c x^{2} e^{6} + 3 \, A c^{2} x^{2} e^{6} - 36 \, B b c d x e^{5} - 18 \, A c^{2} d x e^{5} + 6 \, B b^{2} x e^{6} + 12 \, B a c x e^{6} + 12 \, A b c x e^{6}\right )} e^{\left (-9\right )} - \frac{{\left (9 \, B c^{2} d^{5} - 14 \, B b c d^{4} e - 7 \, A c^{2} d^{4} e + 5 \, B b^{2} d^{3} e^{2} + 10 \, B a c d^{3} e^{2} + 10 \, A b c d^{3} e^{2} - 6 \, B a b d^{2} e^{3} - 3 \, A b^{2} d^{2} e^{3} - 6 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + 2 \, A a b d e^{4} + A a^{2} e^{5} + 2 \,{\left (5 \, B c^{2} d^{4} e - 8 \, B b c d^{3} e^{2} - 4 \, A c^{2} d^{3} e^{2} + 3 \, B b^{2} d^{2} e^{3} + 6 \, B a c d^{2} e^{3} + 6 \, A b c d^{2} e^{3} - 4 \, B a b d e^{4} - 2 \, A b^{2} d e^{4} - 4 \, A a c d e^{4} + B a^{2} e^{5} + 2 \, A a b e^{5}\right )} x\right )} e^{\left (-6\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(10*B*c^2*d^3 - 12*B*b*c*d^2*e - 6*A*c^2*d^2*e + 3*B*b^2*d*e^2 + 6*B*a*c*d*e^2
+ 6*A*b*c*d*e^2 - 2*B*a*b*e^3 - A*b^2*e^3 - 2*A*a*c*e^3)*e^(-6)*ln(abs(x*e + d))
+ 1/6*(2*B*c^2*x^3*e^6 - 9*B*c^2*d*x^2*e^5 + 36*B*c^2*d^2*x*e^4 + 6*B*b*c*x^2*e
^6 + 3*A*c^2*x^2*e^6 - 36*B*b*c*d*x*e^5 - 18*A*c^2*d*x*e^5 + 6*B*b^2*x*e^6 + 12*
B*a*c*x*e^6 + 12*A*b*c*x*e^6)*e^(-9) - 1/2*(9*B*c^2*d^5 - 14*B*b*c*d^4*e - 7*A*c
^2*d^4*e + 5*B*b^2*d^3*e^2 + 10*B*a*c*d^3*e^2 + 10*A*b*c*d^3*e^2 - 6*B*a*b*d^2*e
^3 - 3*A*b^2*d^2*e^3 - 6*A*a*c*d^2*e^3 + B*a^2*d*e^4 + 2*A*a*b*d*e^4 + A*a^2*e^5
+ 2*(5*B*c^2*d^4*e - 8*B*b*c*d^3*e^2 - 4*A*c^2*d^3*e^2 + 3*B*b^2*d^2*e^3 + 6*B*
a*c*d^2*e^3 + 6*A*b*c*d^2*e^3 - 4*B*a*b*d*e^4 - 2*A*b^2*d*e^4 - 4*A*a*c*d*e^4 +
B*a^2*e^5 + 2*A*a*b*e^5)*x)*e^(-6)/(x*e + d)^2