3.1850 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=304 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{e^5 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^5 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)*(d + e*x)^(5/2)) + (2*(b*d - a*e
)^2*(4*b*B*d - 3*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*x)^(3/2)) - (6*b*(b*d -
a*e)*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*Sqrt[d + e*x]) - (2*b^2*(4*b*B*d
- A*b*e - 3*a*B*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)) + (2*b^3*B*(d + e*x)^(3/2)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x))

________________________________________________________________________________________

Rubi [A]  time = 0.137475, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {770, 77} \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{e^5 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{e^5 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^3*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)*(d + e*x)^(5/2)) + (2*(b*d - a*e
)^2*(4*b*B*d - 3*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d + e*x)^(3/2)) - (6*b*(b*d -
a*e)*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*Sqrt[d + e*x]) - (2*b^2*(4*b*B*d
- A*b*e - 3*a*B*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)) + (2*b^3*B*(d + e*x)^(3/2)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x))

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) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3 (A+B x)}{(d+e x)^{7/2}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^3 (b d-a e)^3 (-B d+A e)}{e^4 (d+e x)^{7/2}}+\frac{b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^{5/2}}-\frac{3 b^4 (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^{3/2}}+\frac{b^5 (-4 b B d+A b e+3 a B e)}{e^4 \sqrt{d+e x}}+\frac{b^6 B \sqrt{d+e x}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{2 (b d-a e)^3 (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac{2 (b d-a e)^2 (4 b B d-3 A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac{6 b (b d-a e) (2 b B d-A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt{d+e x}}-\frac{2 b^2 (4 b B d-A b e-3 a B e) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac{2 b^3 B (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.162908, size = 244, normalized size = 0.8 \[ -\frac{2 \sqrt{(a+b x)^2} \left (3 a^2 b e^2 \left (A e (2 d+5 e x)+B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )+a^3 e^3 (3 A e+2 B d+5 B e x)-3 a b^2 e \left (3 B \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )-A e \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )+b^3 \left (B \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )-3 A e \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )\right )\right )}{15 e^5 (a+b x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^(7/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(a^3*e^3*(2*B*d + 3*A*e + 5*B*e*x) + 3*a^2*b*e^2*(A*e*(2*d + 5*e*x) + B*(8*d^2 + 20*d*e*
x + 15*e^2*x^2)) - 3*a*b^2*e*(-(A*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) + 3*B*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2
+ 5*e^3*x^3)) + b^3*(-3*A*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + B*(128*d^4 + 320*d^3*e*x + 240
*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4))))/(15*e^5*(a + b*x)*(d + e*x)^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 317, normalized size = 1. \begin{align*} -{\frac{-10\,B{x}^{4}{b}^{3}{e}^{4}-30\,A{x}^{3}{b}^{3}{e}^{4}-90\,B{x}^{3}a{b}^{2}{e}^{4}+80\,B{x}^{3}{b}^{3}d{e}^{3}+90\,A{x}^{2}a{b}^{2}{e}^{4}-180\,A{x}^{2}{b}^{3}d{e}^{3}+90\,B{x}^{2}{a}^{2}b{e}^{4}-540\,B{x}^{2}a{b}^{2}d{e}^{3}+480\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+30\,Ax{a}^{2}b{e}^{4}+120\,Axa{b}^{2}d{e}^{3}-240\,Ax{b}^{3}{d}^{2}{e}^{2}+10\,Bx{a}^{3}{e}^{4}+120\,Bx{a}^{2}bd{e}^{3}-720\,Bxa{b}^{2}{d}^{2}{e}^{2}+640\,Bx{b}^{3}{d}^{3}e+6\,A{a}^{3}{e}^{4}+12\,Ad{e}^{3}{a}^{2}b+48\,Aa{b}^{2}{d}^{2}{e}^{2}-96\,A{b}^{3}{d}^{3}e+4\,Bd{e}^{3}{a}^{3}+48\,B{a}^{2}b{d}^{2}{e}^{2}-288\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{15\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(-5*B*b^3*e^4*x^4-15*A*b^3*e^4*x^3-45*B*a*b^2*e^4*x^3+40*B*b^3*d*e^3*x^3+45*A*a*b^2*e^4*x^
2-90*A*b^3*d*e^3*x^2+45*B*a^2*b*e^4*x^2-270*B*a*b^2*d*e^3*x^2+240*B*b^3*d^2*e^2*x^2+15*A*a^2*b*e^4*x+60*A*a*b^
2*d*e^3*x-120*A*b^3*d^2*e^2*x+5*B*a^3*e^4*x+60*B*a^2*b*d*e^3*x-360*B*a*b^2*d^2*e^2*x+320*B*b^3*d^3*e*x+3*A*a^3
*e^4+6*A*a^2*b*d*e^3+24*A*a*b^2*d^2*e^2-48*A*b^3*d^3*e+2*B*a^3*d*e^3+24*B*a^2*b*d^2*e^2-144*B*a*b^2*d^3*e+128*
B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

