3.1759 \(\int \frac{(A+B x) (d+e x)^3}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=248 \[ \frac{(a+b x) (d+e x)^3 (A b-a B)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2 (A b-a B) (b d-a e)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e x (a+b x) (A b-a B) (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (A b-a B) (b d-a e)^3 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B (a+b x) (d+e x)^4}{4 b e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.138712, antiderivative size = 248, 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, Rules used = {770, 77} \[ \frac{(a+b x) (d+e x)^3 (A b-a B)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2 (A b-a B) (b d-a e)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e x (a+b x) (A b-a B) (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (A b-a B) (b d-a e)^3 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B (a+b x) (d+e x)^4}{4 b e \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^3}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{(A+B x) (d+e x)^3}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{(A b-a B) e (b d-a e)^2}{b^5}+\frac{(A b-a B) (b d-a e)^3}{b^5 (a+b x)}+\frac{(A b-a B) e (b d-a e) (d+e x)}{b^4}+\frac{(A b-a B) e (d+e x)^2}{b^3}+\frac{B (d+e x)^3}{b^2}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) e (b d-a e)^2 x (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) (b d-a e) (a+b x) (d+e x)^2}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) (a+b x) (d+e x)^3}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B (a+b x) (d+e x)^4}{4 b e \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) (b d-a e)^3 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.134415, size = 185, normalized size = 0.75 \[ \frac{(a+b x) \left (b x \left (6 a^2 b e^2 (2 A e+6 B d+B e x)-12 a^3 B e^3-2 a b^2 e \left (3 A e (6 d+e x)+B \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+b^3 \left (2 A e \left (18 d^2+9 d e x+2 e^2 x^2\right )+3 B \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )\right )\right )+12 (A b-a B) (b d-a e)^3 \log (a+b x)\right )}{12 b^5 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*(b*x*(-12*a^3*B*e^3 + 6*a^2*b*e^2*(6*B*d + 2*A*e + B*e*x) - 2*a*b^2*e*(3*A*e*(6*d + e*x) + B*(18*d^
2 + 9*d*e*x + 2*e^2*x^2)) + b^3*(2*A*e*(18*d^2 + 9*d*e*x + 2*e^2*x^2) + 3*B*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 +
e^3*x^3))) + 12*(A*b - a*B)*(b*d - a*e)^3*Log[a + b*x]))/(12*b^5*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.011, size = 356, normalized size = 1.4 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( -3\,B{x}^{4}{b}^{4}{e}^{3}-4\,A{x}^{3}{b}^{4}{e}^{3}+4\,B{x}^{3}a{b}^{3}{e}^{3}-12\,B{x}^{3}{b}^{4}d{e}^{2}+6\,A{x}^{2}a{b}^{3}{e}^{3}-18\,A{x}^{2}{b}^{4}d{e}^{2}-6\,B{x}^{2}{a}^{2}{b}^{2}{e}^{3}+18\,B{x}^{2}a{b}^{3}d{e}^{2}-18\,B{x}^{2}{b}^{4}{d}^{2}e+12\,A\ln \left ( bx+a \right ){a}^{3}b{e}^{3}-36\,A\ln \left ( bx+a \right ){a}^{2}{b}^{2}d{e}^{2}+36\,A\ln \left ( bx+a \right ) a{b}^{3}{d}^{2}e-12\,A\ln \left ( bx+a \right ){b}^{4}{d}^{3}-12\,Ax{a}^{2}{b}^{2}{e}^{3}+36\,Axa{b}^{3}d{e}^{2}-36\,Ax{b}^{4}{d}^{2}e-12\,B\ln \left ( bx+a \right ){a}^{4}{e}^{3}+36\,B\ln \left ( bx+a \right ){a}^{3}bd{e}^{2}-36\,B\ln \left ( bx+a \right ){a}^{2}{b}^{2}{d}^{2}e+12\,B\ln \left ( bx+a \right ) a{b}^{3}{d}^{3}+12\,Bx{a}^{3}b{e}^{3}-36\,Bx{a}^{2}{b}^{2}d{e}^{2}+36\,Bxa{b}^{3}{d}^{2}e-12\,Bx{b}^{4}{d}^{3} \right ) }{12\,{b}^{5}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3/((b*x+a)^2)^(1/2),x)

[Out]

