3.1766 \(\int \frac{A+B x}{(d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=271 \[ \frac{b (a+b x) (A b-a B)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{(a+b x) (A b-a B)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}-\frac{(a+b x) (B d-A e)}{3 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}+\frac{b^2 (a+b x) (A b-a B) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2 (a+b x) (A b-a B) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]

[Out]

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

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Rubi [A]  time = 0.451746, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{b (a+b x) (A b-a B)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{(a+b x) (A b-a B)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}-\frac{(a+b x) (B d-A e)}{3 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}+\frac{b^2 (a+b x) (A b-a B) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2 (a+b x) (A b-a B) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 64.0054, size = 250, normalized size = 0.92 \[ \frac{b^{2} \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{b^{2} \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{b e \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{4}} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right )}{4 \left (d + e x\right )^{2} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right )}{6 e \left (d + e x\right )^{3} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*(A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/((a + b*x)*(a*e -
b*d)**4) - b**2*(A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/((a +
b*x)*(a*e - b*d)**4) - b*e*(A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/((d + e
*x)*(a*e - b*d)**4) + (2*a + 2*b*x)*(A*b - B*a)/(4*(d + e*x)**2*(a*e - b*d)**2*s
qrt(a**2 + 2*a*b*x + b**2*x**2)) - (2*a + 2*b*x)*(A*e - B*d)/(6*e*(d + e*x)**3*(
a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.286497, size = 168, normalized size = 0.62 \[ \frac{(a+b x) \left (6 b^2 e (d+e x)^3 (A b-a B) \log (a+b x)-6 b^2 e (d+e x)^3 (A b-a B) \log (d+e x)+3 e (d+e x) (A b-a B) (b d-a e)^2+6 b e (d+e x)^2 (A b-a B) (b d-a e)-2 (b d-a e)^3 (B d-A e)\right )}{6 e \sqrt{(a+b x)^2} (d+e x)^3 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Maple [B]  time = 0.029, size = 545, normalized size = 2. \[{\frac{ \left ( bx+a \right ) \left ( 18\,B\ln \left ( ex+d \right ){x}^{2}a{b}^{2}d{e}^{3}+18\,B\ln \left ( ex+d \right ) xa{b}^{2}{d}^{2}{e}^{2}+6\,B{x}^{2}{a}^{2}b{e}^{4}+6\,A{x}^{2}{b}^{3}d{e}^{3}-Bd{e}^{3}{a}^{3}-18\,Aa{b}^{2}{d}^{2}{e}^{2}-3\,Bx{a}^{3}{e}^{4}-6\,B{x}^{2}a{b}^{2}d{e}^{3}+11\,A{b}^{3}{d}^{3}e-18\,Axa{b}^{2}d{e}^{3}-2\,A{a}^{3}{e}^{4}-2\,B{b}^{3}{d}^{4}-18\,B\ln \left ( bx+a \right ){x}^{2}a{b}^{2}d{e}^{3}-18\,B\ln \left ( bx+a \right ) xa{b}^{2}{d}^{2}{e}^{2}+15\,Ax{b}^{3}{d}^{2}{e}^{2}+6\,B\ln \left ( ex+d \right ) a{b}^{2}{d}^{3}e+18\,Bx{a}^{2}bd{e}^{3}-15\,Bxa{b}^{2}{d}^{2}{e}^{2}+6\,B{a}^{2}b{d}^{2}{e}^{2}-3\,Ba{b}^{2}{d}^{3}e-18\,A\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{3}-18\,A\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}{e}^{2}-6\,B\ln \left ( bx+a \right ){x}^{3}a{b}^{2}{e}^{4}+18\,A\ln \left ( bx+a \right ){x}^{2}{b}^{3}d{e}^{3}+18\,A\ln \left ( bx+a \right ) x{b}^{3}{d}^{2}{e}^{2}-6\,B\ln \left ( bx+a \right ) a{b}^{2}{d}^{3}e+6\,B\ln \left ( ex+d \right ){x}^{3}a{b}^{2}{e}^{4}-6\,A\ln \left ( ex+d \right ){b}^{3}{d}^{3}e-6\,A{x}^{2}a{b}^{2}{e}^{4}+9\,Ad{e}^{3}{a}^{2}b+6\,A\ln \left ( bx+a \right ){x}^{3}{b}^{3}{e}^{4}+6\,A\ln \left ( bx+a \right ){b}^{3}{d}^{3}e-6\,A\ln \left ( ex+d \right ){x}^{3}{b}^{3}{e}^{4}+3\,Ax{a}^{2}b{e}^{4} \right ) }{6\, \left ( ae-bd \right ) ^{4}e \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^4/((b*x+a)^2)^(1/2),x)