________________________________________________________________________________________

Maxima [A]  time = 1.11238, size = 440, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (5 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} - a^{3} e^{3} + 15 \,{\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (8 \, b^{3} d^{2} e - 4 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} A}{5 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (5 \, b^{3} e^{4} x^{4} - 128 \, b^{3} d^{4} + 144 \, a b^{2} d^{3} e - 24 \, a^{2} b d^{2} e^{2} - 2 \, a^{3} d e^{3} - 5 \,{\left (8 \, b^{3} d e^{3} - 9 \, a b^{2} e^{4}\right )} x^{3} - 15 \,{\left (16 \, b^{3} d^{2} e^{2} - 18 \, a b^{2} d e^{3} + 3 \, a^{2} b e^{4}\right )} x^{2} - 5 \,{\left (64 \, b^{3} d^{3} e - 72 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x\right )} B}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )} \sqrt{e x + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/5*(5*b^3*e^3*x^3 + 16*b^3*d^3 - 8*a*b^2*d^2*e - 2*a^2*b*d*e^2 - a^3*e^3 + 15*(2*b^3*d*e^2 - a*b^2*e^3)*x^2 +
5*(8*b^3*d^2*e - 4*a*b^2*d*e^2 - a^2*b*e^3)*x)*A/((e^6*x^2 + 2*d*e^5*x + d^2*e^4)*sqrt(e*x + d)) + 2/15*(5*b^
3*e^4*x^4 - 128*b^3*d^4 + 144*a*b^2*d^3*e - 24*a^2*b*d^2*e^2 - 2*a^3*d*e^3 - 5*(8*b^3*d*e^3 - 9*a*b^2*e^4)*x^3
- 15*(16*b^3*d^2*e^2 - 18*a*b^2*d*e^3 + 3*a^2*b*e^4)*x^2 - 5*(64*b^3*d^3*e - 72*a*b^2*d^2*e^2 + 12*a^2*b*d*e^
3 + a^3*e^4)*x)*B/((e^7*x^2 + 2*d*e^6*x + d^2*e^5)*sqrt(e*x + d))