-1/12*(b*x+a)*(-3*B*x^4*b^4*e^3-4*A*x^3*b^4*e^3+4*B*x^3*a*b^3*e^3-12*B*x^3*b^4*d*e^2+6*A*x^2*a*b^3*e^3-18*A*x^
2*b^4*d*e^2-6*B*x^2*a^2*b^2*e^3+18*B*x^2*a*b^3*d*e^2-18*B*x^2*b^4*d^2*e+12*A*ln(b*x+a)*a^3*b*e^3-36*A*ln(b*x+a
)*a^2*b^2*d*e^2+36*A*ln(b*x+a)*a*b^3*d^2*e-12*A*ln(b*x+a)*b^4*d^3-12*A*x*a^2*b^2*e^3+36*A*x*a*b^3*d*e^2-36*A*x
*b^4*d^2*e-12*B*ln(b*x+a)*a^4*e^3+36*B*ln(b*x+a)*a^3*b*d*e^2-36*B*ln(b*x+a)*a^2*b^2*d^2*e+12*B*ln(b*x+a)*a*b^3
*d^3+12*B*x*a^3*b*e^3-36*B*x*a^2*b^2*d*e^2+36*B*x*a*b^3*d^2*e-12*B*x*b^4*d^3)/((b*x+a)^2)^(1/2)/b^5

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Maxima [B]  time = 0.997378, size = 725, normalized size = 2.92 \begin{align*} \frac{13 \, B a^{4} e^{3} \log \left (x + \frac{a}{b}\right )}{6 \,{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{13 \, B a^{3} e^{3} x}{6 \,{\left (b^{2}\right )}^{\frac{3}{2}} b} + \frac{13 \, B a^{2} e^{3} x^{2}}{12 \, \sqrt{b^{2}} b^{2}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B e^{3} x^{3}}{4 \, b^{2}} + A \sqrt{\frac{1}{b^{2}}} d^{3} \log \left (x + \frac{a}{b}\right ) - \frac{7 \, B a^{4} \sqrt{\frac{1}{b^{2}}} e^{3} \log \left (x + \frac{a}{b}\right )}{6 \, b^{4}} - \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B a e^{3} x^{2}}{12 \, b^{3}} - \frac{5 \,{\left (3 \, B d e^{2} + A e^{3}\right )} a^{3} b \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{3 \,{\left (B d^{2} e + A d e^{2}\right )} a^{2} b^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{3} e^{3}}{6 \, b^{5}} + \frac{5 \,{\left (3 \, B d e^{2} + A e^{3}\right )} a^{2} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{3 \,{\left (B d^{2} e + A d e^{2}\right )} a b x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (B d^{2} e + A d e^{2}\right )} x^{2}}{2 \, \sqrt{b^{2}}} - \frac{5 \,{\left (3 \, B d e^{2} + A e^{3}\right )} a x^{2}}{6 \, \sqrt{b^{2}} b} + \frac{2 \,{\left (3 \, B d e^{2} + A e^{3}\right )} a^{3} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{3 \, b^{3}} - \frac{{\left (B d^{3} + 3 \, A d^{2} e\right )} a \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{b} + \frac{{\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} x^{2}}{3 \, b^{2}} - \frac{2 \,{\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{3 \, b^{4}} + \frac{{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13/6*B*a^4*e^3*log(x + a/b)/(b^2)^(5/2) - 13/6*B*a^3*e^3*x/((b^2)^(3/2)*b) + 13/12*B*a^2*e^3*x^2/(sqrt(b^2)*b^
2) + 1/4*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*e^3*x^3/b^2 + A*sqrt(b^(-2))*d^3*log(x + a/b) - 7/6*B*a^4*sqrt(b^(-2)
)*e^3*log(x + a/b)/b^4 - 7/12*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a*e^3*x^2/b^3 - 5/3*(3*B*d*e^2 + A*e^3)*a^3*b*lo
g(x + a/b)/(b^2)^(5/2) + 3*(B*d^2*e + A*d*e^2)*a^2*b^2*log(x + a/b)/(b^2)^(5/2) + 7/6*sqrt(b^2*x^2 + 2*a*b*x +
a^2)*B*a^3*e^3/b^5 + 5/3*(3*B*d*e^2 + A*e^3)*a^2*x/(b^2)^(3/2) - 3*(B*d^2*e + A*d*e^2)*a*b*x/(b^2)^(3/2) + 3/
2*(B*d^2*e + A*d*e^2)*x^2/sqrt(b^2) - 5/6*(3*B*d*e^2 + A*e^3)*a*x^2/(sqrt(b^2)*b) + 2/3*(3*B*d*e^2 + A*e^3)*a^
3*sqrt(b^(-2))*log(x + a/b)/b^3 - (B*d^3 + 3*A*d^2*e)*a*sqrt(b^(-2))*log(x + a/b)/b + 1/3*(3*B*d*e^2 + A*e^3)*
sqrt(b^2*x^2 + 2*a*b*x + a^2)*x^2/b^2 - 2/3*(3*B*d*e^2 + A*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2/b^4 + (B*d^3
+ 3*A*d^2*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)/b^2