[Out]

1/6*(b*x+a)*(18*B*ln(e*x+d)*x^2*a*b^2*d*e^3+18*B*ln(e*x+d)*x*a*b^2*d^2*e^2+6*B*x
^2*a^2*b*e^4+6*A*x^2*b^3*d*e^3-B*d*e^3*a^3-18*A*a*b^2*d^2*e^2-3*B*x*a^3*e^4-6*B*
x^2*a*b^2*d*e^3+11*A*b^3*d^3*e-18*A*x*a*b^2*d*e^3-2*A*a^3*e^4-2*B*b^3*d^4-18*B*l
n(b*x+a)*x^2*a*b^2*d*e^3-18*B*ln(b*x+a)*x*a*b^2*d^2*e^2+15*A*x*b^3*d^2*e^2+6*B*l
n(e*x+d)*a*b^2*d^3*e+18*B*x*a^2*b*d*e^3-15*B*x*a*b^2*d^2*e^2+6*B*a^2*b*d^2*e^2-3
*B*a*b^2*d^3*e-18*A*ln(e*x+d)*x^2*b^3*d*e^3-18*A*ln(e*x+d)*x*b^3*d^2*e^2-6*B*ln(
b*x+a)*x^3*a*b^2*e^4+18*A*ln(b*x+a)*x^2*b^3*d*e^3+18*A*ln(b*x+a)*x*b^3*d^2*e^2-6
*B*ln(b*x+a)*a*b^2*d^3*e+6*B*ln(e*x+d)*x^3*a*b^2*e^4-6*A*ln(e*x+d)*b^3*d^3*e-6*A
*x^2*a*b^2*e^4+9*A*d*e^3*a^2*b+6*A*ln(b*x+a)*x^3*b^3*e^4+6*A*ln(b*x+a)*b^3*d^3*e
-6*A*ln(e*x+d)*x^3*b^3*e^4+3*A*x*a^2*b*e^4)/((b*x+a)^2)^(1/2)/(a*e-b*d)^4/e/(e*x
+d)^3

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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((B*x + A)/(sqrt((b*x + a)^2)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300232, size = 821, normalized size = 3.03 \[ -\frac{2 \, B b^{3} d^{4} + 2 \, A a^{3} e^{4} +{\left (3 \, B a b^{2} - 11 \, A b^{3}\right )} d^{3} e - 6 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e^{2} +{\left (B a^{3} - 9 \, A a^{2} b\right )} d e^{3} + 6 \,{\left ({\left (B a b^{2} - A b^{3}\right )} d e^{3} -{\left (B a^{2} b - A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \,{\left (5 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{3} +{\left (B a^{3} - A a^{2} b\right )} e^{4}\right )} x + 6 \,{\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x +{\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (b x + a\right ) - 6 \,{\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x +{\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (e x + d\right )}{6 \,{\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5} +{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{3} + 3 \,{\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{2} + 3 \,{\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.1348, size = 818, normalized size = 3.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*(-A*b + B*a)*log(x + (-A*a*b**3*e - A*b**4*d + B*a**2*b**2*e + B*a*b**3*d -
a**5*b**2*e**5*(-A*b + B*a)/(a*e - b*d)**4 + 5*a**4*b**3*d*e**4*(-A*b + B*a)/(a
*e - b*d)**4 - 10*a**3*b**4*d**2*e**3*(-A*b + B*a)/(a*e - b*d)**4 + 10*a**2*b**5
*d**3*e**2*(-A*b + B*a)/(a*e - b*d)**4 - 5*a*b**6*d**4*e*(-A*b + B*a)/(a*e - b*d
)**4 + b**7*d**5*(-A*b + B*a)/(a*e - b*d)**4)/(-2*A*b**4*e + 2*B*a*b**3*e))/(a*e
- b*d)**4 - b**2*(-A*b + B*a)*log(x + (-A*a*b**3*e - A*b**4*d + B*a**2*b**2*e +
B*a*b**3*d + a**5*b**2*e**5*(-A*b + B*a)/(a*e - b*d)**4 - 5*a**4*b**3*d*e**4*(-
A*b + B*a)/(a*e - b*d)**4 + 10*a**3*b**4*d**2*e**3*(-A*b + B*a)/(a*e - b*d)**4 -
10*a**2*b**5*d**3*e**2*(-A*b + B*a)/(a*e - b*d)**4 + 5*a*b**6*d**4*e*(-A*b + B*
a)/(a*e - b*d)**4 - b**7*d**5*(-A*b + B*a)/(a*e - b*d)**4)/(-2*A*b**4*e + 2*B*a*
b**3*e))/(a*e - b*d)**4 + (-2*A*a**2*e**3 + 7*A*a*b*d*e**2 - 11*A*b**2*d**2*e -
B*a**2*d*e**2 + 5*B*a*b*d**2*e + 2*B*b**2*d**3 + x**2*(-6*A*b**2*e**3 + 6*B*a*b*
e**3) + x*(3*A*a*b*e**3 - 15*A*b**2*d*e**2 - 3*B*a**2*e**3 + 15*B*a*b*d*e**2))/(
6*a**3*d**3*e**4 - 18*a**2*b*d**4*e**3 + 18*a*b**2*d**5*e**2 - 6*b**3*d**6*e + x
**3*(6*a**3*e**7 - 18*a**2*b*d*e**6 + 18*a*b**2*d**2*e**5 - 6*b**3*d**3*e**4) +
x**2*(18*a**3*d*e**6 - 54*a**2*b*d**2*e**5 + 54*a*b**2*d**3*e**4 - 18*b**3*d**4*
e**3) + x*(18*a**3*d**2*e**5 - 54*a**2*b*d**3*e**4 + 54*a*b**2*d**4*e**3 - 18*b*
*3*d**5*e**2))