________________________________________________________________________________________

Fricas [A]  time = 1.20606, size = 622, normalized size = 2.05 \begin{align*} \frac{2 \,{\left (5 \, B b^{3} e^{4} x^{4} - 128 \, B b^{3} d^{4} - 3 \, A a^{3} e^{4} + 48 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 24 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \,{\left (8 \, B b^{3} d e^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} - 15 \,{\left (16 \, B b^{3} d^{2} e^{2} - 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 5 \,{\left (64 \, B b^{3} d^{3} e - 24 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*B*b^3*e^4*x^4 - 128*B*b^3*d^4 - 3*A*a^3*e^4 + 48*(3*B*a*b^2 + A*b^3)*d^3*e - 24*(B*a^2*b + A*a*b^2)*d^
2*e^2 - 2*(B*a^3 + 3*A*a^2*b)*d*e^3 - 5*(8*B*b^3*d*e^3 - 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 - 15*(16*B*b^3*d^2*e^2
- 6*(3*B*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 - 5*(64*B*b^3*d^3*e - 24*(3*B*a*b^2 + A*b^3)*d
^2*e^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x)*sqrt(e*x + d)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d
^2*e^6*x + d^3*e^5)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.25758, size = 686, normalized size = 2.26 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{3} e^{10} \mathrm{sgn}\left (b x + a\right ) - 12 \, \sqrt{x e + d} B b^{3} d e^{10} \mathrm{sgn}\left (b x + a\right ) + 9 \, \sqrt{x e + d} B a b^{2} e^{11} \mathrm{sgn}\left (b x + a\right ) + 3 \, \sqrt{x e + d} A b^{3} e^{11} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} B b^{3} d^{2} \mathrm{sgn}\left (b x + a\right ) - 20 \,{\left (x e + d\right )} B b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, B b^{3} d^{4} \mathrm{sgn}\left (b x + a\right ) - 135 \,{\left (x e + d\right )}^{2} B a b^{2} d e \mathrm{sgn}\left (b x + a\right ) - 45 \,{\left (x e + d\right )}^{2} A b^{3} d e \mathrm{sgn}\left (b x + a\right ) + 45 \,{\left (x e + d\right )} B a b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 15 \,{\left (x e + d\right )} A b^{3} d^{2} e \mathrm{sgn}\left (b x + a\right ) - 9 \, B a b^{2} d^{3} e \mathrm{sgn}\left (b x + a\right ) - 3 \, A b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 45 \,{\left (x e + d\right )}^{2} B a^{2} b e^{2} \mathrm{sgn}\left (b x + a\right ) + 45 \,{\left (x e + d\right )}^{2} A a b^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 30 \,{\left (x e + d\right )} B a^{2} b d e^{2} \mathrm{sgn}\left (b x + a\right ) - 30 \,{\left (x e + d\right )} A a b^{2} d e^{2} \mathrm{sgn}\left (b x + a\right ) + 9 \, B a^{2} b d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 9 \, A a b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \,{\left (x e + d\right )} B a^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \,{\left (x e + d\right )} A a^{2} b e^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, B a^{3} d e^{3} \mathrm{sgn}\left (b x + a\right ) - 9 \, A a^{2} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, A a^{3} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*B*b^3*e^10*sgn(b*x + a) - 12*sqrt(x*e + d)*B*b^3*d*e^10*sgn(b*x + a) + 9*sqrt(x*e + d)*B*
a*b^2*e^11*sgn(b*x + a) + 3*sqrt(x*e + d)*A*b^3*e^11*sgn(b*x + a))*e^(-15) - 2/15*(90*(x*e + d)^2*B*b^3*d^2*sg
n(b*x + a) - 20*(x*e + d)*B*b^3*d^3*sgn(b*x + a) + 3*B*b^3*d^4*sgn(b*x + a) - 135*(x*e + d)^2*B*a*b^2*d*e*sgn(
b*x + a) - 45*(x*e + d)^2*A*b^3*d*e*sgn(b*x + a) + 45*(x*e + d)*B*a*b^2*d^2*e*sgn(b*x + a) + 15*(x*e + d)*A*b^
3*d^2*e*sgn(b*x + a) - 9*B*a*b^2*d^3*e*sgn(b*x + a) - 3*A*b^3*d^3*e*sgn(b*x + a) + 45*(x*e + d)^2*B*a^2*b*e^2*
sgn(b*x + a) + 45*(x*e + d)^2*A*a*b^2*e^2*sgn(b*x + a) - 30*(x*e + d)*B*a^2*b*d*e^2*sgn(b*x + a) - 30*(x*e + d
)*A*a*b^2*d*e^2*sgn(b*x + a) + 9*B*a^2*b*d^2*e^2*sgn(b*x + a) + 9*A*a*b^2*d^2*e^2*sgn(b*x + a) + 5*(x*e + d)*B
*a^3*e^3*sgn(b*x + a) + 15*(x*e + d)*A*a^2*b*e^3*sgn(b*x + a) - 3*B*a^3*d*e^3*sgn(b*x + a) - 9*A*a^2*b*d*e^3*s
gn(b*x + a) + 3*A*a^3*e^4*sgn(b*x + a))*e^(-5)/(x*e + d)^(5/2)