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Fricas [A]  time = 1.39505, size = 531, normalized size = 2.14 \begin{align*} \frac{3 \, B b^{4} e^{3} x^{4} + 4 \,{\left (3 \, B b^{4} d e^{2} -{\left (B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 6 \,{\left (3 \, B b^{4} d^{2} e - 3 \,{\left (B a b^{3} - A b^{4}\right )} d e^{2} +{\left (B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 12 \,{\left (B b^{4} d^{3} - 3 \,{\left (B a b^{3} - A b^{4}\right )} d^{2} e + 3 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{2} -{\left (B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x - 12 \,{\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} - 3 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{2} -{\left (B a^{4} - A a^{3} b\right )} e^{3}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.988781, size = 214, normalized size = 0.86 \begin{align*} \frac{B e^{3} x^{4}}{4 b} - \frac{x^{3} \left (- A b e^{3} + B a e^{3} - 3 B b d e^{2}\right )}{3 b^{2}} + \frac{x^{2} \left (- A a b e^{3} + 3 A b^{2} d e^{2} + B a^{2} e^{3} - 3 B a b d e^{2} + 3 B b^{2} d^{2} e\right )}{2 b^{3}} - \frac{x \left (- A a^{2} b e^{3} + 3 A a b^{2} d e^{2} - 3 A b^{3} d^{2} e + B a^{3} e^{3} - 3 B a^{2} b d e^{2} + 3 B a b^{2} d^{2} e - B b^{3} d^{3}\right )}{b^{4}} + \frac{\left (- A b + B a\right ) \left (a e - b d\right )^{3} \log{\left (a + b x \right )}}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3/((b*x+a)**2)**(1/2),x)

[Out]

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

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Giac [B]  time = 1.14902, size = 582, normalized size = 2.35 \begin{align*} \frac{3 \, B b^{3} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + 12 \, B b^{3} d x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 18 \, B b^{3} d^{2} x^{2} e \mathrm{sgn}\left (b x + a\right ) + 12 \, B b^{3} d^{3} x \mathrm{sgn}\left (b x + a\right ) - 4 \, B a b^{2} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 4 \, A b^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 18 \, B a b^{2} d x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 18 \, A b^{3} d x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 36 \, B a b^{2} d^{2} x e \mathrm{sgn}\left (b x + a\right ) + 36 \, A b^{3} d^{2} x e \mathrm{sgn}\left (b x + a\right ) + 6 \, B a^{2} b x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 6 \, A a b^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 36 \, B a^{2} b d x e^{2} \mathrm{sgn}\left (b x + a\right ) - 36 \, A a b^{2} d x e^{2} \mathrm{sgn}\left (b x + a\right ) - 12 \, B a^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 12 \, A a^{2} b x e^{3} \mathrm{sgn}\left (b x + a\right )}{12 \, b^{4}} - \frac{{\left (B a b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) - A b^{4} d^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, B a^{2} b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 3 \, A a b^{3} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 3 \, B a^{3} b d e^{2} \mathrm{sgn}\left (b x + a\right ) - 3 \, A a^{2} b^{2} d e^{2} \mathrm{sgn}\left (b x + a\right ) - B a^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + A a^{3} b e^{3} \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*B*b^3*x^4*e^3*sgn(b*x + a) + 12*B*b^3*d*x^3*e^2*sgn(b*x + a) + 18*B*b^3*d^2*x^2*e*sgn(b*x + a) + 12*B*
b^3*d^3*x*sgn(b*x + a) - 4*B*a*b^2*x^3*e^3*sgn(b*x + a) + 4*A*b^3*x^3*e^3*sgn(b*x + a) - 18*B*a*b^2*d*x^2*e^2*
sgn(b*x + a) + 18*A*b^3*d*x^2*e^2*sgn(b*x + a) - 36*B*a*b^2*d^2*x*e*sgn(b*x + a) + 36*A*b^3*d^2*x*e*sgn(b*x +
a) + 6*B*a^2*b*x^2*e^3*sgn(b*x + a) - 6*A*a*b^2*x^2*e^3*sgn(b*x + a) + 36*B*a^2*b*d*x*e^2*sgn(b*x + a) - 36*A*
a*b^2*d*x*e^2*sgn(b*x + a) - 12*B*a^3*x*e^3*sgn(b*x + a) + 12*A*a^2*b*x*e^3*sgn(b*x + a))/b^4 - (B*a*b^3*d^3*s
gn(b*x + a) - A*b^4*d^3*sgn(b*x + a) - 3*B*a^2*b^2*d^2*e*sgn(b*x + a) + 3*A*a*b^3*d^2*e*sgn(b*x + a) + 3*B*a^3
*b*d*e^2*sgn(b*x + a) - 3*A*a^2*b^2*d*e^2*sgn(b*x + a) - B*a^4*e^3*sgn(b*x + a) + A*a^3*b*e^3*sgn(b*x + a))*lo
g(abs(b*x + a))/b^5