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GIAC/XCAS [A]  time = 0.293752, size = 667, normalized size = 2.46 \[ -\frac{{\left (B a b^{3}{\rm sign}\left (b x + a\right ) - A b^{4}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac{{\left (B a b^{2} e{\rm sign}\left (b x + a\right ) - A b^{3} e{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac{{\left (2 \, B b^{3} d^{4}{\rm sign}\left (b x + a\right ) + 3 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) - 11 \, A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) - 6 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 18 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) - 9 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 2 \, A a^{3} e^{4}{\rm sign}\left (b x + a\right ) + 6 \,{\left (B a b^{2} d e^{3}{\rm sign}\left (b x + a\right ) - A b^{3} d e^{3}{\rm sign}\left (b x + a\right ) - B a^{2} b e^{4}{\rm sign}\left (b x + a\right ) + A a b^{2} e^{4}{\rm sign}\left (b x + a\right )\right )} x^{2} + 3 \,{\left (5 \, B a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 5 \, A b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 6 \, B a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 6 \, A a b^{2} d e^{3}{\rm sign}\left (b x + a\right ) + B a^{3} e^{4}{\rm sign}\left (b x + a\right ) - A a^{2} b e^{4}{\rm sign}\left (b x + a\right )\right )} x\right )} e^{\left (-1\right )}}{6 \,{\left (b d - a e\right )}^{4}{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(B*a*b^3*sign(b*x + a) - A*b^4*sign(b*x + a))*ln(abs(b*x + a))/(b^5*d^4 - 4*a*b
^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4) + (B*a*b^2*e*sign(b*
x + a) - A*b^3*e*sign(b*x + a))*ln(abs(x*e + d))/(b^4*d^4*e - 4*a*b^3*d^3*e^2 +
6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5) - 1/6*(2*B*b^3*d^4*sign(b*x + a) +
3*B*a*b^2*d^3*e*sign(b*x + a) - 11*A*b^3*d^3*e*sign(b*x + a) - 6*B*a^2*b*d^2*e^2
*sign(b*x + a) + 18*A*a*b^2*d^2*e^2*sign(b*x + a) + B*a^3*d*e^3*sign(b*x + a) -
9*A*a^2*b*d*e^3*sign(b*x + a) + 2*A*a^3*e^4*sign(b*x + a) + 6*(B*a*b^2*d*e^3*sig
n(b*x + a) - A*b^3*d*e^3*sign(b*x + a) - B*a^2*b*e^4*sign(b*x + a) + A*a*b^2*e^4
*sign(b*x + a))*x^2 + 3*(5*B*a*b^2*d^2*e^2*sign(b*x + a) - 5*A*b^3*d^2*e^2*sign(
b*x + a) - 6*B*a^2*b*d*e^3*sign(b*x + a) + 6*A*a*b^2*d*e^3*sign(b*x + a) + B*a^3
*e^4*sign(b*x + a) - A*a^2*b*e^4*sign(b*x + a))*x)*e^(-1)/((b*d - a*e)^4*(x*e +
d)^3)